Optimal Temperature Distribution for a Nonisothermal Cahn–Hilliard System with Source Term

Abstract

In this note, we study the optimal control of a nonisothermal phase field system of Cahn–Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. The system couples a Cahn–Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails additional mathematical difficulties because the mass conservation of the order parameter, typical of the classic Cahn–Hilliard equation, is no longer satisfied. In this paper, we analyze the case that the double-well potential driving the evolution of the phase transition is differentiable, either (in the regular case) on the whole set of reals or (in the singular logarithmic case) on a finite open interval; nondifferentiable cases like the double obstacle potential are excluded from the analysis. We prove the Fréchet differentiability of the control-to-state operator between suitable Banach spaces for both the regular and the logarithmic cases and establish the solvability of the corresponding adjoint systems in order to derive the associated first-order necessary optimality conditions for the optimal control problem. Crucial for the whole analysis to work is the boundedness property stating that the order parameter attains its values in a compact subset of the interior of the effective domain of the nonlinearity. While this property turns out to be generally valid for regular potentials in three dimensions of space, it can be shown for the logarithmic case only in two dimensions.

1 Introduction

Let $\Omega \subset \mathbb {R}^d$, $d\in \{2,3\}$, be some open, bounded, and connected set having a smooth boundary $\Gamma :=\partial \Omega$ and the outward unit normal field $\,{\textbf {n}}$. Denoting by $\partial _{{\textbf {n}}}$ the directional derivative in the direction of ${\textbf {n}}$, and putting, with a fixed final time $T>0$,

$$\begin{aligned} Q_t:= & {} \Omega \times (0,t) \, \text{ and }\,\Sigma _t:=\Gamma \times (0,t)\, \text{ for } \,t\in (0,T],\\{} & {} \text{ as } \text{ well } \text{ as } \, Q:=Q_T \,\text{ and } \,\Sigma :=\Sigma _T, \end{aligned}$$

we study in this paper as state system the following initial-boundary value problem:

$$\begin{aligned}&\partial _t\phi - \Delta \mu + \gamma \phi = f \quad{} & {} \text {in } {Q}, \end{aligned}$$

(1.1)

$$\begin{aligned}&\mu = - \Delta \phi + F'(\phi ) + a - b \partial _tw \quad{} & {} \text {in } Q, \end{aligned}$$

(1.2)

$$\begin{aligned}&\partial _t^2w - \Delta ({\kappa _1}\partial _tw + {\kappa _2}w) + \lambda \partial _t\phi = u \quad{} & {} \text {in }Q, \end{aligned}$$

(1.3)

$$\begin{aligned}&\partial _{{\textbf {n}}}\phi = \partial _{{\textbf {n}}}\mu = \partial _{{\textbf {n}}}({\kappa _1}\partial _tw + {\kappa _2}w) = 0 \quad{} & {} \text {on }{\Sigma }, \end{aligned}$$

(1.4)

$$\begin{aligned}&\phi (0) = \phi _0, \quad w(0) = w_0, \quad \partial _tw(0) = w_1\quad{} & {} \text {in }\Omega . \end{aligned}$$

(1.5)

The cost functional under consideration is given by

$$\begin{aligned} { \mathcal{J}((\phi , w), u) := }{}&\ {\frac{{\alpha _1}}{2} \int _0^T\!\!\!\int _\Omega |{\phi - \phi _Q}|^2 + \frac{{\alpha _2}}{2} \, \int _\Omega |{\phi (T) - \phi _\Omega }|^2} \nonumber \\&{{} + \frac{{\alpha _3}}{2} \int _0^T\!\!\! \int _\Omega |{w - w_Q}|^2 + \frac{{\alpha _4}}{2} \, \int _\Omega |{w(T) - w_\Omega }|^2} \nonumber \\&{} + \frac{{\alpha _5}}{2} \int _0^T\!\!\!\int _\Omega |{\partial _tw - w'_Q}|^2 + \frac{{\alpha _6}}{2} \, \int _\Omega |{\partial _tw(T) - w'_\Omega }|^2 \nonumber \\&+ \frac{\nu }{2} \int _0^T\!\!\!\int _\Omega |{u}|^2, \end{aligned}$$

(1.6)

with nonnegative constants ${\alpha _i}$, $1\le i\le 6$, which are not all zero, and where $\phi _\Omega , w_\Omega , w'_\Omega \in L^{2}(\Omega )$ and $\phi _Q,w_Q,w'_Q\in L^2(Q)$ denote given target functions.

For the control variable u, we choose as control space

$$\begin{aligned} \mathcal U:= L^{\infty }(Q), \end{aligned}$$

(1.7)

and the related set of admissible controls is given by

$$\begin{aligned} \mathcal{U}_\textrm{ad}: = \big \{ u \in \mathcal{U}: u_\textrm{min} \le u \le u_\textrm{max} \quad \hbox {a.e. in Q}\big \}, \end{aligned}$$

(1.8)

where $u_\textrm{min},u_\textrm{max}\in {L^\infty (Q)}$ satisfy $u_\textrm{min}\le u_\textrm{max}$ almost everywhere in Q.

In summary, the control problem under investigation can be reformulated as follows:

$$\begin{aligned} {{\textbf {(P)}}} \quad \min _{u \in \mathcal{U}_\textrm{ad}}\mathcal{J}((\phi , w), u) \ \text {subject to the constraint that } (\phi ,\mu , w) \text { solves }(1.1)-(1.5). \end{aligned}$$

The state system (1.1)–(1.5) is a formal extension of the nonisothermal Cahn–Hilliard system introduced by Caginalp in [4] to model the phenomenon of nonisothermal phase segregation in binary mixtures (see also [3, 5] and the derivation in [2, Example 4.4.2, (4.44), (4.46)]); it corresponds to the Allen–Cahn counterpart analyzed recently in [13]. The state system is controlled through the control $\,u\,$ occurring on the right-hand side of (1.3) which stands for a distributed heat source or sink. We remark at this place that in materials science thermal control is an important means to monitor the evolution of phase separation processes in many industrial applications of metallic alloys. In this connection, the general aim is to achieve a favorable distribution of the metallic components within the spatial domain where the process takes place. A particular such class is given by the solder alloys which have a fundamental importance for the proper functioning of electronic devices.

The unknowns in the state system have the following physical meaning: $\phi$ is a normalized difference between the volume fractions of pure phases in the binary mixture (the dimensionless order parameter of the phase transformation, which should attain its values in the interval $[-1,1]$), $\mu$ is the associated chemical potential, and $\,w\,$ is the so-called thermal displacement (or freezing index), which is directly connected to the temperature $\theta$ (which in the case of the Caginalp model is actually a temperature difference) through the relation

$$\begin{aligned} w (\cdot , t) = w_0 + \int _0^t\theta (\cdot , s) \,ds, \quad t \in [0,T]. \end{aligned}$$

(1.9)

For a more detailed physical overview, the reader may see the papers [11, 13]. In this respect, ${\kappa _1}$ and ${\kappa _2}$ in (1.3) stand for prescribed positive coefficients related to the heat flux, which is here assumed in the Green–Naghdi form (see [19,20,21, 26])

$$\begin{aligned} \textbf{q}=-{\kappa _1}\nabla (\partial _tw )- {\kappa _2}\nabla w \quad \text{ where } \kappa _i>0, i=1,2. \end{aligned}$$

(1.10)

This setting accounts for a possible previous thermal history of the phenomenon. Moreover, $\gamma$ is a positive physical constant related to the intensity of the mass absorption or production of the source, where the source term in (1.1) is $S:=f- \gamma \phi$. This term reflects the fact that the system may not be isolated and the loss or production of mass is possible, which happens, e.g., in numerous liquid–liquid phase transition problems that arise in cell biology [15] and in tumor growth models [17]. Notice that the presence of the source term entails that the property of mass conservation of the order parameter is no longer valid; in fact, from (1.1) it directly follows that the mass balance has the form

$$\begin{aligned} \frac{d}{dt} \, \Bigg (\frac{1}{|\Omega |}\int _\Omega \phi (t) \Bigg ) = \frac{1}{|\Omega |} \int _\Omega {S(t)}, \quad \text{ for } \text{ a.e. } t\in (0,T), \end{aligned}$$

(1.11)

where $\,|\Omega |\,$ denotes the volume of $\Omega$. To this concern, we would like to quote the paper [8], where a comparable Cahn–Hilliard system without mass conservation was examined from the viewpoint of optimal control. Moreover, we refer to [6, 7, 12, 23, 25, 27, 31], where similar systems have been analyzed. We also note that the presence of a source function in the Cahn–Hilliard equation, which modifies the mass conservation property, is connected to several models related to biology and, in particular, to tumor growth models.

An important tool for the practical numerical solution of optimal control problems is sparsity, i.e., the addition of a nondifferentiable $L^1$-type penalization of controls to the cost functional (1.6), which enhances the occurrence of subdomains of the space-time cylinder $\,Q\,$ where optimal controls vanish. Sparsity of controls is of particular importance in tumor growth models, see, e.g., [14, 16, 28, 30]. We claim that it should be possible, by using a similar approach as in these works, to incorporate sparsity effects also in our model, but we leave this open for possible further investigations.

Also, let us incidentally point out that the differential structure of equation (1.3), with respect to w, is sometimes also referred to as the strongly damped wave equation, see, e.g., [24] and the references therein.

In addition to the quantities already introduced, $\lambda$ stands for the latent heat of the phase transformation, a, b are physical constants related to some positive critical temperature $\theta _c$ (cf. [13]), and the control variable $\,u\,$ is a distributed heat source/sink. Besides, $\phi _0,w_0,$ and $w_1$ indicate some given initial values. Finally, the function F is assumed to have a double-well shape. Prototypical choices for the double-well shaped nonlinearity F are the regular and singular logarithmic potential and its common (nonsingular) polynomial approximation, the regular potential. In the order, they are defined as

$$\begin{aligned}&F_{log}(r) := \left\{ \begin{array}{ll} (1+r)\ln (1+r)+(1-r)\ln (1-r) - c_1 r^2 &{} \quad \hbox {if } |r|\le 1, \\ +\infty &{} \quad \hbox {otherwise}, \end{array} \right. \end{aligned}$$

(1.12)

$$\begin{aligned}&F_{reg}(r) := \frac{1}{4} \, (r^2-1)^2 \,, \quad r \in \mathbb {R}, \end{aligned}$$

(1.13)

with the convention that $0\ln (0):=\lim _{r\searrow 0}r\ln (r)=0$ and $c_1>1$ so that $F_{log}$ is nonconvex. Another important example is the nonregular and singular double obstacle potential, given by

$$\begin{aligned} F_{2obs}(r) := - c_2 r^2 \quad \hbox {if }|r|\le 1 \quad \hbox {and}\quad F_{2obs}(r) := +\infty \quad \hbox {if }|r|>1, \end{aligned}$$

(1.14)

with $c_2>0$. However, the double obstacle case is not included in the subsequent analysis, although we expect that, with similar techniques as those employed in [10], it is possible to extend the analysis also to this kind of nonregular potentials.

The state system (1.1)–(1.5) was recently analyzed in [11], concerning well-posedness and regularity (see the results cited below in Sect. 2), where also the double obstacle case was included. Here, we concentrate on the optimal control problem. While the existence of optimal controls is not too difficult to show, the derivation of first-order necessary optimality conditions is a much more challenging task, since it makes the derivation of differentiability properties of the associated control-to-state operator necessary. This, however, requires that the order parameter $\,\phi \,$ satisfies a separation property, with the meaning that $\,\phi \,$ attains its values in a compact subset of the interior of the effective domain of the derivative $\,F'\,$ of $\,F$. While for regular potentials this condition turns out to be generally satisfied, it cannot be guaranteed for singular potentials. In fact, following the ideas of the recent paper [9] on the isothermal case, one is just able to ensure the validity of the strict separation property for the logarithmic potential $F_\textrm{log}$ in the two-dimensional case $d=2$. Correspondingly, the analysis leading to first-order necessary optimality conditions will be restricted to either the regular case for $d\le 3$ or the logarithmic case in two dimensions of space. In this sense, our results apply to similar cases as those studied in [9] in the isothermal situation. Observe, however, that the control problem considered in [9] differs from that studied in this paper: indeed, in [9] the control $\,u\,$ occurs in the order parameter equation resembling (1.1), while in our case it appears in the energy balance (1.3); for this reason, the set of admissible controls $\mathcal{U}_\textrm{ad}$ had to be assumed in [9] as a subset of the space $H^1(0,T;L^{2}(\Omega ))\cap {L^\infty (Q)}$, which is cumbersome from the viewpoint of optimal control, instead of the much better space ${L^\infty (Q)}$ used here.

The plan of the paper is as follows. The next section is devoted to collect previous results concerning the well-posedness of the state system (1.1)–(1.5). Then, under suitable conditions, we provide some stronger analytic results in terms of regularity and stability properties of the state system with respect to the control variable u appearing in (1.3). The proof of these new results are addressed in Sect. 3. Then, using these results, we analyze in Sect. 4 the optimal control problem (P).

2 Notation, Assumptions and Analytic Results

First, let us set some notation and general assumptions. For any Banach space X, we employ the notation $\mathopen \Vert \cdot \mathclose \Vert _X$, $X^*$, and $\mathopen \langle \cdot , \cdot \mathclose \rangle _X$, to indicate the corresponding norm, its dual space, and the related duality pairing between $X^*$ and X. For two Banach spaces X and Y continuously embedded in some topological vector space Z, we introduce the linear space $X\cap Y$, which becomes a Banach space when equipped with its natural norm $\mathopen \Vert v\mathclose \Vert _{X\cap Y}:=\mathopen \Vert v\mathclose \Vert _X+\mathopen \Vert v\mathclose \Vert _Y\,$, for $v\in X\cap Y$.

