THERMAL CONTROL OF THE SOUZA-AURICCHIO MODEL FOR SHAPE MEMORY ALLOYS

We address the thermal control of the quasi-static evolution of a polycrystalline shape memory alloy specimen. The thermomechanical evolution of the body is described by means of the phenomenological SouzaAuricchio model [6, 53]. By assuming to be able to control the temperature of the body in time we determine the corresponding quasi-static evolution in the energetic sense. By recovering in this context a result by Rindler [49, 50] we prove the existence of optimal controls for a suitably large class of cost functionals and comment on their possible approximation. Dedicated to Professor Michel Frémond on the occasion of his 70th birthday.


Introduction
Shape memory alloys (SMAs) are examples of active materials: comparably large strains can be induced (activated) by means of external mechanical, thermal, or magnetic stimuli. At suitably high temperatures SMAs completely recover strains as large as 8% during loading-unloading cycles (note that conventional steels plasticize around 1% strains). This is the so-called super-elastic SMA behavior. At lower temperatures permanent deformations remain under unloading. Still, the specimen can be forced to recover its original shape by heating: this is the so called shape-memory effect. Finally, some specific SMAs are ferro-magnetic: completely recoverable strains can be induced by the action of an external magnetic field. This amazing macroscopic behavior is the effect of an abrupt and diffusionless solid-solid phase transformation between different crystallographic configurations (phases): the austenite (mostly cubic, predominant at high temperature and low stresses) and the martensites (lower symmetry variants, favored at low temperature or high stresses) [21,22,23].
The remarkable thermo-magneto-mechanical behavior of SMAs is at the basis of a variety of innovative applications ranging from sensors and actuators, to a SMA body is relatively thin in at least one direction and undergoes relatively lowfrequency loading-unloading cycles. In this case the heat produced via deformation and phase-change can be assumed to be (almost) instantaneously dissipated in the environment (in contrast, note that high-frequency mechanical probing on wires reveals that the heat production due to the dissipative phase-transformation is not at all negligible [46]).
A second Ansatz of our analysis is that the temperature is space-homogeneous in the bulk. We concentrate on this setting as we assume that the temperature of the specimen is here the control variable and we see little motivation (if any) in considering a control device able to prescribe a non-homogeneous temperature in a metallic bulk. On the other hand, in case of a fully coupled thermomechanical system (a case which is not under consideration here as the temperature is given) an inhomogeneous temperature evolution is generically to be expected, even in the simple one-dimensional geometry of a wire. Although we stick here to the homogeneous situation, we have however to stress that the non-homogeneous temperature case would also be amenable from the mathematical viewpoint under minor modifications.
The novelty of our contribution is twofold. At first, we establish a novel existence result for the so-called state problem, namely, given the temperature, to determine the quasi-static mechanical evolution of the SMA specimen. As we comment below, our existence result is somewhat stronger than the available ones from [37,40,44]. In particular, in contrast to the above-mentioned papers, the original nonregularized formulation of the Souza-Auricchio model can be directly accommodated in our setting. Moreover, less regularity is required on the given temperature. This last aspect turns out to be crucial with respect to optimal control as we aim at considering the largest possible set of control temperatures.
Our second novel point is that of proving the existence of an optimal control for a suitably large class of cost functionals depending on both mechanics and temperature. The applicative interest in this perspective resides in the possibility of efficiently activating SMA devices by controlling the temperature of the specimen via Joule's heating. This is one of the basic technological activation mechanisms currently exploited in real applications [16]. This is to our knowledge the first control result in the specific setting of the Souza-Auricchio model and, more generally, of (thermo-)generalized plasticity. Our argument is basically the concrete application of the abstract theory developed by Rindler [49] on existence of optimal controls in the frame of rate-independent systems. Note that, as the state problem is nonsmooth and, in particular, not even known to admit a unique solutions, the quest for necessary optimality conditions seems presently out of reach. We collect some comments on possible future developments of this investigation in Subsection 4.1 below.
