Feedback Control of Nonlinear Dissipative Systems by Finite Determining Parameters - A Reaction-diffusion Paradigm

We introduce here a simple finite-dimensional feedback control scheme for stabilizing solutions of infinite-dimensional dissipative evolution equations, such as reaction-diffusion systems, the Navier-Stokes equations and the Kuramoto-Sivashinsky equation. The designed feedback control scheme takes advantage of the fact that such systems possess finite number of determining parameters (degrees of freedom), namely, finite number of determining Fourier modes, determining nodes, and determining interpolants and projections. In particular, the feedback control scheme uses finitely many of such observables and controllers. This observation is of a particular interest since it implies that our approach has far more reaching applications, in particular, in data assimilation. Moreover, we emphasize that our scheme treats all kinds of the determining projections, as well as, the various dissipative equations with one unified approach. However, for the sake of simplicity we demonstrate our approach in this paper to a one-dimensional reaction-diffusion equation paradigm.

There has been some interesting work on reduction methods, with applications focused on scientific computing and feedback control theory, taking advantage of the finite-dimensional asymptotic behavior of these dissipative dynamical systems (see, e.g., [1,11,12,25,35] and references therein) . However, there has been very little rigorous analytical work, in particular in the context of feedback control theory, justifying these applications. In the case of separation of spatial scales, and hence the existence of inertial manifolds, the authors of [32] and [33] provide an example of finite-dimensional feedback control (lumped feedback control) that drives the dynamics of one-dimensional reaction-diffusion system to an a priori specified finite-dimensional dynamics. It is worth stressing again that in the case of inertial manifold the dynamics of the underlying evolution equation is equivalent to that of an ordinary differential equations to begin with. However, the main challenge is in being able to provide a representation of this ODE system in the relevant parameters dictated by the applications. In [23] and [7] the authors have shown that if a certain dissipative system has separation of scales, and hence an inertial manifold, then such a manifold can be parameterized by any set of adequate parameters, e.g. Fourier modes, nodal values, local volume averages, etc... In the above mentioned work of [32] and [33] the authors employed such an equivalence in the parameterization of the inertial manifolds to show their results.
In this paper we propose a new feedback control for controlling general dissipative evolution equations using any of the determining systems of parameters (modes, nodes, volume elements, etc...) without requiring the presence of separation in spatial scales, i.e. without assuming the existence of an inertial manifold. To fix ideas we demonstrate our idea for a simple reaction diffusion equation, the Chafee-Infante equation, which is the real Ginzburg-Landau equation. It is worth mentioning, however, that this new idea has a far more reaching areas of applications, other than feedback control, such as in data assimilations for weather prediction [2,4]. In addition, one can use this approach to show that the long time-time dynamics of the underlying dissipative evolution equation, such as the two-dimensional Navier-Stokes equations, can be imbedded in an infinite-dimensional dynamical system that is induced by an ordinary differential equations, named determining form, which is governed by a globally Lipschitz vector field, cf. [13,14] and [15].
In this paper we will use the Chafee-Infante reaction-diffusion equation for α > 0, large enough, as a paradigm to fix ideas and to use the notions of finite number of determining modes, nodes and volume elements to design feedback control to stabilize the v(x) ≡ 0 unstable steady state solution of (1)- (2). Indeed, by linearizing equation (1) about v ≡ 0 one obtains the linear equation . Therefore, we obtaiṅ and whose solution is Therefore, for α > 0, large enough, all the low wave numbers k 2 < α L 2 π 2 ν are unstable. Consequently, the dimension of the unstable manifold of v ≡ 0 behaves like α L 2 ν (see, for instance, [3,24] and [36] for a similar analysis). The aim of this paper is to design a feedback control that stabilizes v ≡ 0, for example, either by observing the values of the solutions at certain nodal points, local averages of the solutions in subintervals of [0, L], or by observing finitely many of their Fourier modes. Based on the above discussion, a naive analysis would suggest that one would need about L 2 α ν feedback controllers to stabilize v ≡ 0. In this paper we will give a rigorous justification to this assertion. First, we demonstrate our result for the case of local averages, which is the most straightforward approach. Later, we present a more general abstract result, that unifies our approach, utilizing all sorts of approximate interpolant operators, as observables and controllers, and show that this abstract approach applies to the Fourier modes, local volumes (i.e. local averages) and nodal values as particular examples. It is worth mentioning that the same feedback control scheme can be used to stabilize any other time-dependent solution, v(x, t), of (1)- (2). The details of the proof are similar to the ones presented here for stabilizing the zero solution; thus, for the sake of simplicity they will not be provided. Furthermore, similar scheme can be also implemented for feedback control of other nonlinear dissipative dynamical systems, such as the two-dimensional Navier-Stokes equations, the Kuramoto-Sivashinsky equation and reaction-diffusion systems. A computational study concerning the implementation of this feedback control scheme for various nonlinear dissipative equations will be reported in a forthcoming work [30]. In addition, one can design similar feedback control algorithm with stochastically noisy observables and controllers to stabilize, in the average, given solutions; within errors that are determined by the standard deviation of the noise. This can be achieved by combining some of the ideas presented in [4] with those presented in the present paper, a subject of future work.