A special notation is used for the standard Lebesgue and Sobolev spaces defined on $\Omega$. For every $1 \le p \le \infty$ and $k \ge 0$, they are denoted by $L^p(\Omega )$ and $W^{k,p}(\Omega )$, with the associated norms $\mathopen \Vert \,\cdot \,\mathclose \Vert _{L^p(\Omega )}=\mathopen \Vert \,\cdot \,\mathclose \Vert _{p}$ and $\mathopen \Vert \,\cdot \,\mathclose \Vert _{W^{k,p}(\Omega )}$, respectively. If $p=2$, they become Hilbert spaces, and we employ the standard convention $H^k(\Omega ):= W^{k,2}(\Omega )$. For convenience, we also set

$$\begin{aligned}&H := L^{2}(\Omega ), \quad V := H^{1}(\Omega ), \quad W := \{v\in H^{2}(\Omega ): \ \partial _{{\textbf {n}}}v=0 \, \text{ on } \,\Gamma \}. \end{aligned}$$

For simplicity, we use the symbol $\mathopen \Vert \,\cdot \,\mathclose \Vert$ for the norm in H and in any power of it. Observe that the embeddings $\, W \hookrightarrow V \hookrightarrow H \hookrightarrow V^* {{}\hookrightarrow W^*{}}\,$ are dense and compact. As usual, H is identified with a subspace of ${V^*}$ to have the Hilbert triplet $(V,H,{V^*})$ along with the identity

$$\begin{aligned} \mathopen \langle u,v\mathclose \rangle =(u,v) \quad \text {for every }u\in H \text {and }v\in V, \end{aligned}$$

where we employ the special notation $\mathopen \langle \cdot ,\cdot \mathclose \rangle := \mathopen \langle \cdot ,\cdot \mathclose \rangle _V$.

Next, for a generic element $v\in {V^*}$, we define its generalized mean value $\overline{v}$ by

$$\begin{aligned} \overline{v}:= \frac{1}{|\Omega |} \, \mathopen \langle v , \varvec{1} \mathclose \rangle , \end{aligned}$$

(2.1)

where $\varvec{1}$ stands for the constant function that takes the value 1 in $\Omega$. It is clear that $\overline{v}$ reduces to the usual mean value if $v\in H$. The same notation $\overline{v}$ is employed also if v is a time-dependent function.

To conclude, for normed spaces $\,X\,$ and $\,v\in L^1(0,T;X)$, we define the convolution products

$$\begin{aligned} (\varvec{1} * v)(t):=\int _0^t v(s)\,\textrm{ds}, \quad (\varvec{1} \circledast v)(t):=\int _t^T v(s)\,\textrm{ds} , \qquad \hbox {t }\in [0,T]. \end{aligned}$$

(2.2)

We recall that the 2D or 3D set $\Omega$ is assumed to be open, bounded, connected and have a smooth boundary $\Gamma :=\partial \Omega$. For the remainder of this paper, we make the following general assumptions.

• (A1) The structural constants $\gamma$, a, b, ${\kappa _1}$, ${\kappa _2}$, and $\lambda$ are positive.

• (A2) The double-well potential F can be written as $F= \hat{\beta }+ \hat{\pi }$, where

  $$\begin{aligned} \hat{\beta }:\mathbb {R}\rightarrow [0,+\infty ]~\text {is convex and lower semicontinuous with }\hat{\beta }(0)=0. \end{aligned}$$

  This entails that $\beta := \partial \hat{\beta }$ is a maximal monotone graph with $\beta (0) \ni 0$. Moreover, we assume that

  $$\begin{aligned} \hat{\pi }\in C^3(\mathbb {R}), \text {where } \pi := \hat{\pi }': \mathbb {R}\rightarrow \mathbb {R}~\text {is a }\textrm{Lipschitz}\,\, \text {continuous function.} \end{aligned}$$

  Besides, denoting the effective domain of $\beta$ by $D(\beta )$, we assume that $D(\beta ) = (r_-,r_+)\,$ with $\,-\infty \le r_-<0<r_+\le +\infty \,$ and that the restriction of $\hat{\beta }$ to $\,(r_-,r_+)\,$ belongs to $\,{C^3}(r_-,r_+)$. There, $\beta$ reduces to the derivative of $\hat{\beta }$, and we require that

  $$\begin{aligned} \lim _{r\searrow r_-} \beta (r)=-\infty ~\,\text {and }\, \lim _{r\nearrow r_+}\beta (r)=+\infty . \end{aligned}$$

  Please note that $F'$ in (1.2) has to be understood as $\beta + \pi$.

• (A3) Let $f\in {L^\infty (Q)}$. We set $\rho := \frac{\mathopen \Vert f\mathclose \Vert _\infty }{\gamma }$ and assume the compatibility condition that all of the quantities

  $$\begin{aligned}&{\inf _{x\in \Omega }\phi _0(x) , \ \sup _{x\in \Omega }\phi _0(x)}, \ - \rho - (\overline{\phi _0})^- \,, \ \rho + (\overline{\phi _0})^+ \quad \hbox {belong to the interior of }D(\beta ), \end{aligned}$$

where $(\cdot )^+$ and $(\cdot )^-$ denote the positive and negative part functions, respectively.

We remark that the last conditions in (A3) allow us to prove that the mean value of $\phi$ belongs to the interior of $D(\beta )$; the same assumption was prescribed in [11], where the above system (1.1)–(1.5) has been analyzed and weak and strong well-posedness results have been shown for general potentials and source terms. Since here we aim at solving the optimal control problem (P), we are forced to work under the framework of strong solutions. This, in particular, forces us to restrict the investigation to differentiable potentials, more precisely, to either regular ones like (1.13) or, under the further restriction that $d=2$, to the logarithmic potential from (1.12). Since we are going to assume (A1)–(A3) in any case, we state the following results under these assumptions, even if some of the conditions may be relaxed (cf. [11]).

As a consequence of [11, Theorems 2.2, 2.3, and 2.5], we have the following well-posedness result for the initial-boundary value problem (1.1)–(1.5).

Theorem 2.1

(Well-posedness of the state system) Suppose that (A1)–(A3) hold true, and let the data of the system fulfill

$$\begin{aligned} f&\in H^{1}(0,T;{V^*}), \quad u \in L^{2}(0,T;H), \end{aligned}$$

(2.3)

$$\begin{aligned} \phi _0&\in H^{3}(\Omega ) \cap W , \quad w_0\in V, \quad w_1\in V . \end{aligned}$$

(2.4)

Then, there exists a unique solution $(\phi ,\mu ,w)$ to the system (1.1)–(1.5) satisfying

$$\begin{aligned}&\phi \in H^{1}(0,T;V) \cap L^{\infty }(0,T;W^{2,6}(\Omega )) {\quad \hbox {with} \quad \beta (\phi ) \in L^{\infty }(0,T;L^{6}(\Omega ))}, \end{aligned}$$

(2.5)

$$\begin{aligned}&\mu \in L^{\infty }(0,T;V), \end{aligned}$$

(2.6)

$$\begin{aligned}&w \in H^{2}(0,T;H) \cap {{} C^{1}([0,T];V)}, \end{aligned}$$

(2.7)

as well as the estimate

$$\begin{aligned}&\mathopen \Vert \phi \mathclose \Vert _{H^{1}(0,T;V) \cap L^{\infty }(0,T;W^{2,6}(\Omega )) } +\mathopen \Vert \mu \mathclose \Vert _{L^{\infty }(0,T;V)} + \mathopen \Vert \beta (\phi )\mathclose \Vert _{L^{\infty }(0,T;L^{6}(\Omega ))} \nonumber \\&\quad + \mathopen \Vert w\mathclose \Vert _{H^{2}(0,T;H) \cap {C^{1}([0,T];V)} } \le K_1\,, \end{aligned}$$

(2.8)

with some constant $K_1>0$ that depends only on the structure of the system, $\Omega$, T, and upper bounds for the norms of the data and the quantities related to assumptions (2.3)–(2.4). Besides, let $u_i \in L^{2}(0,T;H)$, $i=1,2$, and let $(\phi _i,\mu _i,w_i)$ be the corresponding solutions. Then it holds that

$$\begin{aligned}&\mathopen \Vert \phi _1-\phi _2\mathclose \Vert _{L^{\infty }(0,T;{V^*})\cap L^{2}(0,T;V)} + \mathopen \Vert w_1-w_2\mathclose \Vert _{H^{1}(0,T;H)\cap L^{\infty }(0,T;V)} \nonumber \\&\quad \le K_2 \mathopen \Vert {\varvec{1}*}(u_1-u_2)\mathclose \Vert _{L^{2}(0,T;H)}\,, \end{aligned}$$

(2.9)

with some $K_2>0$ that depends only on the structure of the system, $\Omega$, T, and an upper bound for the norms of $\beta (\phi _1)$ and $\beta (\phi _2)$ in $L^{1}(Q)$.

Let us remark that, due to (2.5), the compact embedding $W^{2,6}(\Omega ) \hookrightarrow C^0(\overline{\Omega })$, and classical compactness results (see, e.g., [29, Section 8, Corollary 4]), it follows that $\phi \in C^{0}([0,T];X)$ whenever $W^{2,6}(\Omega )$ is compactly embedded in X, in particular with $X= C^0(\overline{\Omega })$, whence $\phi \in C^0(\overline{Q})$.

Remark 2.2

The above well-posedness result refers to the natural variational form of the homogeneous Neumann problem for equation (1.1), due to the low regularity of $\mu$ specified in (2.6). However, it is clear that, thanks to (2.5), (A3) and the elliptic regularity theory, we also have that $\mu \in L^{2}(0,T;W)$, so that we actually can write (1.1) in its strong form. A similar consideration can be repeated for the linear combination ${\kappa _1}\partial _tw +{\kappa _2}w$ in (1.3) as you can find in the remark below.

Remark 2.3

We point out that the regularity $C^{1}([0,T];V)$ for the variable w stated in (2.7) does not directly follow from [11, Theorems 2.2, 2.3, 2.5], where just the regularity $W^{1,\infty }(0,T;V)$ was noticed. This, however, can be deduced with the help of (1.3), rewritten as the parabolic equation

$$\begin{aligned}&{\frac{1}{{\kappa _1}} \partial _ty - \Delta y = {f_w},} \quad \text {with }\, y:={\kappa _1}\partial _tw + {\kappa _2}w \, \text {and } \,\nonumber \\&f_w:= u - \lambda \partial _t\phi + \frac{{\kappa _2}}{{\kappa _1}} \partial _tw, \end{aligned}$$

(2.10)

where, due to the previous results, it readily follows that $f_w \in L^{2}(0,T;H)$. Note that y satisfies (2.10) along with the Neumann homogeneous boundary condition in (1.4), and the initial condition (cf. (1.5))

$$\begin{aligned} {y(0)={}}({\kappa _1}\partial _tw + {\kappa _2}w) (0) = {\kappa _1}w_1 + {\kappa _2}w_0 \in V. \end{aligned}$$

Then, by a straightforward application of the parabolic regularity theory (see, e.g., [1, 22]), it turns out that

$$\begin{aligned} {y ={}} {\kappa _1}\partial _tw + {\kappa _2}w \in H^{1}(0,T;H)\cap C^{0}([0,T];V)\cap L^{2}(0,T;W) {.} \end{aligned}$$

At this point, it is not difficult to check that $w\in C^{1}([0,T];V)$, due to the solvability in V of the Cauchy problem for the linear ordinary differential equation $\, {\kappa _1}\partial _tw + {\kappa _2}w=y$.

As will be clear in the forthcoming Sect. 4, the analytic framework encapsulated in Theorem 2.1 does not suffice to rigorously prove the Fréchet differentiability of the solution operator associated with the system (1.1)–(1.5) (cf. Theorem 4.4 further on), which is a key point to formulate the first-order necessary conditions for optimality addressed in Sect. 4.3. For this reason, before entering the study of the optimal control problem (P), we present some refined analytic results which are now possible by virtue of the more restricting condition we are assuming on the potentials. In particular, a key regularity property to include singular and regular potentials in the analysis of the optimal control problem is the so-called strict separation property for the order parameter $\phi$. This means that the values of $\phi$ are always confined in a compact subset of the interior of $D(\beta )$. Notice that, if $D(\beta )=\mathbb {R}$, then the boundedness of $\phi$ that follows from the previous theorem already guarantees this property. For singular potentials, when $D(\beta )$ is an open interval, that means that the singularities of the potential at the end-points of $D(\beta )$ are not reached by $\phi$ at any time. Consequently, the potential and its derivatives can be considered as globally Lipschitz continuous functions. The proof of the following result, sketched in Sect. 3, is derived with minor modifications arguing as done in [9, Proposition 2.6]. It ensures both more regularity for the solution and the desired separation property in the important case of the logarithmic potential (1.12) in two dimensions.

Theorem 2.4

(Regularity and separation principle) Suppose that (A1)–(A3) hold, let $d=2$, and F be the logarithmic potential defined in (1.12). Moreover, in addition to (2.3)–(2.4), let f and the auxiliary datum $\mu _0$ fulfill

$$\begin{aligned} f&\in {H^{1}(0,T;H)}, \quad {\mu _0:= - \Delta \phi _0+ F'(\phi _0) +a - bw_1 \in W}. \end{aligned}$$

(2.11)

Then, the unique solution $(\phi ,\mu ,w)$ obtained from Theorem 2.1 additionally enjoys the regularity properties

$$\begin{aligned} \partial _t\phi \in L^{\infty }(0,T;H) \cap L^{2}(0,T;W), \quad \mu \in L^\infty (Q), \quad \beta (\phi ) \in L^\infty (Q), \end{aligned}$$

(2.12)

as well as

$$\begin{aligned} \mathopen \Vert \partial _t\phi \mathclose \Vert _{L^{\infty }(0,T;H) \cap L^{2}(0,T;W)} + \mathopen \Vert \mu \mathclose \Vert _{L^\infty (Q)} + \mathopen \Vert \beta (\phi )\mathclose \Vert _{L^\infty (Q)} \le K_4, \end{aligned}$$

for some $K_4>0$ that depends only on the structure of the system, the initial data, $\Omega$, and T. Furthermore, assume that

$$\begin{aligned} r_-< \min _{x \in \overline{\Omega }} \phi _0(x) \le \max _{x \in \overline{\Omega }} \phi _0(x) < r_+. \end{aligned}$$

Then, the order parameter $\phi$ enjoys the strict separation property, that is, there exist real numbers $r_*$ and $r^*$ depending only on the structure of the system such that

$$\begin{aligned} r_-< r_* \le \phi (x,t)\le r^* < r_+ \quad \text { for }a.e. \,\,(x,t) \in Q. \end{aligned}$$

Remark 2.5

We point out that the regularity for $\mu$ in (2.12) is a consequence of the regularity $\mu \in L^{\infty }(0,T;W)$ (which follows from equation (1.1) and the known regularity of $\phi$ and f) and of the Sobolev embedding $W{\hookrightarrow }L^\infty (\Omega )$, which holds up to the three-dimensional case. Notice also that a class of potentials slightly more general than the logarithmic one in (1.12) may be possibly considered: for this aim we refer to [18, Theorem 5.1], where a strict separation property has been derived in a suitable framework.