This is the plan of the paper. In Section 2 we recall the basic features of the Souza-Auricchio model, collect our assumptions, and state the main results. Section 3 reports on the analysis of the state problem whereas the existence proof for optimal controls is developed in Section 4 together with some comments for possible future research.

Main results
2.1. Mechanical problem formulation. We shall start by recalling some ingredients of the Souza-Auricchio model. The Reader is however referred to the original contributions in [6,7,8,53] for further motivation, detail, numerical illustration, and validation.
We denote by R d×d sym (d = 2, 3) the space of symmetric 2-tensors in R d endowed with the usual scalar (contraction) product a:b = tr(ab) := a ij b ij (summation convention) and the corresponding norm |a| = √ a:a. The space R d×d sym is orthogonally decomposed as R d×d sym = R d×d dev ⊕ R1 2 , where R1 2 is the subspace spanned by the identity 2-tensor 1 2 , while R d×d dev is the subspace of all deviatoric symmetric tensors. The non-empty, connected, bounded, and open subset Ω ⊂ R d represents the reference configuration of the body. Given the displacement u : Ω → R d from the reference configuration with u ∈ H 1 (Ω; R d ) we denote the symmetric gradient (∇u + ∇u )/2 of u by ε(u). In particular, throughout the paper we will make tacit use of the well-known Korn inequality for any u ∈ H 1 (Ω; R d ) and some constant c Korn > 0 depending just on Ω.
Within the small-strain regime, the linearized strain ε(u) is additively decomposed into an elastic part ε el and an inelastic (or transformation) part z as The elastic part of the linearized strain is directly related to the stress exerted on the body. In particular, we assume the elastic material response σ = Cε el where σ stands for the stress and C is the positive-definite and symmetric elasticity 4-tensor.
The internal variable z ∈ R d×d dev will be regarded as the tensorial descriptor of the internal phase distribution in the material. The deviatoric nature of z reflects the fact that martensitic transitions are assumed to be volume preserving. The norm |z| represents a measure of product phase (detwinned martensite) vs. parent phase (austenite and twinned martensite). The direction z/|z| instead describes the predominant orientation of the product phase.
In order to better illustrate the role of the internal variable z let us momentarily leave our polycrystalline framework and consider the single-crystal case of a cubictetragonal martensitic system (as such of NiMgGa, FePd, and FePt alloys, among many others). By fixing a reference frame (e 1 , e 2 , e 3 ) aligned with the edges of the cubic austenitic crystal, the corresponding inelastic strain tensors z i relative to the i-th tetragonal martensitic phase (i = 1, 2, 3) read Here, ε L > 0 denotes the maximal strain which is obtainable via reorientation of martensitic variants. In particular, |z i | = ε L . By indicating with λ i the local phase proportion of the i-th martensitic phase (0 ≤ λ i ≤ 1 and λ 1 +λ 2 +λ 3 ≤ 1), one gets the overall inelastic strain to be of the form z = λ 1 z 1 + λ 2 z 2 + λ 3 z 3 so that |z| = ε L (λ 2 1 +λ 2 2 +λ 2 3 ) and .
In particular, the first equality above qualifies |z| as an indicator of the total proportion of (detwinned) martensitic phase in the material whereas the director z/|z| distinguishes among the different martensites as we have that We assume the boundary ∂Ω to be Lipschitz and we let Γ Dir ∪ Γ tr = ∂Ω where Γ Dir ∩ Γ tr = ∅ and H d−1 (Γ Dir ) > 0. The body will be subject to a given surface traction on the part Γ tr of the boundary. On the other hand, non-homogeneous Dirichlet conditions for the displacement will be prescribed on Γ Dir . More precisely, by letting be given, the trace of u Dir on Γ Dir plays the role of the prescribed boundary value for the displacement u. In particular, for all given times t ∈ [0, T ], the set of admissible states (u(t), z(t)) is given by Y(u Dir (t)) wherē : u =ū on Γ Dir . We assume each phase to be isotropic and described by the same elasticity tensor. This is of course a crude simplification but still does not jeopardize the overall output of the model [6,53]. In particular, we denote the elastic energy functional C : Hence, the elastic contribution to the stored energy of the material is simply given by C(ε el ) = C(ε(u)−z).