Finite volume elements feedback control
To fix ideas we propose the following feedback control system for (1)-(2) in order to stabilize the steady state solution v ≡ 0, where is the characteristic function of the interval J k , for k = 1, . . . , N , Here, the local averages of the solution, u k , for k = 1, ..., N, are the observables, and they are also used as the feedback controllers in (4). It is easy to observe v ≡ 0 is also a steady state solution for (4)- (5).
Before showing that (4)-(5) globally stabilizes the steady state v ≡ 0, one has to prove first the global existence and uniqueness of the feedback system (4)-(5).
In section 4, we will show in Theorem 4.1 a result concerning global existence, uniqueness and stabilization for a general family of finite-dimensional feedback control that includes system (4)-(5) as a particular case. Therefore, we will postpone this task of proving the global existence and uniqueness until section 4, and we only show here the global stability of v ≡ 0. This is in order to fix ideas and to demonstrate our general approach.
Assuming the global existence and uniqueness of (4)-(5), we will show in this section that every solution u of (4)-(5) tends to zero, as t → ∞, under specific explicit assumptions on N, ν, α, L and µ (see Theorem 2.1 for details). But first we need the following proposition to prove our result. We observe that similar propositions were introduced and proved in [8,23,26,27] and [28] (see also [32] and [33]). We adapt here similar ideas from [8] for our proof.
where h = L N . Moreover, where Proof.
where in the last equality we used the fact that N k=1 By virtue of Poincaré inequality we have Thus, (9) and (10) imply which proves inequality (7) in the Proposition 2.1. Next, we prove inequaliy (8). From the Poincaré inequality (10) we have Thus, by summing over k = 1, . . . , N, in the above inequality we conclude inequality (8) of the Proposition 2.1.
Proof. We take the L 2 inner product of equation (4) with u, and integrate by parts to obtain Using (8), from Proposition 2.1, and the assumption that µ ≥ ν 2π Substituting (14) in (13) we obtain Therefore, by virtue of Gronwall's inequality and the assumption that ν > α h 2 ; and the Theorem follows.

Remark 2.1
It is worth mentioning that the assumptions of Theorem 2.1, in particular, that N > L 2 α 4π 2 ν , is consistent with the fact that the dimension of the unstable manifold about v ≡ 0 is of order of L 2 α ν . That is, one needs at least this number of parameters to stabilize v ≡ 0. In Theorem 4.1 we give a different and more general proof, that illustrates this point further. As we have mentioned earlier, one can use the same idea to stabilize any other solution, v(x, t), of (1)-(2) by using a slightly modified feedback control in the right-hand side of (4)-

Interpolant operators as feedback controllers
In this section we will consider a general linear map I h : which is an interpolant operator that approximates identity with error of order h. Specifically, it approximates the inclusion map i : holds, for every ϕ ∈ H 1 ([0, L]). The last inequality is a version of the wellknown Bramble-Hilbert inequality, that usually appears in the context of finite elements [5]. We propose here to consider the following general feedback system, to stabilize the solution v(x, t) of (1)- (2), of the form To fix ideas we focus on stabilizing the steady state solution v ≡ 0 of (1)- (2).
Here one can think of I h (u) as the observables and controllers that will be used to stabilize our system. Before we state and prove our general theorems concerning system (16)- (17), we will give some examples of the approximate interpolant I h (ϕ) which satisfy the approximation property (15). In particular, we are interested in interpolant operators, I h , of finite-rank, and whose rank is of the order O(1/h).