As a straightforward consequence of the above results, we have the following.

Corollary 2.6

Assume that (A1)–(A3) and (2.3)–(2.4) hold. In addition, suppose that either $D(\beta )=\mathbb {R}$ or that the assumptions of Theorem 2.4 are fulfilled. Then, there exists a positive constant $K_5$ just depending on the structure and an upper bound for the norms of the data of the system such that

$$\begin{aligned} \mathopen \Vert \phi \mathclose \Vert _{L^\infty (Q)} + \max _{i={0,1,2,3}} \mathopen \Vert F^{(i)}(\phi )\mathclose \Vert _{L^\infty (Q)} \le K_5. \end{aligned}$$

(2.13)

With the above regularity improvement, we are now in a position to obtain a stronger continuous dependence estimate concerning the controls.

Theorem 2.7

(Refined continuous dependence result) Suppose that the same assumptions as in Corollary 2.6 are fulfilled. Consider $u_i \in L^{2}(0,T;H)$, $i=1,2$, and let $(\phi _i,\mu _i,w_i)$, $i=1,2$, be the corresponding solutions. Then, it holds that

$$\begin{aligned}&\mathopen \Vert \phi _1-\phi _2\mathclose \Vert _{H^{1}(0,T;{V^*})\cap L^{\infty }(0,T;V)\cap L^{2}(0,T;W)} + \mathopen \Vert \mu _1-\mu _2\mathclose \Vert _{L^{2}(0,T;V)} \nonumber \\&\qquad + \mathopen \Vert w_1-w_2\mathclose \Vert _{H^{2}(0,T;{V^*})\cap W^{1,\infty }(0,T;{V}) \cap H^{1}(0,T;{ W})}\nonumber \\&\quad \le K_6 \mathopen \Vert u_1-u_2\mathclose \Vert _{L^{2}(0,T;H)}{,} \end{aligned}$$

(2.14)

with some $K_6>0$ that depends only on the structure of the system, $\Omega$, and T.

Notice that the above result holds for regular potentials both in dimensions two and three, as for these the Lipschitz continuity of $F'$ follows as a consequence of Theorem 2.1. On the other hand, the logarithmic potential can be considered just in dimension two as a consequence of the separation principle established by Theorem 2.4. It is worth pointing out that the regularity improvement obtained in Theorem 2.4 does not require more regularity of the control variable u. In particular, the strong well-posedness for the system is guaranteed for any control $u \in L^{2}(0,T;H)$ (in which the control space $\mathcal{U}$ is embedded, see (1.7)).

Let us conclude this section by collecting some useful tools that will be employed later on. We often owe to the Young, Poincaré and compactness inequalities:

$$\begin{aligned}&ab \le \delta a^2 + \frac{1}{4\delta } \, b^2 \quad \hbox {for every } a,b\in \mathbb {R}\,and\, \delta >0, \end{aligned}$$

(2.15)

$$\begin{aligned} {}&\mathopen \Vert v\mathclose \Vert _V \le C_\Omega \, \bigl ( \mathopen \Vert \nabla v\mathclose \Vert + |\overline{v}| \bigr ) \quad \hbox {for every }v\in V, \end{aligned}$$

(2.16)

$$\begin{aligned} {}&{\mathopen \Vert v\mathclose \Vert \le \delta \, \mathopen \Vert \nabla v\mathclose \Vert + C_{\Omega ,\delta } \, \mathopen \Vert v\mathclose \Vert _* \quad \hbox {for every }v\in V \text {and } \delta >0, } \end{aligned}$$

(2.17)

where $C_\Omega$ depends only on $\Omega$, $C_{\Omega ,\delta }$ depends on $\delta$, in addition, and $\mathopen \Vert \,\cdot \,\mathclose \Vert _*$ is the norm in ${V^*}$ to be introduced below (see (2.20)).

Next, we recall an important tool which is commonly used when working with problems connected to the Cahn–Hilliard equation. Consider the weak formulation of the Poisson equation $-\Delta z=\psi$ with homogeneous Neumann boundary conditions. Namely, for a given $\psi \in {V^*}$ (and not necessarily in H), we consider the problem:

$$\begin{aligned} \hbox {{find}} \quad z \in V \quad \hbox {such that} \quad \int _\Omega \nabla z \cdot \nabla v = \mathopen \langle \psi , v \mathclose \rangle \quad \hbox {for every }v\in V. \end{aligned}$$

(2.18)

Since $\Omega$ is connected and regular, it is well known that the above problem admits a unique solution z if and only if $\psi$ has zero mean value. Hence, we can introduce the associated solution operator $\mathcal N$, which turns out to be an isomorphism between the following spaces, as

$$\begin{aligned}&\mathcal N: \textrm{dom}(\mathcal N):=\mathopen \{\psi \in {V^*}:\ \overline{\psi }=0\mathclose \} \rightarrow \mathopen \{z\in V:\ \overline{z}=0\mathclose \}, \quad \mathcal{N}: \psi \mapsto z, \end{aligned}$$

(2.19)

where z is the unique solution to (2.18) satisfying $\overline{z}=0$. Moreover, it follows that the formula

$$\begin{aligned} \mathopen \Vert \psi \mathclose \Vert _*^2 := \mathopen \Vert \nabla \mathcal N(\psi -\overline{\psi })\mathclose \Vert ^2 + |\overline{\psi }|^2 \quad \hbox {for every }\psi \in {V^*} \end{aligned}$$

(2.20)

defines a Hilbert norm in ${V^*}$ that is equivalent to the standard dual norm of ${V^*}$. From the above properties, one can obtain the following identities:

$$\begin{aligned}&\int _\Omega \nabla \mathcal N\psi \cdot \nabla v = \mathopen \langle \psi , v \mathclose \rangle \quad \hbox {for every }\psi \in \textrm{dom}(\mathcal N), v\in V, \end{aligned}$$

(2.21)

$$\begin{aligned}&\mathopen \langle \psi , \mathcal N\zeta \mathclose \rangle = \mathopen \langle \zeta , \mathcal N\psi \mathclose \rangle \quad \hbox {for every }\psi ,\zeta \in \textrm{dom}(\mathcal N), \end{aligned}$$

(2.22)

$$\begin{aligned}&\mathopen \langle \psi , \mathcal N\psi \mathclose \rangle = \int _\Omega |\nabla \mathcal N\psi |^2 = \mathopen \Vert \psi \mathclose \Vert _*^2 \quad \hbox {for every }\psi \in \textrm{dom}(\mathcal N), \end{aligned}$$

(2.23)

as well as

$$\begin{aligned} \int _0^t\mathopen \langle \partial _tv(s) , \mathcal Nv(s) \mathclose \rangle \, ds&= \int _0^t\mathopen \langle v(s) , \mathcal N(\partial _tv(s)) \mathclose \rangle \, ds\nonumber \\&= \frac{1}{2} \, \mathopen \Vert v(t)\mathclose \Vert _*^2 - \frac{1}{2} \, \mathopen \Vert v(0)\mathclose \Vert _*^2\,, \end{aligned}$$

(2.24)

which holds for every $t\in [0,T]$ and every $v\in H^{1}(0,T;\textrm{dom}(\mathcal N))$.

Finally, without further reference later on, we are going to employ the following convention: the capital-case symbol C is used to denote every constant that depends only on the structural data of the problem such as $\Omega$, T, a, b, ${\kappa _1}$, ${\kappa _2}$, $\gamma$, $\lambda$, the shape of the nonlinearities, and the norms of the involved functions. Therefore, its meaning may vary from line to line and even within the same line. In addition, when a positive constant $\delta$ enters the computation, the related symbol $C_\delta$, in place of a general C, denotes constants that depend on $\delta$, in addition.

3 Regularity and Continuous Dependence Results

This section is devoted to the proofs of Theorems 2.4 and 2.7. The first result is propedeutic to the second one, which will play a key role in proving that the solution operator associated with the system enjoys some differentiability properties.

Proof of Theorem 2.4

We can follow exactly the same argument as that used in [9, Section 5.2] to prove the analogous result [9, Proposition 2.6]. However, although we should perform the estimates in a rigorous way on a suitable discrete scheme designed on a proper approximating problem as done in the quoted paper, we proceed formally, for simplicity, by directly acting on problem (1.1)–(1.5), and point out the few differences arising from the presence of the additional variable w. We differentiate both (1.1) and (1.2) with respect to time and test the resulting inequalities by $\partial _t\phi$ and $\Delta \partial _t\phi$, respectively. If we sum up and integrate by parts and over (0, t), then a cancellation occurs, and we obtain that

$$\begin{aligned}&\frac{1}{2} \int _\Omega |\partial _t\phi (t)|^2 + \gamma \int _{Q_t}|\partial _t\phi |^2 + \int _{Q_t}|\Delta \partial _t\phi |^2 \nonumber \\&\quad = \frac{1}{2} \int _\Omega |\partial _t\phi (0)|^2 + \int _{Q_t}\partial _t{f} \, \partial _t\phi + \int _{Q_t}(\beta '+\pi ')(\phi ) \, \partial _t\phi \, \Delta \partial _t\phi - b \int _{Q_t}\partial _t^2w \, \Delta \partial _t\phi \,. \nonumber \end{aligned}$$

This is the analogue of [9, formula (5.16)] and essentially differs from it just for the presence of the last term. However, this term can be easily dealt with by using Young’s inequality and the regularity of w ensured by (2.7). Indeed, we have that

$$\begin{aligned} - b \int _{Q_t}\partial _t^2w \, \Delta \partial _t\phi \le \frac{1}{4} \int _{Q_t}|\Delta \partial _t\phi |^2 + C \int _{Q_t}|\partial _t^2w|^2 \le \frac{1}{4} \int _{Q_t}|\Delta \partial _t\phi |^2 + C \,. \nonumber \end{aligned}$$

As the other terms can be treated as in the quoted paper, we arrive at the analogue of [9, formula (5.17)], i.e.,

$$\begin{aligned}&\mathopen \Vert \partial _t\phi \mathclose \Vert _{L^{\infty }(0,T;H)}^2 + \mathopen \Vert \Delta \partial _t\phi \mathclose \Vert _{L^{2}(0,T;H)}^2 \nonumber \\&\quad \le C \, \bigl ( \mathopen \Vert \partial _t\phi (0)\mathclose \Vert ^2 + \mathopen \Vert \beta '(\phi )\mathclose \Vert _{L^{2}(0,T;L^{3}(\Omega ))}^2 + 1 \bigr ) \, e^{C \, \mathopen \Vert \beta '(\phi )\mathclose \Vert _{L^{4}(0,T;L^{3}(\Omega ))}^4} \,. \nonumber \end{aligned}$$

At this point, the new variable w just enters the computation of $\partial _t\phi (0)$. By still proceeding formally, we recover the initial value for $\mu (0) = \mu _0$ from (1.2) at the time $t=0$, then, using the regularity of $\mu _0$ (and f) stated in (2.11), we find out that

$$\begin{aligned} \partial _t\phi (0) = f(0) + \Delta \mu _0 - \gamma \phi _0\in H \end{aligned}$$

from (1.1), also written for $t=0$. Then, we obtain that

$$\begin{aligned} {\mathopen \Vert \partial _t\phi (0)\mathclose \Vert ^2 \le \mathopen \Vert f(0) + \Delta \mu _0 - \gamma \phi _0\mathclose \Vert ^2 \le C.} \nonumber \end{aligned}$$

At this point, w does not enter the argument any longer, and we can proceed as in [9]. We do not repeat here the corresponding computations for convenience, and just point out that the assumption $d=2$ enters when applying the Trudinger inequality to handle the contributions arising from the nonlinearity $\beta$ and its derivatives. $\square$

Proof of Theorem 2.7

To begin with, let us set the following notation for the differences involved in the statement:

$$\begin{aligned} \phi :=\phi _1-\phi _2, \quad \mu := \mu _1-\mu _2, \quad u:= u_1- u_2, \quad w:= w_1 - w_2. \end{aligned}$$

Next, we write the system solved by the differences that, in its strong form, reads as

$$\begin{aligned}&\partial _t\phi - \Delta \mu + \gamma \phi = 0 \quad{} & {} \text { in }Q, \end{aligned}$$

(3.1)

$$\begin{aligned}&\mu = - \Delta \phi + (F'(\phi _1) - F'(\phi _2)) - b \partial _tw \quad{} & {} \hbox { in}\ Q, \end{aligned}$$

(3.2)

$$\begin{aligned}&\partial _t^2w - \Delta ({\kappa _1}\partial _tw + {\kappa _2}w) + \lambda \partial _t\phi = u \quad{} & {} \hbox { in}\ Q, \end{aligned}$$