Following the original modelization from [6,53], the inelastic part of the stored energy of the material is a function of the inelastic strain z and of the temperature θ only. In particular, the inelastic energy density takes the form Here, β is a given Lipschitz continuous nonnegative function describing the temperature dependence of the inelastic response of the medium. In particular, β(θ) corresponds to the austenite-martensite transition critical stress at temperature θ > 0. The original choice β(θ) = b(θ−θ m ) + of the Souza-Auricchio model (b > 0 and θ m being a critical temperature for the martensite-austenite equilibrium in the stress-free configuration) is included in our frame. The constant c h > 0 is a classical linearized kinematic hardening coefficient and I K stands for the indicator function of the closed, convex set K := {z ∈ R d×d dev : |z| ≤ ε L }. In particular, I K (z) = 0 if z ∈ K and I K = ∞ elsewhere. In passing, one shall note that the existence and optimal control issues discussed here do not rely on the particular form of the inelastic energy and could be adapted to much more general situations. We however prefer to stick to the original modeling choice in (2) for the sake of definiteness.
A last term has to be introduced in the overall stored energy of the system in order to penalize martensite-martensite interfaces. Indeed, we include the interfacial energy term where ν > 0 is a scale parameter. In particular, 1/ν represents the overall length of martensite-martensite interfaces within some reference domain. Note that the latter integral bears the meaning of a total variation and, as such, bears also a crucial compactifying effect. The occurrence of this interfacial term however does not prevent z from possibly exhibiting jumps. This is a particularly desirable feature in connection with shape memory alloys where sharp ( often just few atomic layers thick) phase boundaries are usually observed.
By assuming the temperature of the body θ ∈ W 1,1 (0, T ) to be (spatially homogeneous and) prescribed, the stored-energy functional W(·, ·; θ(t)) : Y(u Dir (t)) → [0, +∞] for the body at time t ∈ [0, T ] will be hence given by In particular, the functional E(u, z) collects all terms above which are independent of time (i.e., of the temperature θ) whereas F(θ(t), z) contains the only temperaturedriven term. Note that, along with the above provisions, the stored-energy functional W(·, ·; θ(t)) turns out to be uniformly convex in H 1 (Ω; R d ) × L 2 (Ω; R d×d dev ) albeit non-smooth. In particular, its domain is the convex and closed subset of Y(u Dir (t)) given by {(u, z) ∈ Y(u Dir (t)) : z ∈ K a.e. in Ω}.
In addition to the above-prescribed boundary displacement conditions on Γ Dir , we shall consider some imposed body force f and surface traction g, as well. We assume to be given and define the total load ∈ W 1,1 (0, T ; (H 1 (Ω; R d )) ) as Starting from a suitable initial state (u 0 , z 0 ), the dissipative evolution of the quasi-static system is governed by the occurrence of a dissipation (pseudo-)potential D : for some activation radius R > 0. Note that the dissipation potential acts on the rateż for it encodes the action of dissipative forces. Moreover, we let the total dissipation of the process on the time interval [s, t] ⊆ [0, T ] be given by the sup being taken among all partitions of the interval [s, t].
For the sake of later purposes, we shall define the set of stable states S(t, θ) at time t and for the temperature θ as The relevance of this notion resides on the fact that, given a stable state (u, z), no competitor state (ū,z) can be preferred in terms of balance between total energy and dissipation. This global minimality requirement, sometimes disputable in other situations, is here fully motivated by the convexity of the total energy of the body.