Examples of finite-rank approximate identity interpolant operators
3.1.1 Finite volume elements Using the notation of section 2 we consider the interpolant operator that uses local spatial averages (finite volume elements) for approximating the local values of the underlying function. We observe that the interpolant operator, I h (ϕ), that is introduced in (18) and implemented in (16), is exactly the same one discussed in details in section 2. In particular, one can easily see that approximating inequality (15) holds in this case, thanks to Proposition 2.1.

An interpolant operator based on nodal values
In this example we consider the interpolant operator where J k and χ J k are as in section 2, and the points x k ∈ J k , for k = 1, 2, · · · , N are arbitrary. Next, we show that the interpolant operator given in (19) satisfied the approximation property (15). Here again we adopt ideas from [8,23,26] to prove the next proposition. Proof.
As in the proof of the Proposition 2.1, we observe that and then we obtain Using Cauchy-Schwarz inequality, we get which concludes the proof of Proposition 3.1.
In view of (1), (16) and (19) we propose the following feedback controller for stabilizing v ≡ 0 which is a special case of (16).

Projection onto Fourier modes as an interpolant operator
Here, we consider the following projection onto the first N Fourier modes as an example of an interpolant operator; where the Fourier coefficients are given bŷ Next, we observe that inequality (15) holds for the interpolant operator given in (22).
Proof. The proof of this proposition is a simple exercise in Fourier series. Thus, it will be omitted.

Existence, uniqueness and stabilization using the I h feedback control
In this section we establish the global existence and uniqueness for the general feedback system introduced in (16)- (17); and that the I h feedback control is stabilizing the steady state solution v ≡ 0 of the (1)-(2). This will be accomplished under the assumptions (15) and that µ is large enough, and h is small enough, satisfying: µ ≥ (2α + 3νL −2 ) and ν ≥ µ c 2 h 2 .
To this end one uses the standard Galerkin approximation procedure based on the eigenfunctions of the Laplacian, subject to the Neumann boundary condition, i.e., cos( π k x L ) for k = 1, 2.... We will omit the details of this standard procedure and provide only the formal a-priori estimates (see, e.g., [36]). These estimates can be obtained rigorously through the Galerkin procedure, by passing to the limit while using the relevant compactness theorems. Let us now establish the aformentioned formal a-priori bounds for the solution which are essential for guaranteeing globlal existence and uniqueness. System (16)- (17) can be rewritten as Taking the L 2 -inner product of (25) with u, integrating by parts and using the Neumann boundary conditions, we obtain Writing and applying the Cauchy-Schwarz inequality, we get Using Young's inequality we reach Using (15), and the definition of the H 1 -norm given in (6) we obtain Thanks to the assumption (24) we conclude Therefore, by dropping the u x 2 L 2 term from the left-hand side of (27) and applying Gronwall's inequality we have Notice that from (27) one also concludes that for every τ > 0 Next, we show the continuous dependence of the solutions of (16) on the initial data and the uniqueness, provided the assumptions (15) and (24) hold. Indeed, let u 1 , u 2 be two solutions and w = u 1 − u 2 of (16). From (16) we find that Multiplying by w and integrating with respect to x over [0, L] we get A straightforward computation, using the Cauchy-Schwarz and Young inequalities and assumption (15), yields Using (6), the definition of the H 1 -norm, we reach By assumption (24) the above implies Using assumption (24) one more time the above inequality simplifies to Finally, by Gronwall's inequality we have Thus, if w(0) = 0 then w(t) L 2 ≡ 0. Moreover, inequaliy (30) implies the continuous dependence of the solutions of (16)-(17) on the initial data. In conclusion, from the above, and in particular thanks to (28) and (29), we have the following theorem: In particular, we concluded the under that above assumption the feedback control interpolant operator I h is stabilizing the steady state solution v ≡ 0 of (1)-(2).

Remark 4.1
Let us observe that in order to satisfy assumption (24) one can choose, for small values of ν, µ = O(α). As a result, assumption (24) will hold if we choose h small enough such that N : , that is the number of feedback controllers is comparable to the dimension of the unstable manifold about v ≡ 0. This is consistent with our earlier observation in the introduction and in Remark 2.1.