(3.3)

$$\begin{aligned}&\partial _{{\textbf {n}}}\phi = \partial _{{\textbf {n}}}\mu = \partial _{{\textbf {n}}}({\kappa _1}\partial _tw + {\kappa _2}w) = 0 \quad{} & {} \hbox { on}\ \Sigma , \end{aligned}$$

(3.4)

$$\begin{aligned}&\phi (0) = w(0) = \partial _tw(0) = 0 \quad{} & {} \hbox { in}\ \Omega . \end{aligned}$$

(3.5)

First Estimate

First, we recall that $F'$ is now assumed to be Lipschitz continuous. Then, testing (3.1) by $\phi$, (3.2) by $\mu$, and adding the resulting equations lead us to

$$\begin{aligned}&\frac{1}{2} \, \frac{d}{dt} \, \mathopen \Vert \phi \mathclose \Vert ^2 + \gamma \mathopen \Vert \phi \mathclose \Vert ^2 + \mathopen \Vert \mu \mathclose \Vert ^2 = \int _\Omega (F'(\phi _1)-F'(\phi _2)) \mu - b \int _\Omega \partial _tw \mu \\ {}&\quad {{} \le \frac{1}{2} \, \mathopen \Vert \mu \mathclose \Vert ^2 + C \bigl ( \mathopen \Vert \phi \mathclose \Vert ^2+\mathopen \Vert \partial _tw\mathclose \Vert ^2 \bigl ).} \end{aligned}$$

Now, recalling the continuous dependence estimate already proved in Theorem 2.1, we infer, after integrating over time, that

$$\begin{aligned} \mathopen \Vert \phi _1-\phi _2\mathclose \Vert _{L^{\infty }(0,T;H)} + \mathopen \Vert \mu _1-\mu _2\mathclose \Vert _{L^{2}(0,T;H)} \le C \mathopen \Vert {\mathbf {1 *}} (u_1-u_2)\mathclose \Vert _{L^{2}(0,T;H)}. \end{aligned}$$

(3.6)

Second Estimate

First, let us establish an auxiliary estimate. Since $F'$ and $F''$ are supposed to be Lipschitz continuous and (2.8) ensures a uniform bound for $\mathopen \Vert \nabla \phi _2\mathclose \Vert _\infty$ , since $W^{1,6}(\Omega )\hookrightarrow L^{\infty }(\Omega )$, we have, almost everywhere in (0, T), that

$$\begin{aligned}{} & {} \mathopen \Vert F'(\phi _1)-F'(\phi _2)\mathclose \Vert _V \le \mathopen \Vert F'(\phi _1)-F'(\phi _2)\mathclose \Vert + \mathopen \Vert F''(\phi _1)\nabla \phi _1 - F''(\phi _2)\nabla \phi _2\mathclose \Vert \nonumber \\{} & {} \quad \le C \, \mathopen \Vert \phi \mathclose \Vert + \mathopen \Vert F''(\phi _1)\nabla \phi \mathclose \Vert + \mathopen \Vert (F''(\phi _1)-F''(\phi _2))\nabla \phi _2\mathclose \Vert \nonumber \\{} & {} \quad \le C \, \mathopen \Vert \phi \mathclose \Vert + C \, \mathopen \Vert \nabla \phi \mathclose \Vert \le C \, \mathopen \Vert \phi \mathclose \Vert _V. \nonumber \end{aligned}$$

Next, we multiply (3.1) by ${\textbf{1}/ |\Omega |}$ to obtain that

$$\begin{aligned} \frac{d}{dt} \, \overline{\phi }(t) + \gamma \, \overline{\phi }(t)= 0 \quad \hbox {for a.a.}\, t\in (0,T), \end{aligned}$$

(3.7)

which entails that $\overline{\phi }(t)=0$ for every $t\in [0,T]$ since $\overline{\phi }(0)=0$. In particular, besides $\phi$, even $\partial _t\phi$ has zero mean value. Thus, we are allowed to test (3.1) by $\mathcal{N} (\partial _t\phi )$, (3.2) by $-\partial _t\phi$, and (3.3) by $\frac{b}{\lambda }\partial _tw$, and add the resulting identities. By also accounting for the Lipschitz continuity of $F'$ and the Young inequality, we deduce that, a.e. in (0, T),

$$\begin{aligned}&\mathopen \Vert \partial _t\phi \mathclose \Vert _*^2 + \frac{\gamma }{2} \, \frac{d}{dt} \, \mathopen \Vert \phi \mathclose \Vert _*^2 + \frac{1}{2} \, \frac{d}{dt} \, \mathopen \Vert \nabla \phi \mathclose \Vert ^2 + {\frac{b}{2\lambda }} \, \frac{d}{dt} \, \mathopen \Vert \partial _tw\mathclose \Vert ^2 \\&\qquad \quad + \frac{{\kappa _1}b}{2\lambda } \, \mathopen \Vert \nabla (\partial _tw)\mathclose \Vert ^2 + \frac{{\kappa _2}b}{2 \lambda } \, \frac{d}{dt} \, \mathopen \Vert \nabla w\mathclose \Vert ^2 \\ {}&\quad {} = \int _\Omega (F'(\phi _1) - F'(\phi _2)) \partial _t\phi + \frac{b}{\lambda }\int _\Omega u \, \partial _tw \\ {}&\quad {{} \le C \bigl ( \mathopen \Vert \phi \mathclose \Vert _V\, \mathopen \Vert \partial _t\phi \mathclose \Vert _* + \mathopen \Vert u\mathclose \Vert \, \mathopen \Vert \partial _tw\mathclose \Vert \bigr )} {{} \le \frac{1}{2} \, \mathopen \Vert \partial _t\phi \mathclose \Vert _*^2 + C \bigl ( \mathopen \Vert \phi \mathclose \Vert _V^2 + \mathopen \Vert u\mathclose \Vert ^2 + \mathopen \Vert \partial _tw\mathclose \Vert ^2 \bigr ).} \end{aligned}$$

Hence, integrating over time and using (2.9), we may conclude that

$$\begin{aligned}&\mathopen \Vert \phi _1-\phi _2\mathclose \Vert _{H^{1}(0,T;{V^*})\cap L^{\infty }(0,T;V)} + \mathopen \Vert w_1-w_2\mathclose \Vert _{W^{1,\infty }(0,T;H) \cap H^{1}(0,T;V)} \nonumber \\&\quad {} \le C\mathopen \Vert u_1-u_2\mathclose \Vert _{L^{2}(0,T;H)}. \end{aligned}$$

(3.8)

Third Estimate

By testing (3.1) by $\mu$, we have that

$$\begin{aligned} \int _\Omega \partial _t\phi \, \mu + \int _\Omega |\nabla \mu |^2 + \gamma \int _\Omega \phi \mu = 0 \,. \nonumber \end{aligned}$$

Now, we recall that $\phi$ and $\partial _t\phi$ have zero mean value. Hence, by also accounting for the Poincaré inequality (2.16), we deduce that

$$\begin{aligned} \int _\Omega |\nabla \mu |^2 = - \int _\Omega \partial _t\phi \, (\mu -\overline{\mu }) {\,-\,} \gamma \int _\Omega \phi (\mu -\overline{\mu }) \le \frac{1}{2} \int _\Omega |\nabla \mu |^2 + C \bigl ( \mathopen \Vert \partial _t\phi \mathclose \Vert _*^2 + \mathopen \Vert \phi \mathclose \Vert ^2 \bigr ). \nonumber \end{aligned}$$

Therefore, thanks to (3.6) and (3.8), it readily follows that

$$\begin{aligned} {\mathopen \Vert \mu _1- \mu _2\mathclose \Vert _{L^{2}(0,T;V)}} \le C \mathopen \Vert u_1-u_2\mathclose \Vert _{L^{2}(0,T;H)}. \end{aligned}$$

(3.9)

Fourth Estimate

A simple comparison argument in (3.2), along with the above estimates and elliptic regularity theory, entails that

$$\begin{aligned} \mathopen \Vert \phi _1- \phi _2\mathclose \Vert _{L^{2}(0,T;W)} \le C \mathopen \Vert u_1-u_2\mathclose \Vert _{L^{2}(0,T;H)}. \end{aligned}$$

(3.10)

Fifth Estimate

We take an arbitrary $v\in L^{2}(0,T;V)$, multiply (3.3) by v, and integrate over Q and by parts. By rearranging and estimating, we easily obtain that

$$\begin{aligned} \int _Q\partial _t^2w \, v&\le C \bigl ( \mathopen \Vert u\mathclose \Vert _{L^{2}(0,T;H)} + \mathopen \Vert \partial _tw\mathclose \Vert _{L^{2}(0,T;V)}\bigr ) \mathopen \Vert v\mathclose \Vert _{L^{2}(0,T;V)} \\&\quad + C \bigl (\mathopen \Vert w\mathclose \Vert _{L^{2}(0,T;V)} + \mathopen \Vert \partial _t\phi \mathclose \Vert _{L^{2}(0,T;{V^*})} \bigr ) \mathopen \Vert v\mathclose \Vert _{L^{2}(0,T;V)} \,. \end{aligned}$$

On account of the previous estimates, we conclude that

$$\begin{aligned} \mathopen \Vert \partial _t^2w_1 - \partial _t^2w_2\mathclose \Vert _{L^{2}(0,T;{V^*})} \le C \mathopen \Vert u_1-u_2\mathclose \Vert _{L^{2}(0,T;H)}. \end{aligned}$$

(3.11)

Sixth Estimate

Arguing as in Remark 2.3, we now rewrite (3.3) as a parabolic equation in the auxiliary variable $y:= {\kappa _1}\partial _tw + {\kappa _2}w + {\kappa _1}\lambda \phi$ obtaining that

$$\begin{aligned}&{\frac{1}{{\kappa _1}} \partial _ty - \Delta y } = {u + \displaystyle \frac{{\kappa _2}}{{\kappa _1}} \partial _tw - {\kappa _1}\lambda \Delta \phi .} \end{aligned}$$

Besides, let us underline that y satisfies homogeneous Neumann boundary conditions and null initial conditions, as it can be realized from (3.4) and (3.5). Then, using a well-known parabolic regularity result and the already found estimates (3.8) and (3.10), it is straightforward to deduce that

$$\begin{aligned}&\mathopen \Vert y\mathclose \Vert _{H^{1}(0,T;H)\cap C^{0}([0,T];V)\cap L^{2}(0,T;W)} \\&\quad \le C \Bigl \Vert u + \displaystyle \frac{{\kappa _2}}{{\kappa _1}} \partial _tw - {\kappa _1}\lambda \Delta \phi \Bigr \Vert _{L^{2}(0,T;H)}\, \le C \mathopen \Vert u\mathclose \Vert _{L^{2}(0,T;H)}\,. \end{aligned}$$

Thus, by solving the Cauchy problem for the ordinary differential equation ${\kappa _1}\partial _tw + {\kappa _2}w = y - {\kappa _1}\lambda \phi$ in terms of w, and recalling again (3.8) and (3.10), we find out that

$$\begin{aligned} \mathopen \Vert w_1 - w_2\mathclose \Vert _{{H^{2}(0,T;{V^*})\cap {}} W^{1,\infty }(0,T;V) \cap H^{1}(0,T;W)} \le C \mathopen \Vert u_1-u_2\mathclose \Vert _{L^{2}(0,T;H)}. \end{aligned}$$

(3.12)

This completes the proof, as collecting the above estimates yields (2.14). $\square$

4 The Optimal Control Problem

In this section, we study the optimal control problem introduced at the beginning, which we recall here for the reader’s convenience:

$$\begin{aligned} {\varvec{(\mathrm P)}} \quad \min _{u \in \mathcal{U}_\textrm{ad}}\mathcal{J}((\phi , w), u) \quad \text {subject to the constraint that }(\varphi ,\mu , w)\,\, \text {solves } \end{aligned}$$

(1.1)–(1.5), where the cost functional $\mathcal{J}$ is given by (1.6).

To begin with, let us fix some notation concerning the solution operator $\mathcal{S}$ associated with the system (1.1)–(1.5). As a consequence of the Theorems 2.1, 2.4, and 2.7, the control-to-state operator

$$\begin{aligned} \mathcal{S}{{} = (\mathcal{S}_1,\mathcal{S}_2,\mathcal{S}_3)} : L^2(Q) {{}(\supset \mathcal{U}){}} \rightarrow \mathcal{Y}, \quad \mathcal{S}: u \mapsto (\phi , \mu , w), \end{aligned}$$

is well defined, where $(\phi ,\mu ,w) \in \mathcal Y$ is the unique solution to the state system, and the Banach space $\mathcal Y$, referred to as the state space, is defined by some regularity conditions that are surely implied by (2.5)–(2.7), that is,

$$\begin{aligned} \mathcal{Y}&:= \big ({H^{1}(0,T;V) \cap L^{\infty }(0,T;W^{2,6}(\Omega ))}\big ) \times L^{\infty }(0,T;V) \\ {}&\, \, \quad \times \big ( H^{2}(0,T;H) \cap {C^{1}([0,T];V)} \big ). \end{aligned}$$

Moreover, the continuous dependence estimate provided by Theorem 2.7 allows us to infer that the solution operator is Lipschitz continuous in the sense that, for any pair $(u_1,u_2)$ of controls, it holds that

$$\begin{aligned}&\mathopen \Vert \mathcal{S}(u_1)- \mathcal{S}(u_2)\mathclose \Vert _\mathcal{X} \le K_6\mathopen \Vert u_1-u_2\mathclose \Vert _{L^{2}(0,T;H)}, \end{aligned}$$

where $\mathcal X$ is the space defined by

$$\begin{aligned} \mathcal {X}&:= \big (H^{1}(0,T;{V^*})\cap L^{\infty }(0,T;V) \cap L^{2}(0,T;W)\big )\times L^{2}(0,T;V) \nonumber \\ {}&\quad \, \, \times \big (H^{2}(0,T;{V^*})\cap {W^{1,\infty }(0,T;V) \cap H^{1}(0,T;W)}\big ). \end{aligned}$$

(4.1)

Furthermore, we introduce the reduced cost functional, given by

$$\begin{aligned} {\mathcal{J}}_\textrm{red}: L^2(Q) \subset \mathcal{U} \rightarrow \mathbb {R}, \quad {\mathcal{J}}_\textrm{red}: u \mapsto \mathcal{J}(\mathcal{S}_1( u),\mathcal{S}_3( u), u), \end{aligned}$$

(4.2)

which allows us to reduce the optimization problem (P) to the form

$$\begin{aligned} \min _{u \in \mathcal{U}_\textrm{ad}} {\mathcal{J}}_\textrm{red}(u). \end{aligned}$$

In what follows, we are working in the framework of Theorem 2.1 (and possibly in the sense of Theorem 2.4). For this reason, the following conditions will be in order:

• (C1) The source f fulfills (2.3), and the initial data $\phi _0, w_0,$ and $w_1$ satisfy (2.4). Moreover, if we consider the logarithmic potential and $d=2$, they additionally fulfill (2.11).