2.2.
Energetic solvability of the state problem. Assume that we are given the space-homogeneous temperature θ ∈ W 1,1 (0, T ). We shall be interested in establishing the existence of a suitably weak solution to the quasi-static state problem where σ stands for the stress, n is the outward normal to Γ tr , and ∂ is the (possibly partial) subdifferential in L 2 (Ω; R d×d dev ) in the sense of convex analysis [13]. In particular, we focus on energetic solutionsà la Mielke [36,42], namely trajecto- and t → β (θ(t))θ(t)|z(t)| are integrable, and, for all t ∈ [0, T ], we have the two conditions: Stability: Energy balance: Our result on the state problem reads as follows.
Theorem 2.1 (Existence for the state problem). Assume (1) and (3). Given θ ∈ W 1,1 (0, T ) and an initial value (u 0 , z 0 ) ∈ S(0, θ(0)) there exists an energetic solution (u, z) of the state problem in the sense of (5)- (6). Moreover, all energetic solutions belong to W 1, ). The proof of this result follows via time discretization by applying the general strategy for existence of energetic solutions of rate-independent systems [36]. In particular, we shall implement in our concrete case the improvements introduced by Rindler [49] to the by-now classical existence proof by Francfort & Mielke [20]. In particular, we cannot rely here on the absolute continuity of the power of external actions and shall hence provide a refined discussion of the relative terms into the energy balance equation. A sketch of this argument is given in Section 3 below.
Our result shall be compared with the former analysis by Mielke, Petrov, & Paoli in [37,40] where existence theories for the Souza-Auricchio model in the non-isothermal case have been firstly obtained. These results differ from ours under three aspects. At first, in [37,40] a regularization (smoothing) of the inelastic energy density (both in θ and z) is needed. This in particular prevents the Authors from considering in [37,40] the original inelastic energy density of Souza-Auricchio in (2) which is instead included in our analysis. On the other hand, one shall note that the smooth setting of [37,40] allows for a uniqueness theory which is here out of reach. Secondly, the given temperature in [37,40] was taken to be C 1 -regular in time whereas we just require absolute continuity. This issue is indeed crucial here as we are aiming at optimal controls and compactness of temperatures (controls) will clearly play a role. Finally, we include here the total variation of z into the stored energy instead of its squared gradient as in [37,40]. This has the advantage of allowing discontinuous in space inelastic strains z and of requiring no extra (and somewhat disputable) boundary conditions. 2.3. Optimal control. We shall now come to the controllability statement. As we have already mentioned, we are here focusing on the situation of a SMA specimen which deforms under given mechanical loading under the influence of a controlled space-homogeneous time-dependent temperature θ. In particular, given the temperature θ ∈ W 1,1 (0, T ) let us denote by ) the set of energetic solutions of the state problem from Theorem 2.1. Let the set of admissible temperatures (controls) be denoted by Θ ⊂ W 1,1 (0, T ). Then, the optimal control problem consists in the minimization of a given cost functional which is depending on both the energetic solution and the control. Our problem is to find an optimal control θ * ∈ Θ and a corresponding optimal energetic solution (u * , z * ) ∈ Sol(θ * ) such that (u * , z * ) ∈ Arg Min {J (u, z, θ) | (u, z) ∈ Sol(θ), θ ∈ Θ} .
In order to possibly find optimal controls we shall consider the following standard requirements.
Compatibility of the initial value and the controls: Compactness of controls: Θ is weakly compact in W 1,r (0, T ) for some r > 1. (8) Lower semicontinuity of the cost functional: The compatibility condition in (7) was already presented in [49] and is just intended to ensure that the initial values are stable regardless of the choice of the control. In case all θ ∈ Θ share the same initial value (which is somehow natural in applications where θ(0) is usually room temperature) the compatibility condition (7) can be easily checked. On the other hand, note that if (0) = 0 the choice z = 0 (which corresponds to pure austenite for high temperatures and specific accommodated martensites at low temperatures) fulfills the compatibility (7) under no restrictions on Θ.