Stabilizing in the H 1 -norm
In the previous section we have shown that the feedback system (16)-(17) stabilzes the steady state solution v ≡ 0 in the L 2 -norm, i.e., u L 2 → 0, as t → ∞, provided assumptions (24) holds. Next, we show that we also have u(t) H 1 → 0, as t → ∞. To this end it is enough to show that u x L 2 → 0, as t → ∞. Let us rewrite (16)-(17) as Thanks to the estimate (29) we realize that the solution instantaneously becomes in H 1 . Therefore, without loss of generality we can assume that the initial data u 0 ∈ H 1 . Below we provide the formal arguments and estimates, which, as we have already indicated earlier, can be established rigorously by using a Galerkin approximation procedure. We take the L 2 inner product of (31) with −u xx , integrating by parts, and using the Neumann boundary conditions (17) we obtain: By Cauchy-Schwarz inequality we have Applying Young's inequality we obtain Using property (15) and the definition of the H 1 −norm in (6), we have Thanks to assumption (24) we observe that Therefore, the above implies Since lim t→∞ u(t) 2 = 0, then by Gronwall's inequality it is easy to show that u x 2 L 2 → 0, as t → ∞, (see also special Gronwall's type Lemma in [26]).

Nodal observables and feedback controllers
In this section we propose a different feedback control based on nodal value observables and feedback controllers. Assume that the observables are the values of the solutions u(x k ), at the points x k ∈ J k = [(k − 1) L N , k L N ], k = 1, ..., N, and that the feedback is at some points x k ∈ J k , x k is not necessarily the same as x k . That is the measurements are made at x k , while the feedback controllers are at x k , for k = 1, 2..., N. To avoid technical issues that are dealing with boundary conditions, we focus here on the periodic boundary condition case. In this case the feedback system will read u(x, t) = u(x + L, t), where h = L N ; and δ(x − a) ∈ H −1 per ([0, L]), for a ∈ [0, L], and is extended periodically such that < δ(· − a), ϕ >= ϕ(a) for every ϕ ∈ H 1 per ([0, L]). The feedback control proposed in (33) - (34) is different than that of (16)- (17), since the right-hand side in (33) is a distribution that belongs to H −1 per ([0, L]), while the right-hand side in (16) belongs to L 2 ([0, L]). In this section we will show that, under similar assumptions to those in Theorem 4.1, the proposed feedback system (33) stabilizes the steady state v ≡ 0 in the L 2 −norm. One should not expect here a stronger statement, as the one stated in section 5, in which the stabilizing is also valid in the H 1 −norm. This is because the solutions of (33)- (34) are weaker than those of (16)- (17), since the right-hand side in (33) is less regular than its counterpart in (16). As before, we will show, below, the formal steps, which demonstrate simultaneously the global existence, uniqueness and stabilizing effect. These formal steps and estimates can be justified rigorously by implementing the Galerkin procedure based on the eigenfunction of the Laplacian, subject to periodic boundary conditions, with period L (see, e.g., [36]). First, let us prove the following Lemma, which is basically the embedding of the Hölder space of C 1 2 ⊂ H 1 (see, e.g., [8,36]). and Proof. We prove inequality (36) for ϕ ∈ C 1 ([0, L]), and by the density of C 1 ⊂ H 1 the result follows for every ϕ ∈ H 1 .
By summing the above inequality over k = 1, .., N we conclude (36). To prove (37) we observe that for every x ∈ J k we have and by integrating with respect to x over J k , and using the Cauchy-Schwarz inequality, we obtain Now we conclude (37) by summing over k = 1, . . . , N. Proof. We will show here only the relevant a priori estimates. The rest of the regularity results are standard for nonlinear parabolic equations (see, e.g., [36]). We take the H −1 action of (33) on u ∈ H 1 , and use Lemma of Lions-Magenes (cf. Chap. III-p.169, [37]), to obtain: where in the last step we applied the Young's inequality. Next, we apply (36) and (37) to the right-hand side Since ν ≥ 2µ h 2 and 4α < µ we conclude: Thanks to Gronwall's inequality we conclude from the above (40), and the regularity of the solutions as stated in the theorem. Next, we prove the uniqueness of solutions. Let u 1 and u 2 be any two solutions. Denote by w = u 1 − u 2 . Then w satisfies ∂w ∂t − ν w xx − α w + (u 2 1 + u 1 u 2 + u 2 2 ) w = −µ h N k=1 w(x k ) δ(x − x k ).
Next, we follow the same steps as in the beginning of the proof to obtain Since ν ≥ 2µ h 2 and µ > 4 α we conclude, thanks to Gronwall's inequality, Notice that (41) implies the uniqueness of the solutions and their continuous dependence on the initia data.

Remark 6.1
Here again we observe that for small values of ν by choosing µ = O(α) then the condition of the theorem imply that N := L h = O( α ν ); which is comparable to the dimension of the unstable manifold about the steady state v ≡ 0.