• (C2) The functions $u_\textrm{min}, u_\textrm{max}$ belong to $\mathcal U$ with $u_\textrm{min} \le u_\textrm{max} \,\, \hbox {a.e. in Q}$.

• (C3) ${\alpha _1},\dots ,{\alpha _6},$ and $\nu$ are nonnegative constants, not all zero.

• (C4) The target functions fulfill $\phi _Q,w_Q, w_Q' \in L^2(Q)$, $\phi _\Omega \in V, {w_\Omega }\in H$, and $w_\Omega ' \in V$.

Please note that the restrictions imposed on $\phi _\Omega$ and $w_\Omega '$ are just of technical nature and they are needed for the solvability of the adjoint problem. On the other hand, they can be avoided provided that $\alpha _2=\alpha _6 =0$.

4.1 Existence of Optimal Controls

The first result we are going to address concerns the existence of optimal controls.

Theorem 4.1

(Existence of optimal controls) We suppose that the assumptions (A1)–(A3) and (C1)–(C4) are fulfilled. Then, the optimal control problem (P) admits a solution.

Proof of Theorem 4.1

As the proof is an immediate consequence of the direct method of the calculus of variations, we just briefly outline the crucial steps. Consider a minimizing sequence $\{u_n\}_n \subset \mathcal{U}_\textrm{ad}$ for the reduced cost functional ${\mathcal{J}}_\textrm{red}$ defined by (4.2). Let us introduce also the sequence of the associated states $\{(\phi _n,\mu _n, w_n)\}_n$, where $(\phi _n,\mu _n, w_n) =\mathcal{S}(u_n)$ for every $n \in \mathbb {N}$. Namely, we have that

$$\begin{aligned} \lim _{n \rightarrow \infty } {\mathcal{J}}_\textrm{red}(u_n) = \lim _{n \rightarrow \infty } \mathcal{J}\big ((\mathcal{S}_1(u_n), \mathcal{S}_3(u_n)), u_n \big ) = \inf _{u \in \mathcal{U}_\textrm{ad}} {\mathcal{J}}_\textrm{red}(u) \ge 0. \end{aligned}$$

Thus, as $\mathcal{U}_\textrm{ad}$ is bounded in $\mathcal U$, by standard compactness arguments, using also that $\mathcal{U}_\textrm{ad}$ is closed and convex, we obtain a limit function $u^* \in \mathcal{U}_\textrm{ad}$ and a nonrelabelled subsequence such that, as $n \rightarrow \infty$,

$$\begin{aligned} u_n \rightarrow u^* \quad \hbox { weakly-star in}\ L^\infty (Q). \end{aligned}$$

On the other hand, by the boundedness property (2.8) stated in Theorem 2.1, along with standard compactness results (see, e.g., [29, Section 8, Corollary 4]), we also have that

$$\begin{aligned} \phi _n&\rightarrow \phi ^* \quad{} & {} \text {weakly-star in }H^{1}(0,T;V) \cap L^{\infty }(0,T;W^{2,6}(\Omega )), \\ {}{} & {} {}&\quad \hbox { and strongly in}\ C^0(\overline{Q}), \\ \mu _n&\rightarrow \mu ^* \quad{} & {} \text {weakly-star in}\ L^{\infty }(0,T;V), \\ w_n&\rightarrow w^* \quad{} & {} \text {weakly-star in}\, H^{2}(0,T;H) \cap W^{1,\infty }(0,T;V), \\ {}{} & {} {}&\quad \hbox { and strongly in }{}\ C^{1}([0,T];H), \\ {F'(\phi _n)}&{\rightarrow \xi ^* \quad }{} & {} \text {weakly-star in}\, L^{\infty }(0,T;L^{6}(\Omega )), \end{aligned}$$

for some limits $\phi ^*,\mu ^*, w^*$, and $\xi ^*$. The first strong convergence follows from the compact embedding $W^{2,6}(\Omega ) \hookrightarrow C^0(\overline{\Omega })$. Besides, as

$$\begin{aligned} \pi (\phi _n) \rightarrow \pi (\phi ^*) \quad \hbox {strongly in }C^0(\overline{Q}) \ \ \text { and} \quad \beta (\phi _n) \rightarrow \xi ^* - \pi (\phi ^*) \quad \hbox {weakly in }L^1(Q), \end{aligned}$$

by maximal monotonicity arguments it is not difficult to conclude that $\xi ^* = F'(\phi ^*)$. Then, using the above weak, weak-star and strong convergence properties, it is a standard matter to pass to the limit as n tends to infinity in the variational formulation associated with system (1.1)–(1.5), written for $\phi _n,\mu _n, w_n,$ and $u_n$. This will also prove that $(\phi ^*, \mu ^*, w^*)$ is nothing but $\mathcal{S}(u^*)$. Finally, the lower semicontinuity of norms entails that

$$\begin{aligned} {\mathcal{J}}_\textrm{red}(u^*) \le \liminf _{n\rightarrow \infty } {\mathcal{J}}_\textrm{red}(u_n ) = \lim _{n\rightarrow \infty } {\mathcal{J}}_\textrm{red}(u_n ) = \inf _{ u\in \mathcal{U}_\textrm{ad}} {\mathcal{J}}_\textrm{red}(u)\,, \end{aligned}$$

meaning that $u^*$ is a global minimizer for ${\mathcal{J}}_\textrm{red}$. $\square$

4.2 Differentiability of the Solution Operator

In the following, we are going to prove some differentiability properties for the solution operator $\mathcal{S}$. Since these have to be analyzed in open sets, let us take an open ball in the $L^\infty$-topology that contains the set of admissible controls $\mathcal{U}_\textrm{ad},$ namely, let $R>0$ be chosen such that

$$\begin{aligned} { \mathcal{U}_R:= \{ u \in \mathcal{U} : \mathopen \Vert u\mathclose \Vert _\mathcal{U} <R\} \supset \mathcal{U}_\textrm{ad}.} \end{aligned}$$

Now, we fix $u \in \mathcal{U}_R$ and denote by $(\phi , \mu , w )= \mathcal{S}(u)$ the unique corresponding state. Then, the linearized system to (1.1)–(1.5) at the fixed control u is given, for any $h \in L^2(Q)$, as follows:

$$\begin{aligned}&\partial _t\xi - \Delta \eta + \gamma \xi = 0 \quad{} & {} \hbox {in}\ Q, \end{aligned}$$

(4.3)

$$\begin{aligned}&\eta = - \Delta \xi + F''(\phi )\xi - b \partial _t\zeta \quad{} & {} \text {in } Q, \end{aligned}$$

(4.4)

$$\begin{aligned}&\partial _t^2\zeta - \Delta ({\kappa _1}\partial _t\zeta + {\kappa _2}\zeta ) + \lambda \partial _t\xi = h \quad{} & {} \hbox {in}\ Q, \end{aligned}$$

(4.5)

$$\begin{aligned}&\partial _{{\textbf {n}}}\xi = \partial _{{\textbf {n}}}\eta = \partial _{{\textbf {n}}}({\kappa _1}\partial _t\zeta + {\kappa _2}\zeta ) = 0 \quad{} & {} \hbox {on}\ \Sigma , \end{aligned}$$

(4.6)

$$\begin{aligned}&\xi (0) = \zeta (0) = \partial _t\zeta (0) = 0 \quad{} & {} \text {in } \Omega . \end{aligned}$$

(4.7)

The proof of the well-posedness of the above system is very similar (and, in fact, easier, as the system is linear) to the proof of Theorem 2.1. We have the following result.

Theorem 4.2

(Well-posedness of the linearized system) Assume that (A1)–(A3) and (C1) hold, and let $u \in \mathcal{U}_R$ with associated state $(\phi , \mu , w )= \mathcal{S}(u)$ be given. Then, for every $h \in L^2(Q)$, the linearized system (4.3)–(4.7) admits a unique solution $(\xi ,\eta ,\zeta ) \in \mathcal{X}$, where $\mathcal X$ is the Banach space introduced by (4.1). Furthermore, there exists some $K_7>0$, which depends only on the structure of the system and an upper bound for the norm of f and those of the initial data, such that

$$\begin{aligned}&\mathopen \Vert \xi \mathclose \Vert _{H^{1}(0,T;{V^*})\cap L^{\infty }(0,T;V) \cap L^{2}(0,T;W)} + \mathopen \Vert \eta \mathclose \Vert _{L^{2}(0,T;V)} \nonumber \\ {}&\quad + \mathopen \Vert \zeta \mathclose \Vert _{ H^{2}(0,T;{V^*})\cap {{}W^{1,\infty }(0,T;V) \cap H^{1}(0,T;W){}}} \le {K_7 \mathopen \Vert h\mathclose \Vert _{L^{2}(0,T;H)}}. \end{aligned}$$

(4.8)

Remark 4.3

Due to the low regularity level given by the definition (4.1) of $\mathcal X$, the above result must refer to a proper variational formulation of the linearized problem. For instance, (4.3) with the homogeneous Neumann boundary condition for $\eta$ has to be read as

$$\begin{aligned} \mathopen \langle \partial _t\xi , v \mathclose \rangle + \int _\Omega \nabla \eta \cdot \nabla v + \gamma \int _\Omega \xi v = 0 \quad \hbox {a.e. in (0,T)}, \hbox {for every }v\in V. \nonumber \end{aligned}$$

Proof of Theorem 4.2

As the system is linear, the uniqueness of solutions readily follows once (4.8) has been shown for a special solution. Indeed, suppose that there are two solutions $(\xi _1,\eta _1,\zeta _1)$ and $(\xi _2,\eta _2,\zeta _2).$ It is then enough to repeat the procedure used below with $\xi = \xi _1-\xi _2$, $\eta =\eta _1-\eta _2$ and $\zeta =\zeta _1-\zeta _2$ to realize that the same as (4.8) holds with the right-hand side equal to 0 so that $(\xi _1,\eta _1,\zeta _1)\equiv (\xi _2,\eta _2,\zeta _2)$, i.e., the uniqueness.

Since the proof of existence is standard, we avoid introducing any approximation scheme and just provide formal estimates. The rigorous argument can be straightforwardly reproduced, e.g., on a suitable Faedo–Galerkin scheme.

First Estimate

We aim at proving that

$$\begin{aligned}&\mathopen \Vert \xi \mathclose \Vert _{L^{\infty }(0,T;V)} + \mathopen \Vert \eta \mathclose \Vert _{L^{2}(0,T;V)} + \mathopen \Vert \zeta \mathclose \Vert _{W^{1,\infty }(0,T;H) \cap H^{1}(0,T;V)}\nonumber \\&\quad \le C \mathopen \Vert h\mathclose \Vert _{L^{2}(0,T;H)} \,. \end{aligned}$$

(4.9)

We preliminarily observe that

$$\begin{aligned} \mathopen \Vert \partial _t\xi \mathclose \Vert _{L^{2}(0,t;{V^*})} \le C \bigl ( \mathopen \Vert \xi \mathclose \Vert _{L^{2}(0,t;H)} + \mathopen \Vert \eta \mathclose \Vert _{L^{2}(0,t;V)} \bigr ) \quad \hbox {for every }t\in (0,T]\,, \end{aligned}$$

(4.10)

as one immediately sees by multiplying (4.3) by any $v\in L^{2}(0,t;V)$ and integrating over $Q_t$ and by parts. Moreover, we recall (2.8) and (2.13) and observe that the former yields a uniform $L^\infty$ bound for $\nabla \phi$ since $W^{1,6}(\Omega )\hookrightarrow L^{\infty }(\Omega )$. It then follows that

$$\begin{aligned} \mathopen \Vert F''(\phi )\xi \mathclose \Vert _V \le C \mathopen \Vert \xi \mathclose \Vert _V\quad \hbox {a.e. in (0,T)}\,. \end{aligned}$$

(4.11)

At this point, we are ready to perform the desired estimate. We test (4.3) by $\eta +\xi$, (4.4) by $-\partial _t\xi +\eta$, (4.5) by $\frac{b}{\lambda }\partial _t\zeta$, and add the resulting equalities to infer that a.e. in (0, T) it holds

$$\begin{aligned}&\mathopen \Vert \eta \mathclose \Vert ^2_V + \frac{1}{2} \, \frac{d}{dt} \, \mathopen \Vert \xi \mathclose \Vert ^2_V + \gamma \mathopen \Vert \xi \mathclose \Vert ^2 + \frac{b}{2\lambda } \, \frac{d}{dt} \, \mathopen \Vert \partial _t\zeta \mathclose \Vert ^2 + \frac{{\kappa _1}b}{\lambda } \, \mathopen \Vert \nabla \partial _t\zeta \mathclose \Vert ^2 + \frac{{\kappa _2}b}{2\lambda } \, \frac{d}{dt} \, \mathopen \Vert \nabla \zeta \mathclose \Vert ^2 \\ {}&\quad = - \gamma \int _\Omega \xi \eta + \int _\Omega F''(\phi ) \xi \, (\eta - \partial _t\xi ) - b \int _\Omega \partial _t\zeta \eta + \frac{b}{\lambda }\int _\Omega h \, \partial _t\zeta \,, \end{aligned}$$

thanks to a number of cancellations. Now, the whole right-hand side can easily be bounded from above by

$$\begin{aligned}&\frac{1}{4} \, \mathopen \Vert \eta \mathclose \Vert ^2_V + C \bigl ( \mathopen \Vert \xi \mathclose \Vert _V^2 + \mathopen \Vert \partial _t\zeta \mathclose \Vert ^2 + \mathopen \Vert h\mathclose \Vert ^2) - \int _\Omega F''(\phi ) \xi \, \partial _t\xi \,, \end{aligned}$$

and it is clear that (4.9) follows upon integrating in time and invoking Gronwall’s lemma provided we can properly estimate the time integral of the last term. Using also (4.10) and (4.11), we have that

$$\begin{aligned}{} & {} - \int _{Q_t}F''(\phi ) \xi \, \partial _t\xi \le C \mathopen \Vert F''(\phi )\xi \mathclose \Vert _{L^{2}(0,t;V)} \mathopen \Vert \partial _t\xi \mathclose \Vert _{L^{2}(0,t;{V^*})} \nonumber \\{} & {} \quad \le C \mathopen \Vert \xi \mathclose \Vert _{L^{2}(0,t;V)} \bigl ( \mathopen \Vert \xi \mathclose \Vert _{L^{2}(0,t;H)} + \mathopen \Vert \eta \mathclose \Vert _{L^{2}(0,t;V)} \bigr ) \le \frac{1}{4} \, \mathopen \Vert \eta \mathclose \Vert _{L^{2}(0,t;V)}^2 + C \mathopen \Vert \xi \mathclose \Vert _{L^{2}(0,t;V)}^2, \nonumber \end{aligned}$$

and this is sufficient to conclude.