The compactness of Θ from (8) is here chosen just for the sake of simplicity. In particular it can be relaxed by asking for some extra coercivity with respect to θ on the functional J . We stick to assumption (8) for the sake of definiteness only.
The lower semicontinuity requirement in (9) is standard. For the sake of illustration, let us remark that a possible quadratic cost functional covered by our theory is are given desired displacement and inelastic strain profiles whereas u f ∈ L 2 (Ω; R d ) and z f ∈ L 2 (Ω; R d×d ) are given target states. Note that the latter functional is not lower semicontinuous with respect to the weak-star topology of ). Still, it fulfills (9) as an effect of the requirement (u n , z n ) ∈ Sol(θ n ) which indeed provides extra compactness.
Our main result is the following.
We shall give a direct proof of this theorem in Section 4 for the sake of completeness. Still, let us mention that the result may be equivalently obtained by applying in the present situation the abstract theory developed by Rindler [49].

State problem
This section provides a proof of Theorem 2.1. As already mentioned, the argument follows the by-now classical proof of convergence of time discretization for rate-independent evolution problems [36]. Still, exactly in the same spirit of [49], some specific care is devoted to ascertain the convergence of the work of external actions. In particular, these work read here (10)  Note that the former existence results for given temperature in [37,40] by-pass this problem by requiring the temperature to be C 1 . This would however be not satisfactory here as we are indeed interested in optimal control via θ. Namely, we shall better consider the least possible time-regularity for the temperature θ entailing the solvability of the state problem. We circumvent here the lack of absolute continuity of the above work by directly considering the concrete from (10). For the sake of completeness we aim at providing here a sketch of the proof.
Let us start by changing variables in order to reduce to a time-independent state space. In particular, we let v = u − u Dir and focus on the pair (v, z) taking values in the space Y 0 := Y(0). We easily compute that Let now L : [0, T ] → Y 0 be given by and notice that L ∈ W 1,1 (0, T ; Y 0 ). Eventually, (u, z) is an energetic solution of the quasi-static evolution problem (5) , v(t) and t → β (θ(t))θ(t)|z(t)| are integrable, and we have, for all t ∈ [0, T ], Stability: Energy balance: We shall prove the existence of such (v, z) via time discretization. Time discretization. Assume to be given a sequence of partitions {0 = t n 0 < t n 1 < · · · < t n N n −1 < t n N n = T } with diameter τ n = max i=1,...,N n (t n i −t n i−1 ) going to 0 as n → ∞. We inductively define (a sequence of) unique solutions for i = 1, . . . , N n with (v n 0 , z n 0 ) = (v 0 , z 0 ). To this aim it suffices to observe that the map (u, z) → E(u, z) + F(θ(t), z) − L(t), (u, z) + D(z−z) is uniformly convex and lower semicontinuous in Y 0 for any given θ(t) ∈ R andz ∈ L 1 (Ω; R d×d dev ). Next, we denote by (v n , z n ) the right-continuous and piecewise constant interpolant of the values {(v n i , z n i )} N n i=0 on the partition. Moreover, we let t n : [0, T ] → [0, T ] be given by t n (t) = t n i−1 for t ∈ [t n i−1 , t n i ) for i ∈ 1, . . . , N n .
Stability at the discrete level. The minimality in (13) entails that (v n i , z n i ) is stable at time t n i , that is (v n i , z n i ) ∈ S (t n i , θ(t n i )), for all i = 1, . . . , N . Indeed, for any (v,z) ∈ Y 0 , we get and the term D(z n i − z n i−1 ) cancels out. Convergence to a time-continuous evolution. Taking into account the minimality (13) of (v n i , z n i ) we deduce that By exploiting the discrete Gronwall Lemma and the coercivity of E we deduce that sup t∈[0,T ] E(v n (t), z n (t))+F(θ(t), z n (t)) and Diss D (z n , [0, T ]) are bounded independently of n.
We make use of the generalization of Helly's selection principle from [35], and find a (not relabeled) subsequence and a non-decreasing function φ : Moreover, by exploiting the quadratic character of E(·, z) we readily have from minimality (13) that v n = L(z n , L • t n ) where L : L 1 (Ω; R d×d dev )×Y 0 → H 1 (Ω; R d ) is a linear and continuous operator. In particular v n = L(z n , L • t n ) → L(z, L) =: v. Stability of the limit trajectory. We now prove that the set is closed with respect to the weak topology of R×Y 0 . Let (t k , v k , z k ) ∈ S with t k → t and (v k , z k ) → (v, z) weakly in Y 0 . By the lower semicontinuity of E and F and taking into account the strong continuity of D in L 1 (Ω; R d×d dev ) and the continuity of θ and L we get The aim is now to exploit the latter closure property in order to prove that (v(t), z(t)) is a stable state, i.e. (11) holds. Note that t → t n (t) converges uniformly to the identity and (v n (t), z n (t)) = (v n (t n (t)), z n (t n (t))) converges to (v(t), z(t)). Hence, the stability (11) follows since we have (t n (t), v n (t n (t)), z n (t n (t))) ∈ S and the latter is closed. Upper energy estimate. We can rewrite the inequality (14) in the following way We now pass to the lim inf in the latter relation by exploiting the lower semicontinuity of E and F, the integrability of˙ and of (β • θ)˙, and the boundedness of v n from (15)- (16). By the Lebesgue Dominated Convergence Theorem, we deduce that for all t ∈ [0, T ], i.e., the upper energy estimate. Lower energy estimate. Let us now check the converse inequality with respect to (17). Summing up for j = 0, . . . , M m we deduce that We can handle the first term in the last line of (18) as follows where we used some obvious notation for the piecewise mean on the partition. As Diss D (z, [0, t]) < ∞ we have that z is continuous in L 1 (Ω, R d×d dev ) with the exception of at most a countable number of times. This in particular entails that z m → z pointwise almost everywhere in [0, t]. Moreover, β • θ ∈ W 1,1 (0, T ) and one has that Hence, by Dominated Convegence we can conclude that From the fact that (v(s), z(s)) ∈ S (s, θ(s)) for all s ∈ [0, t] we readily deduce that v = L(z, L). In particular, v has at most a countable number of discontinuity points in time and v m → v pointwise almost everywhere. Eventually, we can pass to the limit into inequality (18) and conclude that Absolute continuity of the evolution. Let us now prove that indeed t → (v(t), z(t)) is absolutely continuous in L 2 (Ω; R d )×H 1 (Ω; R d×d sym ). In order to do so, it is notationally convenient to switch back to the original variables (u, z) = (v + u Dir , z). From the stability (5) at time s and the uniform convexity of E + F of constant α > 0 we get that Hence, by means of Gronwall's Lemma one checks that for some suitable constant c depending just on α and T . In particular, the absolute continuity of t → (v(t), z(t)) ensues.

Existence of an optimal control
We shall finally turn to the proof of Theorem 2.2. Let (u n , z n , θ n ) with (u n , z n ) ∈ Sol(θ n ) be a minimizing sequence for J , namely J (u n , z n , θ n ) → inf J (u, z, θ) : (u, z) ∈ Sol(θ), θ ∈ Θ .
Owing to the compactness (8), we can extract a not relabeled subsequence in such a way that both θ n → θ and β • θ n → β • θ weakly in W 1,r (0, T ) and uniformly. By exploiting the energy balance (6) we readily get that sup t∈[0,T ] E(u n (t), z n (t))+F(θ n (t), z n (t)) and Diss D (z n , [0, T ]) are bounded independently of n.