Second Estimate

We now readily deduce from (4.10) that

$$\begin{aligned} \mathopen \Vert \partial _t\xi \mathclose \Vert _{L^{2}(0,T;{V^*})} \le C \mathopen \Vert h\mathclose \Vert _{L^{2}(0,T;H)}\,. \end{aligned}$$

(4.12)

On the other hand, by comparing the terms in (4.4) and taking advantage of (4.9) and (4.11), well-known elliptic regularity results allow us to infer that

$$\begin{aligned} \Vert {\xi }\Vert _{L^{2}(0,T;W)} \le C \mathopen \Vert h\mathclose \Vert _{L^{2}(0,T;H)}\,. \end{aligned}$$

(4.13)

Third Estimate

Now, let us rewrite equation (4.5) in terms of the auxiliary variable $z: = {\kappa _1}\partial _t\zeta + {\kappa _2}\zeta + {\kappa _1}\lambda \xi$. We obtain

$$\begin{aligned}&\frac{1}{{\kappa _1}} \partial _tz - \Delta z = h + \frac{{\kappa _2}}{{\kappa _1}} \partial _t\zeta - {\kappa _1}\lambda \Delta \xi , \end{aligned}$$

and observe that, in view of (4.6)–(4.7), z satisfies Neumann homogeneous boundary conditions and null initial conditions. Then, by known parabolic regularity results, (4.9), and (4.13), we easily deduce that

$$\begin{aligned}&\mathopen \Vert z\mathclose \Vert _{H^{1}(0,T;H)\cap C^{0}([0,T];V)\cap L^{2}(0,T;W)} \le C \Bigl \Vert h + \frac{{\kappa _2}}{{\kappa _1}} \partial _t\zeta - {\kappa _1}\lambda \Delta \xi \Bigr \Vert _{L^{2}(0,T;H)}\, \\&\quad \le C \mathopen \Vert h\mathclose \Vert _{L^{2}(0,T;H)}\,. \end{aligned}$$

Hence, by recalling the definition of z and the already proved bounds (4.9), (4.12), and (4.13), we arrive at

$$\begin{aligned} \mathopen \Vert \zeta \mathclose \Vert _{H^{2}(0,T;{V^*})\cap W^{1,\infty }(0,T;V) \cap H^{1}(0,T;W)} \le C \mathopen \Vert h\mathclose \Vert _{L^{2}(0,T;H)}\,. \end{aligned}$$

(4.14)

Due to the embeddings $V^* \hookrightarrow W^*$ and $W \hookrightarrow H\equiv H^* \hookrightarrow W^*$, by interpolation we have that

$$\begin{aligned} H^{2}(0,T;{V^*})\cap H^{1}(0,T;W) \hookrightarrow C^{1}([0,T];H), \end{aligned}$$

whence (4.14) entails, in particular, that

$$\begin{aligned} \mathopen \Vert \zeta \mathclose \Vert _{C^{1}([0,T];H)} \le C \mathopen \Vert h\mathclose \Vert _{L^{2}(0,T;H)}. \end{aligned}$$

(4.15)

This concludes the sketch of the proof. $\square$

We now expect that - provided we select the correct Banach spaces - the linearized system encapsulates the behavior of the Fréchet derivative of the solution operator $\mathcal{S}$. This is stated rigorously in the next theorem, but prior to this, let us introduce the following Banach space:

$$\begin{aligned} \mathcal{Z}&:= \big ({{}H^{1}(0,T;W^*) \cap C^{0}([0,T];H){}} \cap L^{2}(0,T;W) \big ) \times L^{2}(0,T;H) \nonumber \\&\ \quad {}\times \big ({ H^{2}(0,T;W^*) \cap C^{1}([0,T];H) \cap H^{1}(0,T;W) }\big ). \end{aligned}$$

(4.16)

Theorem 4.4

(Fréchet differentiability of the solution operator) Let the set of assumptions (A1)–(A3) and (C1) be fulfilled. Then, the control-to-state operator $\mathcal{S}$ is Fréchet differentiable at any $u\in \mathcal{U}_R$ as a mapping from $L^2(Q)$ into $\mathcal Z$. Moreover, for $u\in \mathcal{U}_R$, the mapping $D\mathcal{S}(u)\in \mathcal L(L^2(Q),\mathcal Z)$ acts as follows: for every $h\in L^2(Q)$, $D\mathcal{S}(u)h$ is the unique solution $(\xi ,\eta ,\zeta )$ to the linearized system (4.3)–(4.7) associated with h.

Proof of Theorem 4.4

We fix $u\in \mathcal{U}_R$ and first notice that the map $h\mapsto (\xi ,\eta ,\zeta )$ of the statement actually belongs to $\mathcal L(L^2(Q),\mathcal Z)$ as a consequence of (4.8). Then, we proceed with a direct check of the claim by showing that

$$\begin{aligned} \frac{\mathopen \Vert \mathcal{S}(u+h) - \mathcal{S}(u) - (\xi ,\eta , \zeta )\mathclose \Vert _\mathcal{Z}}{\mathopen \Vert h\mathclose \Vert _{L^2(Q)}} \rightarrow 0 \quad \text {as }\mathopen \Vert h\mathclose \Vert _{L^2(Q)} \rightarrow 0. \end{aligned}$$

(4.17)

This will imply both the Fréchet differentiability of $\mathcal{S}$ in the sense specified in the statement and the validity of the identity $D\mathcal{S}(u)h = (\xi ,\eta , \zeta )$.

At this place, we remark that the following argumentation will be formal, because of the low regularity of the linearized variables (recall Remark 4.3). Nevertheless, we adopt it for brevity, in order to avoid any approximation, like a Faedo–Galerkin scheme based on the eigenfunctions of the Laplace operator with homogeneous Neumann boundary conditions (in which case, e.g., the Laplacian of the components of the discrete solution could actually be used as test functions).

Without loss of generality, we may assume that $\mathopen \Vert h\mathclose \Vert _{L^{2}(Q)}$ is small enough. In particular, we owe to the estimates proved for the solutions to the nonlinear problem corresponding to both u and $u+h$. For convenience, let us set

$$\begin{aligned} \psi := \phi ^h - \phi - \xi , \quad \sigma := \mu ^h - \mu - \eta , \quad \omega := w^h - w - \zeta , \end{aligned}$$

with $(\phi ^h, \mu ^h, w^h):=\mathcal{S}(u + h)$, $(\phi , \mu , w):=\mathcal{S}(u)$, and where $(\xi , \eta , \zeta )$ is the unique solution to (4.3)–(4.7) associated with h. Due to the previous results, we already know that $(\psi , \sigma , \omega ) \in \mathcal{X} \hookrightarrow \mathcal{Z}$ and that, by difference, it yields a weak solution to the system

$$\begin{aligned}&\partial _t\psi - \Delta \sigma + \gamma \psi = 0 \quad{} & {} \hbox { in}\ Q, \end{aligned}$$

(4.18)

$$\begin{aligned}&\sigma = - \Delta \psi + [F'(\phi ^h) - F'(\phi ) - F''(\phi )\xi ] - b \partial _t\omega \quad{} & {} \hbox { in}\ Q, \end{aligned}$$

(4.19)

$$\begin{aligned}&\partial _t^2\omega - \Delta ({\kappa _1}\partial _t\omega + {\kappa _2}\omega ) + \lambda \partial _t\psi = 0 \quad{} & {} \hbox { in}\ Q, \end{aligned}$$

(4.20)

$$\begin{aligned}&\partial _{{\textbf {n}}}\psi = \partial _{{\textbf {n}}}\sigma = \partial _{{\textbf {n}}}({\kappa _1}\partial _t\omega + {\kappa _2}\omega ) = 0 \quad{} & {} \hbox { on}\ \Sigma , \end{aligned}$$

(4.21)

$$\begin{aligned}&\psi (0) = \omega (0) = \partial _t\omega (0) = 0 \quad{} & {} \text { in } \Omega . \end{aligned}$$

(4.22)

Besides, with the above notation, (4.17) amounts to show that

$$\begin{aligned} \mathopen \Vert (\psi , \sigma , \omega )\mathclose \Vert _\mathcal{Z} = o (\mathopen \Vert h\mathclose \Vert _{L^2(Q)}) \quad \hbox { as}\ \mathopen \Vert h\mathclose \Vert _{L^2(Q)} \rightarrow 0. \end{aligned}$$

(4.23)

Moreover, Theorems 2.1 and 2.7 entail that

$$\begin{aligned}&\mathopen \Vert \phi ^h\mathclose \Vert _{H^{1}(0,T;V) \cap L^{\infty }(0,T;W^{2,6}(\Omega )) } +\mathopen \Vert \mu ^h\mathclose \Vert _{L^{\infty }(0,T;V)}\nonumber \\&\qquad +\mathopen \Vert w^h\mathclose \Vert _{H^{2}(0,T;H) \cap W^{1,\infty }(0,T;V)} \le K_1, \end{aligned}$$

(4.24)

as well as

$$\begin{aligned}&\mathopen \Vert \phi ^h - \phi \mathclose \Vert _{H^{1}(0,T;{V^*})\cap L^{\infty }(0,T;V)\cap L^{2}(0,T;W)} + \mathopen \Vert \mu ^h-\mu \mathclose \Vert _{L^{2}(0,T;V)} \nonumber \\ {}&\quad + \mathopen \Vert w^h - w \mathclose \Vert _{H^{2}(0,T;{V^*})\cap W^{1,\infty }(0,T;H) \cap H^{1}(0,T;V)} \le K_6 \mathopen \Vert h\mathclose \Vert _{L^{2}(0,T;H)}. \end{aligned}$$

(4.25)

Actually, for the logarithmic potential in the two-dimensional setting, we also have a stronger version of (4.24) arising as a consequence of Theorem 2.4.

Before entering the details, we recall that Taylor’s formula yields that

$$\begin{aligned} F'(\phi ^h) - F'(\phi ) - F''(\phi )\xi&= F''(\phi ) \psi + R^h \,(\phi ^h-\phi )^2, \end{aligned}$$

(4.26)

where the remainder $R^h$ is given by

$$\begin{aligned} R^h= \int _0^1 F^{(3)}\big ( \phi +s (\phi ^h-\phi ) \big ) (1-s)\,ds\,. \end{aligned}$$

Due to (2.13), we have that

$$\begin{aligned} \mathopen \Vert R^h\mathclose \Vert _{L^\infty (Q)} \le C. \end{aligned}$$

(4.27)

First Estimate

We notice that $\psi$ has zero mean value as can be easily checked by testing (4.18) by $1/|\Omega |$ and using (4.22). Hence, we can test (4.18) by $\mathcal N\psi$ and (4.19) by $-\psi$. Moreover, we integrate (4.20) in time and test the resulting equation by $\frac{b}{\lambda }\partial _t\omega$. Finally, we sum up and add the same term $\frac{{\kappa _1}b}{2\lambda } \frac{d}{dt}\mathopen \Vert \omega \mathclose \Vert ^2=\frac{{\kappa _1}b}{2\lambda }\int _\Omega \omega \,\partial _t\omega$ to both sides. We obtain that

$$\begin{aligned}&\frac{1}{2} \, \frac{d}{dt} \, \mathopen \Vert \psi \mathclose \Vert ^2_{*} + \gamma \mathopen \Vert \psi \mathclose \Vert ^2_* + \mathopen \Vert \nabla \psi \mathclose \Vert ^2 + \frac{b}{\lambda } \, \mathopen \Vert \partial _t\omega \mathclose \Vert ^2 + \frac{{\kappa _1}b}{2\lambda } \, \frac{d}{dt} \, \mathopen \Vert \omega \mathclose \Vert ^2_V \\ {}&\quad = \int _\Omega [F'(\phi ^h) - F'(\phi ) - F''(\phi )\xi ] \psi {{}-{}} \frac{ b{\kappa _2}}{\lambda }\int _\Omega {\nabla ({\varvec{1}*} \omega ) \cdot \nabla \partial _t\omega } +\frac{{\kappa _1}b}{2\lambda } \int _\Omega \omega \,\partial _t\omega . \end{aligned}$$

Since we aim at applying the Gronwall lemma, we should integrate over (0, t) with respect to time. However, for brevity, we just estimate the first two terms of the right-hand side obtained by integration (the last one can be trivially handled by the Young inequality) and avoid writing the integration variable s in the integrals over (0, t). The first one can be controlled by using the Hölder and Young inequalities, (4.25), the continuous embedding $V\hookrightarrow L^{4}(\Omega )$, (4.26), (4.27), and the compactness inequality (2.17) as follows:

$$\begin{aligned}&\int _{Q_t}[F'(\phi ^h) - F'(\phi ) - F''(\phi )\xi ] \psi = \int _{Q_t}[F''(\phi ) \psi + R^h \,(\phi ^h-\phi )^2] \psi \\&\quad \le C \int _0^t\mathopen \Vert \psi \mathclose \Vert ^2 \, ds + C \int _0^t\mathopen \Vert \phi ^h-\phi \mathclose \Vert ^2_4\mathopen \Vert \psi \mathclose \Vert \, ds \le C \int _0^t\mathopen \Vert \psi \mathclose \Vert ^2 \, ds + C \int _0^t\mathopen \Vert \phi ^h-\phi \mathclose \Vert _V^4 \, ds \\ {}&\quad \le C \int _0^t\mathopen \Vert \psi \mathclose \Vert ^2 \, ds + {{}C \hspace{1.0pt}T \mathopen \Vert h\mathclose \Vert _{L^2(Q)}^4{}} {{}\le {} } \frac{1}{2} \int _0^t\mathopen \Vert \nabla \psi \mathclose \Vert ^2 \, ds + C \int _0^t\mathopen \Vert \psi \mathclose \Vert _*^2 \, ds + {{} C \mathopen \Vert h\mathclose \Vert _{L^2(Q)}^4{}}. \end{aligned}$$

As for the second term, we integrate by parts both in space and time. By also accounting for the Young inequality, we find that

$$\begin{aligned}&{{} - \frac{b{\kappa _2}}{\lambda }\int _{Q_t}\nabla ({\varvec{1}*}\omega ) \cdot \nabla \partial _t\omega = - \frac{b{\kappa _2}}{\lambda }\int _\Omega \nabla ({\varvec{1}*}\omega )(t) \cdot \nabla \omega (t) + \frac{b{\kappa _2}}{\lambda }\int _{Q_t}|\nabla \omega |^2} \\&\quad \le \frac{{\kappa _1}b}{4\lambda } \int _\Omega |\nabla \omega (t)|^2 + C \int _\Omega \Bigl | \int _0^t\nabla \omega \, ds \Bigr |^2 + C \int _{Q_t}|\nabla \omega |^2 \\&\quad \le \frac{{\kappa _1}b}{4\lambda } \int _\Omega |\nabla \omega (t)|^2 + C \int _{Q_t}|\nabla \omega |^2. \end{aligned}$$

Thus, we can apply the Gronwall lemma and conclude that

$$\begin{aligned} \mathopen \Vert \psi \mathclose \Vert _{L^{\infty }(0,T;{V^*})\cap L^{2}(0,T;V)} + \mathopen \Vert \omega \mathclose \Vert _{H^{1}(0,T;H) \cap L^{\infty }(0,T;V)} \le C \mathopen \Vert h\mathclose \Vert ^2_{L^2(Q)}. \end{aligned}$$

(4.28)

Second Estimate

We test (4.18) by $\psi$, (4.19) by $\Delta \psi$, and add the resulting equalities to find that

$$\begin{aligned}&\frac{1}{2} \, \frac{d}{dt} \, \mathopen \Vert \psi \mathclose \Vert ^2 + \mathopen \Vert \Delta \psi \mathclose \Vert ^2 + \gamma \mathopen \Vert \psi \mathclose \Vert ^2 \\&\quad = \int _\Omega [F'(\phi ^h) - F'(\phi ) - F''(\phi )\xi ] \Delta \psi {\,-\,} b \int _\Omega \partial _t\omega \Delta \psi . \end{aligned}$$

As above, we only estimate the right-hand side of the equality obtained by integrating over (0, t). By also accounting for the previous estimate, we have that

$$\begin{aligned}&\int _{Q_t}[F'(\phi ^h) - F'(\phi ) - F''(\phi )\xi ] \Delta \psi {\,-\,} b \int _{Q_t}\partial _t\omega \Delta \psi \\ {}&\quad \le \int _{Q_t}|F''(\phi )| \, |\psi | \, |\Delta \psi | + \int _{Q_t}|R^h| \, |\phi ^h-\phi |^2 \, |\Delta \psi | + C \int _0^t\mathopen \Vert \partial _t\omega \mathclose \Vert \, \mathopen \Vert \Delta \psi \mathclose \Vert \, ds \\ {}&\quad \le \frac{1}{2} \int _0^t\mathopen \Vert \Delta \psi \mathclose \Vert ^2 \, ds + C \int _0^t(\mathopen \Vert \psi \mathclose \Vert ^2 + \mathopen \Vert \partial _t\omega \mathclose \Vert ^2) \, ds + C \mathopen \Vert h\mathclose \Vert _{L^{2}(Q)}^4 \\ {}&\quad \le \frac{1}{2} \int _0^t\mathopen \Vert \Delta \psi \mathclose \Vert ^2 \, ds + C \mathopen \Vert h\mathclose \Vert _{L^{2}(Q)}^4. \end{aligned}$$

Thus, owing also to the elliptic regularity theory, we conclude that

$$\begin{aligned} \mathopen \Vert \psi \mathclose \Vert _{{L^{\infty }(0,T;H)\cap {}}L^{2}(0,T;W)} \le C \mathopen \Vert h\mathclose \Vert ^2_{L^2(Q)}. \end{aligned}$$

(4.29)

Third Estimate

Next, we test (4.19) by $\sigma$ and, arguing as above, we obtain that

$$\begin{aligned} \mathopen \Vert \sigma \mathclose \Vert _{L^{2}(0,T;H) } \le C \mathopen \Vert h\mathclose \Vert ^2_{L^2(Q)}. \end{aligned}$$

(4.30)

Fourth Estimate

We can now test (4.18) by an arbitrary function $v\in L^{2}(0,T;W)$ and, in view of (4.29) and (4.30), easily infer that

$$\begin{aligned} \Bigl | \int _0^T \langle \partial _t\psi , v \rangle _W \Bigl |&\le \mathopen \Vert \sigma \mathclose \Vert _{L^{2}(0,T;H) }\mathopen \Vert \Delta v\mathclose \Vert _{L^{2}(0,T;H) } + \gamma \mathopen \Vert \psi \mathclose \Vert _{L^{2}(0,T;H) }\mathopen \Vert v\mathclose \Vert _{L^{2}(0,T;H) } \\&\le C \mathopen \Vert h\mathclose \Vert ^2_{L^2(Q)}\mathopen \Vert v\mathclose \Vert _{L^{2}(0,T;W)} \quad \hbox { for all } \, v\in L^{2}(0,T;W). \end{aligned}$$

Hence, $\mathopen \Vert \partial _t\psi \mathclose \Vert _{L^{2}(0,T;W^*)}$ is uniformly bounded by a quantity proportional to $\mathopen \Vert h\mathclose \Vert ^2_{L^2(Q)}$, so that from (4.29) and an interpolation argument we recover that

$$\begin{aligned} \mathopen \Vert \psi \mathclose \Vert _{{H^{1}(0,T;W^*)\cap C^{0}([0,T];H)\cap {}}L^{2}(0,T;W)} \le C \mathopen \Vert h\mathclose \Vert ^2_{L^2(Q)}. \end{aligned}$$

(4.31)

Fifth Estimate

Next, we rewrite equation (4.20) in terms of the auxiliary variable $\tau : = {\kappa _1}\partial _t\omega + {\kappa _2}\omega + {\kappa _1}\lambda \psi$ to obtain

$$\begin{aligned}&\frac{1}{{\kappa _1}} \partial _t\tau - \Delta \tau = \frac{{\kappa _2}}{{\kappa _1}} \partial _t\omega - {\kappa _1}\lambda \Delta \psi {.} \end{aligned}$$

Thanks to (4.21)–(4.22), it turns out that $\tau$ satisfies Neumann homogeneous boundary conditions and null initial conditions. Then, by virtue of parabolic regularity results along with (4.28) and (4.31), we have that

$$\begin{aligned}&\mathopen \Vert \tau \mathclose \Vert _{H^{1}(0,T;H)\cap C^{0}([0,T];V)\cap L^{2}(0,T;W)}\\&\quad \le C \Bigl \Vert \frac{{\kappa _2}}{{\kappa _1}} \partial _t\omega - {\kappa _1}\lambda \Delta \psi \Bigr \Vert _{L^{2}(0,T;H)}\,\le C \mathopen \Vert h\mathclose \Vert ^2_{L^2(Q)}\,. \end{aligned}$$

Therefore, observing that ${\kappa _1}\partial _t\omega + {\kappa _2}\omega = \tau - {\kappa _1}\lambda \psi$, it follows that both $\omega$ and $\partial _t\omega$ satisfy (at least) the same estimate as (4.31), which yields

$$\begin{aligned} \mathopen \Vert \omega \mathclose \Vert _{{H^{2}(0,T;W^*)\cap C^{1}([0,T];H)\cap {}}H^{1}(0,T;W)} \le C \mathopen \Vert h\mathclose \Vert ^2_{L^2(Q)}. \end{aligned}$$

(4.32)

This concludes the proof since the estimates (4.30)–(4.32) directly lead to (4.23). $\square$

4.3 Adjoint System and First-Order Optimality Conditions

As a final step, we now introduce a suitable adjoint system to (1.1)–(1.5) in order to recover a more practical form of the optimality conditions for (P). Let $u \in \mathcal{U}_\textrm{ad}$ be given with its associated state $(\phi ,\mu ,w)$. In a strong formulation, the adjoint system is expressed by the backward-in-time parabolic system

$$\begin{aligned}&- \partial _tp - \Delta q + \gamma p + F''(\phi ) q - \lambda \partial _tr = {\alpha _1} (\phi - \phi _Q) \quad{} & {} \hbox { in}\ Q, \end{aligned}$$

(4.33)

$$\begin{aligned}&q = - \Delta p \quad{} & {} \hbox { in}\ Q, \\ \nonumber&-\partial _tr - \Delta ({\kappa _1}r - {\kappa _2}({\textbf {1}} \circledast r)) - b q \end{aligned}$$

(4.34)

$$\begin{aligned}&\quad = {\alpha _3} ({\textbf {1}} \circledast (w- w_Q)) + {\alpha _4} (w(T) - w_\Omega ) + {\alpha _5} (\partial _tw - w_Q') \quad{} & {} \hbox { in}\ Q, \end{aligned}$$

(4.35)

$$\begin{aligned}&\partial _{{\textbf {n}}}p = \partial _{{\textbf {n}}}q = \partial _{{\textbf {n}}}({\kappa _1}r - {\kappa _2}({\textbf {1}} \circledast r) ) = 0 \quad{} & {} \hbox { on}\ \Sigma , \end{aligned}$$

(4.36)

$$\begin{aligned}&p(T) = {\alpha _2} (\phi (T)- \phi _\Omega ) - \lambda {\alpha _6} (\partial _tw(T)- w_\Omega ') ,{} & {} \nonumber \\ {}&r(T) = {\alpha _6} (\partial _tw(T)- w_\Omega ') \quad{} & {} \hbox { in}\ \Omega , \end{aligned}$$

(4.37)

where the convolution product $\circledast$ has been introduced in (2.2). Concerning this product, note in particular that $\partial _t(\textbf{1} \circledast r) = -r.$ Let us introduce the following shorthand for the right-hand side of (4.35),

$$\begin{aligned} f_r := {\alpha _3} ({\textbf {1}} \circledast (w- w_Q)) + {\alpha _4} (w(T) - w_\Omega ) + {\alpha _5} (\partial _tw - w_Q')\,. \end{aligned}$$

We also notice that the second term is independent of time. Due to the regularity properties in (2.7) and (C4), it holds that

$$\begin{aligned} \mathopen \Vert f_r\mathclose \Vert _{L^{2}(0,T;H)} \le C (\mathopen \Vert w\mathclose \Vert _{H^{2}(0,T;H) \cap W^{1,\infty }(0,T;V) }+1) \le C. \end{aligned}$$

(4.38)

Let us remark that, in the adjoint system, the variable r does not correspond to the freezing index w, but to its time derivative $\partial _tw$ (the temperature), as is readily seen by checking the computations to follow in (4.42). Consequently, equation (4.35) is of first order in time instead of second order: actually, (4.35) may be rewritten as of second order equation in time in terms of the time-integrated variable $\textbf{1} \circledast r$, since it holds that $- \partial _tr = \partial _t^2(\textbf{1} \circledast r)$.

Theorem 4.5

(Well-posedness of the adjoint system) Let the assumptions (A1)–(A3) and (C1)–(C4) hold, and let $u \in \mathcal{U}_\textrm{ad}$ with associated state $(\phi ,\mu ,w){=\mathcal{S}(u)}$ be given. Then, the adjoint system (4.33)–(4.37) admits a unique weak solution (p, q, r) such that

$$\begin{aligned} p&\in H^{1}(0,T;{V^*})\cap L^{\infty }(0,T;V) \cap L^{2}(0,T;W), \\ q&\in L^{2}(0,T;V), \\ r&\in H^{1}(0,T;H) \cap L^{\infty }(0,T;V). \end{aligned}$$

Remark 4.6

Similarly as in Remark 4.3, we should here speak of a proper variational formulation. For instance, (4.33) with the homogeneous Neumann boundary condition for q has to be read as

$$\begin{aligned}{} & {} - \mathopen \langle \partial _tp, v \mathclose \rangle + \int _\Omega \nabla q \cdot \nabla v + \int _\Omega \bigl ( \gamma p + F''(\phi ) q - \lambda \partial _tr \bigr ) v \nonumber \\{} & {} \quad = \int _\Omega {\alpha _1}(\phi -\phi _Q) v \quad \hbox {a.e. in (0,T)}, \hbox {for every }v\in V. \nonumber \end{aligned}$$

Proof of Theorem 4.5

Again, for existence, we proceed formally, but let us underline that the following computations can however be reproduced in a rigorous framework.