We can hence extract again (still not relabeling) in order to get that z n → z pointwise in BV (Ω; R d×d dev ), u n → u pointwise in H 1 (Ω; R d ) (by linearity), and (u n , z n ) → (u, z) weakly-star in L ∞ (0, T ; H 1 (Ω; R d )×L 2 (Ω; R d×d dev )). The proof of Theorem 2.1 can be adapted to the present situation in order to ensure that (u, z) ∈ Sol(θ). To this aim, the differences arise solely in the treatment of those terms containing θ n . In particular, the above-mentioned convergences of θ n and z n entail directly the convergence Ω β(θ n (t))|z n (t)| dx → Ω β(θ(t))|z(t)| dx ∀t ∈ [0, T ].
Moreover, one can also check for the limit t 0 Ω β (θ n )θ n |z n | dx ds → t 0 Ω β (θ)θ|z| dx ds as we have that β (θ n )θ n → β (θ)θ weakly in L r (0, T ) and (by possibly extracting again) z n → z strongly in L p (Ω×[0, T ]; R d×d dev ) for all p ∈ [1, ∞) (recall that z n are uniformly bounded in R d×d dev ). As we now have that (u, z) ∈ Sol(θ), the assertion follows directly from the lower semicontinuity assumption (9). 4.1. Some comments on the approximation of optimal controls. The present existence result represents just a first step in the direction of optimally control the complex thermomechanical behavior of SMA via the Souza-Auricchio model. Indeed, one would also be interested in taking this investigation further into possibly computing optimal controls. However, this seems to be a quite delicate task due to the crucially non-smooth nature of the Souza-Auricchio model.
As already mentioned, energetic solutions to the Souza-Auricchio model are not known to be unique. This would call rigth from the beginning for trying to establish necessary optimality conditions in the frame of set-valued state-to-control mappings. One possible way to restore uniqueness at the state-problem level seems that of regularizing the Souza-Auricchio model by smoothing the inelastic energy density in (2). This approach has been followed by Mielke, Paoli, & Petrov [37,40] for existence (but note that here the temperature is continuously differentiable with respect to time) and, in the isothermal case, regularizations have been proved to converge to the original non-smooth setting in [5].
Hence, a possible strategy in order to compute optimal controls could be that of considering this regularized situation, optimally controlling the regularized problem, and showing the approximation of the limiting non-smooth case via regularized optimal controls. This kind of argument would quite naturally fall within the general frame introduced by Rindler [50]. However, the specific form of the work of external actions in (10) once again prevents from applying the general results form [50] and some extension of the latter seems rather necessary. Moreover, the possibility of obtaining necessary conditions even at the regularized case seems presently demanding. In this respect, the reader can consider the recent and quite technical analysis by Wachsmuth [56,57] on the comparably simpler linearized-kinematichardening elastoplasticity case.
From the mechanical viewpoint the above-mentioned regularization is rather arguable for it crucially affects the predictive capabilities of the model. Indeed, as an effect of smoothing we loose the possibility of predicting both initiation of the parent-to-product phase transformation under tension and the saturation effect on inelastic evolution. These two are indeed crucial features as they allow at the algorithmic level to efficiently distinguish between purely elastic and inelastic evolution. This distinction turns out to contribute to the remarkable robustness of the Souza-Auricchio model with respect to approximations. We hence prefer to stick here to the original non-smooth situation in order to maximally preserve these distinctive traits of the Souza-Auricchio model.
A second possible strategy for computing optimal control would be that of considering some time or even space-time discretized (approximate) optimal control procedure in the spirit of [50]. This development also appears not at all trivial. On the one hand, one should note that the derivation of necessary optimality conditions at the discrete level appears to be possibly amenable under some regularization only (see above). Moreover, the already mentioned obstruction in applying the available general results due to the specific form of the work of external actions would demand for some significant adaptation of the theory from [41,49,50]. We shall hence resort in reconsidering these opportunities elsewhere.