First Estimate

We test (4.33) by $p+q$, (4.34) by $\partial _tp + (K_5 +1) q$, where $K_5$ is the positive constant arising from (2.13), (4.35) by $- \frac{\lambda }{b} \partial _tr$ and add the resulting identities to each other. Then, after cancellations, we infer that

$$\begin{aligned}&- \frac{1}{2} \, \frac{d}{dt} \, \mathopen \Vert p\mathclose \Vert _V^2 + (K_5 +1) \mathopen \Vert q\mathclose \Vert ^2 + \mathopen \Vert \nabla q\mathclose \Vert ^2 + \gamma \mathopen \Vert p\mathclose \Vert ^2 + \frac{\lambda }{b} \, \mathopen \Vert \partial _tr\mathclose \Vert ^2 \nonumber \\ {}&\quad {{}-{}} \frac{{\kappa _1}\lambda }{{2}b} \, \frac{d}{dt} \, \mathopen \Vert \nabla r\mathclose \Vert ^2 + {\frac{{\kappa _2}\lambda }{b} \int _\Omega \nabla (\textbf{1}\circledast r) \cdot \nabla \partial _tr} \nonumber \\ {}&= - \gamma \int _\Omega p q - \int _\Omega F''(\phi )q (p+q) + \lambda \int _\Omega \partial _tr \, p + {\alpha _1} \int _\Omega (\phi - \phi _Q) (p+q) \nonumber \\ {}&\qquad {+ K_5 \int _\Omega \nabla p \cdot \nabla q} - \frac{\lambda }{b} \int _\Omega f_r \, \partial _tr. \end{aligned}$$

(4.39)

Now, recalling (2.13), the second term on the right-hand side can be bounded from above as

$$\begin{aligned}&- \int _\Omega F''(\phi )q (p+q) \le \mathopen \Vert F''(\phi )\mathclose \Vert _\infty \mathopen \Vert p\mathclose \Vert \mathopen \Vert q\mathclose \Vert + \mathopen \Vert F''(\phi )\mathclose \Vert _\infty \mathopen \Vert q\mathclose \Vert ^2 \\&\quad \le \Big (\frac{1}{2} + K_5 \Big ) \mathopen \Vert q\mathclose \Vert ^2 + C \mathopen \Vert p\mathclose \Vert ^2, \end{aligned}$$

and the first term appearing on the right can be absorbed by the corresponding contribution appearing on the left of (4.39). By the Young inequality, we see that the remaining terms on the right-hand side are bounded above by

$$\begin{aligned} {\frac{\lambda }{2b}} \, \mathopen \Vert \partial _tr\mathclose \Vert ^2 +\frac{1}{2} \, \mathopen \Vert \nabla q\mathclose \Vert ^2 +\frac{1}{4} \, \mathopen \Vert q\mathclose \Vert ^2 + C (\mathopen \Vert p\mathclose \Vert ^2_V+ 1 )\,, \end{aligned}$$

thanks to (2.8) and the estimate (4.38) of $f_r$. Next, we integrate over (t, T), for any $t\in (0,T)$, and notice that (C4) provide uniform bounds for $\mathopen \Vert p(T)\mathclose \Vert _V^2$ and $\mathopen \Vert r(T)\mathclose \Vert _V^2$ using their explicit form given by (4.37). Moreover, we treat the integral deriving from the last term on the left-hand side of (4.39) as follows. With the notation $Q^t:=\Omega \times (t,T)$, we have that

$$\begin{aligned} \frac{{\kappa _2}\lambda }{b} \int _{Q^t}\nabla (\textbf{1}\circledast r) \cdot \nabla \partial _tr = - \frac{{\kappa _2}\lambda }{b} \int _\Omega \nabla (\textbf{1}\circledast r)(t) \cdot \nabla r(t) + \frac{{\kappa _2}\lambda }{b} \int _{Q^t}{|\nabla r|^2} \,. \nonumber \end{aligned}$$

On the other hand, we also have that

$$\begin{aligned}{} & {} \Bigl | -\frac{{\kappa _2}\lambda }{b} \int _\Omega \nabla (\textbf{1}\circledast r)(t) \cdot \nabla r(t) \Bigr | \le \frac{{\kappa _1}\lambda }{{4}b} \int _\Omega |\nabla r(t)|^2 + C \int _\Omega |\nabla (\textbf{1}\circledast r)(t)|^2 \nonumber \\{} & {} \quad \le \frac{{\kappa _1}\lambda }{{4}b} \int _\Omega |\nabla r(t)|^2 + C \int _{Q^t}|\nabla r|^2 . \nonumber \end{aligned}$$

Thus, from the (backward) Gronwall lemma and the obvious subsequent inequality

$$\begin{aligned} \mathopen \Vert r(t)\mathclose \Vert \le C \, \mathopen \Vert \partial _tr\mathclose \Vert _{L^{2}(Q)} + C \quad \hbox {for every }t\in [0,T], \nonumber \end{aligned}$$

we infer that

$$\begin{aligned} \mathopen \Vert p\mathclose \Vert _{L^{\infty }(0,T;V)} + \mathopen \Vert q\mathclose \Vert _{L^{2}(0,T;V)} + \mathopen \Vert r\mathclose \Vert _{H^{1}(0,T;H) \cap L^{\infty }(0,T;V)} \le C. \end{aligned}$$

Second Estimate

Elliptic regularity theory applied to (4.34) then produces

$$\begin{aligned} \mathopen \Vert p\mathclose \Vert _{L^{2}(0,T;W)} \le C. \end{aligned}$$

Third Estimate

Finally, it is a standard matter to infer from a comparison argument in (4.33), along with the above estimates, that

$$\begin{aligned} \mathopen \Vert \partial _tp\mathclose \Vert _{L^{2}(0,T;{V^*})} \le C. \end{aligned}$$

This concludes the (formal) proof of the existence of a solution. By performing the same estimates in the case of vanishing right-hand side and final data, we see that the solution must vanish, whence uniqueness in the general case follows by linearity. $\square$

Finally, using the adjoint variables, we present the first-order necessary conditions for an optimal control $u^*$ solving (P). In the following, (p, q, r) and $(\xi ,\eta ,\zeta )$ denote the solutions of the respective adjoint problem and linearized problem, but written in terms of the associated state $(\phi ^*,\mu ^*,w^*) =\mathcal{S}(u^*)$ that replaces $(\phi ,\mu ,w)$ in systems (4.3)–(4.7) and (4.33)–(4.37).

Theorem 4.7

(First-order optimality conditions) Suppose that (A1)–(A3) and (C1)–(C4) hold. Let $u^*$ be an optimal control for (P) with associated state $(\phi ^*,\mu ^*,w^*)=\mathcal{S}(u^*)$ and adjoint (p, q, r). Then, it necessarily fulfills the variational inequality

$$\begin{aligned} \int _Q (r + \nu u^*)(u-u^*) \ge 0 \quad {\hbox { for every}}~ u\in \mathcal{U}_\textrm{ad}. \end{aligned}$$

(4.40)

Proof of Theorem 4.7

From standard results of convex analysis, the first-order necessary optimality condition for every optimal control $u^*$ of (P) is expressed in the abstract form as

$$\begin{aligned} \mathopen \langle D{\mathcal{J}}_\textrm{red}(u^*), u- u^*\mathclose \rangle \ge 0 \quad \forall u \in \mathcal{U}_\textrm{ad}, \end{aligned}$$

where $D{\mathcal{J}}_\textrm{red}$ denotes the Fréchet derivative of the reduced cost functional $\mathcal{J}$. As a consequence of the Fréchet differentiability of the control-to-state operator established in Theorem 4.4, and the form of the cost functional $\mathcal{J}$ in (1.6), this entails that any optimal control $u^*$ necessarily fulfills

$$\begin{aligned}&{{\alpha _1}}\int _Q (\phi ^* - \phi _Q)\xi + {{\alpha _2}}\int _\Omega (\phi ^*(T) - \phi _\Omega )\xi (T) + {{\alpha _3}}\int _Q (w^* - w_Q)\zeta \nonumber \\ {}&\quad + {{\alpha _4}}\int _\Omega (w^*(T) - w_\Omega )\zeta (T) + {{\alpha _5}} \int _Q (\partial _tw^* - w'_Q) \partial _t\zeta \nonumber \\ {}&\quad + {{\alpha _6}} \int _\Omega (\partial _tw^*(T) - w'_\Omega ) \partial _t\zeta (T) + \nu \int _Q u^* (u-u^*) \ge 0 \quad \forall u \in \mathcal{U}_\textrm{ad}, \end{aligned}$$

(4.41)

where $(\xi ,\eta ,\zeta )$ is the unique solution to the linearized system as obtained from Theorem 4.2 associated with $(\phi ,\mu ,w)=(\phi ^*,\mu ^*,w^*)=\mathcal{S}(u^*)$ and $h=u-u^*$. Unfortunately, the above formulation is not very useful in numerical applications as it depends on the linearized variables. However, with the help of the adjoint variables, playing the role of Lagrangian multipliers, the above variational inequality can be simplified. In this direction, we test (4.3) by p, (4.4) by q, (4.5) by r, and add the resulting equalities and integrate over time and by parts. More precisely, we should consider the variational formulations of the linearized and adjoint systems mentioned in Remarks 4.3 and 4.6 in order to avoid writing some Laplacian that does not exist in the usual sense, and we should also owe to (well-known) generalized versions of the integration by parts in time. However, for shortness, we proceed as said above and obtain

$$\begin{aligned} 0 =&\int _Q [\partial _t\xi - \Delta \eta + \gamma \xi ] p + \int _Q [-\eta - \Delta \xi + F''(\phi )\xi - b \partial _t\zeta ] q \nonumber \\ {}&\quad + \int _Q [\partial _t^2\zeta - \Delta ({\kappa _1}\partial _t\zeta + {\kappa _2}\zeta ) + \lambda \partial _t\xi - h] r \nonumber \\ {}=&\int _Q \xi [- \partial _tp - \Delta q + \gamma p + F''(\phi ) q - \lambda \partial _tr ] \nonumber \\ {}&\quad + \int _Q \eta [-\Delta p -q] + \int _Q \partial _t\zeta [- \partial _tr - \Delta ({\kappa _1}r - {\kappa _2}(\textbf{1} \circledast r)) - b q ] \nonumber \\ {}&\quad + \int _\Omega [\xi (T) p(T) + \partial _t\zeta (T) r(T) + \lambda \xi (T) r(T)] - \int _Q h r. \end{aligned}$$

(4.42)

Using the adjoint system (4.33)–(4.37) and the associated final conditions, and integrating by parts as well, we infer that

$$\begin{aligned}&\int _Qr (u-u^*) = \int _Qhr = \int _Q\xi \, {\alpha _1} (\phi ^*-\phi _Q) + \int _Q\partial _t\zeta \, [ {\alpha _3} ({\textbf {1}}\circledast (w^*-w_Q) \nonumber \\&\qquad + {\alpha _4} (w^*(T)-w_\Omega ) + {\alpha _5} (\partial _tw^*-w'_Q) ] \nonumber \\&\qquad + \int _\Omega \xi (T) \, [ {\alpha _2} (\phi ^*(T)-\phi _\Omega ) - \lambda {\alpha _6} (\partial _tw^*(T)-w'_\Omega ) ] \nonumber \\&\qquad + \int _\Omega \partial _t\zeta (T) \, {\alpha _6} (\partial _tw^*(T)-w'_\Omega ) + \int _\Omega \lambda \xi (T) \, {\alpha _6} (\partial _tw^*(T)-w'_\Omega ) \nonumber \\&\quad = {{\alpha _1}}\int _Q (\phi ^* - \phi _Q)\xi {+ {{\alpha _2}}\int _\Omega (\phi ^*(T) - \phi _\Omega )\xi (T)} + {{\alpha _3}}\int _Q (w^* - w_Q)\zeta \\ {}&\qquad {+ {{\alpha _4}}\int _\Omega (w^*(T) - w_\Omega )\zeta (T)} + {{\alpha _5}} \int _Q (\partial _tw^* - w'_Q) \partial _t\zeta \\&\qquad + {{\alpha _6}} \int _\Omega (\partial _tw^*(T) - w'_\Omega ) \partial _t\zeta (T), \end{aligned}$$

so that (4.41) entails (4.40), and this concludes the proof. $\square$

Corollary 4.8

Suppose the assumptions of Theorem 4.7 are fulfilled, and let $u^*$ be an optimal control with associated state $(\phi ^*,\mu ^*,w^*)=\mathcal{S}(u^*)$ and adjoint (p, q, r). Then, whenever $\nu >0$, $u^*$ is the $L^2$-orthogonal projection of $- \frac{1}{\nu }r$ onto $\mathcal{U}_\textrm{ad}$. Besides, we have the pointwise characterization of the optimal control $u^*$ as

$$\begin{aligned} u^*(x,t)=\max \Big \{ u_\textrm{min}(x,t), \min \{u_\textrm{max}(x,t),-\frac{1}{\nu } \, r(x,t)\} \Big \} \quad \hbox {for } a.e. \,(x,t) \in Q. \end{aligned}$$

Remark 4.9

It is worth pointing out that Theorem 4.7 and Corollary 4.8 even hold for any locally optimal control $u^*$. Indeed, we note that, as the control-to-state map is nonlinear, the optimization problem turns out to be nonconvex. Therefore, numerical methods for the discretization of the system will in general only be able to detect locally optimal controls.