Optimal control problem for Lengyel – Epstein model with obstacles and state constraints

Abstract. This paper considers the state constrained optimal control problem for Lengyel–Epstein model with obstacles. We prove existence and regularity results for this model by applying the standard methods. We show the relationship between the control problem and its approximation. Moreover, we derive the necessary conditions for the optimal control of our original problem by using the approximate problem.


Introduction
This paper is concerned with the state constrained optimal control problem for the Lengyel-Epstein model min L(u, v, w) = T 0 g t, u(t) + h w(t) dt (1) subject to and where Ω is a bounded domain in R N (N = 1, 2, 3) with a smooth boundary ∂Ω, say of class C 2 , u and v are the dimensionless concentration for activator and inhibitor, c Vilnius University, 2016 respectively; a, b, c and θ are dimensionless parameters of the chemical system; δ > 0 is proportional to the ratio of the diffusion coefficients of the main species.The obstacle ∂I [σ * ,σ * ] (u) is the subdifferential of the indicator function I [σ * ,σ * ] (u) on the closed interval [σ * , σ * ]; κ > 0, σ * , σ * ∈ R are the given constants.u 0 (x), h 0 (x) and φ(x, t) are given functions and Bw is the control term.Here F (u) ⊂ S is the state constraint, which can be regarded as the description of the physical background of the Lengyel-Epstein model.Equation (2) without the control term Bw and κ = 0 is the classical Lengyel-Epstein model (see [4,6,10,11,13,15,16,20,26]).It comes from the reaction between chlorine dioxide, iodine and malonic acid (CDIMA reaction), and is one of the most thoroughly studied oscillatory chemical systems both in experiment and in numeric.In [10], the photosensitive CDIMA reaction was investigated by using the Lengyel-Epstein model modified to include the effect of external illumination.Jensen et al. studied the localized structures and front propagation in the Lengyel-Epstein model [13].Recently, based on Runge-Kutta method, Bastian, Kartawidjaja [4] solved the parallel performance of the Lengyel-Epstein model.More recently, Váquez et al. [10] studied the chaotic behaviors induced by modulated illumination in the Lengyel-Epstein model under Turing considerations.As we all know, in some physical examples, the range of the activator u would not be the whole real numbers R, but often be a bounded closed interval [σ * , σ * ].
Here we are going to pay attention to this point and give an adequate mathematical treatment to it.Note that ∂I [σ * ,σ * ] (u) is a multi-valued and maximal monotone graph in R, which can coincide with the subdifferential of I [σ * ,σ * ] (u).Namely, I [σ * ,σ * ] (u) is assumed to be +∞ out of a bounded interval.
A pair (u, v) is said to be a weak solution of (2) if and only if in the sense of D (0, T ).Let U be a real Hilbert space and B : U → H be a linear continuous operator.Assume that Z is a Banach space with the dual Z * , which is strictly convex, and S ⊂ Z is a closed convex subset with finite codimensionality.
The following items are the assumptions on data: ) and for every Λ > 0, there exists L Λ > 0 independent on t such that for t ∈ [0, T ] and (H3) U → R is lower semicontinuous and convex with the following growth property: there exist c 1 > 0 and c 2 ∈ R such that and F (u) ⊂ S is the state constraint.In this paper, we consider the following optimal control problem: Minimize (P): L(u, v, w) over all (u, v, w) ∈ A ad .
It is known that for each w ∈ L 2 (Q), u 0 ∈ V and v 0 ∈ V , system (2) has a unique solution u, v ∈ Y (see [9]).The first question regarding problem (P) is if there is an admissible solution, namely, if the set A ad is nonempty.Taking into account the arguments in the proof of the main results in [3], we may assume in the sequel that problem (P) admits at least one admissible solution.
In the past decades, much attention has been paid to the optimal control problems governed by nonlinear parabolic system including semilinear equations, variational inequalities and system with phase transitions [5,7,8,12,14,18,21,22,23,27,28].In particular, the optimal control for semilinear parabolic system without state constraint was discussed in [14,21,25,29].Recently, in [23], based on the energy estimates and the compact methods, Ryu and Yagi considered the optimal control problems of adsorbateinduced phase transition model.More recently, a first order optimality condition for nonhomogeneous Cauchy-Neumann boundary optimal control problem of non-linear phasefield system was derived in [5].In [24], the authors studied Pontryagin's maximum principle for optimal control problems (with a state constraint) governed by the 3-dimensional Navier-Stokes equations.In order to overcome the problem associated with the state constraint, the authors first defined a new penalty functional depending on a small parameter ε with which they approximated the original problem with a family of optimal control problems (P ε ) without state constraints.Pontryagin's maximum principle is derived for the approximate problem (P ε ) and the limit as ε goes to 0 yields an optimality condition http://www.mii.lt/NA for the original control problem with a state constraint.These are the steps followed in this article.The main differences between the present work and works mentioned above are as follows.In this paper, the nonlinearity involved in the Lengyel-Epstein model is stronger than that in the 3-dimensional Navier-Stokes equations, which makes the analysis of the optimal control problems in this article more involved.Moreover, because of the obstacle ∂I [σ * ,σ * ] (u) in the first equation of system (2), we cannot obtain the optimal conditions of problem (2) directly.In this paper, we derive the necessary conditions for problem (P) by showing the relation between approximation problem (P ε ) (problem (P ε ) contains the approximation of ∂I [σ * ,σ * ] (u)) and problem (P).
In order to give the necessary conditions for problem (P), we specify our notion of a strong solution to problem (2).
The main purpose of this paper is to derive the necessary optimal conditions for (P) governed by the Lengyel-Epstein model with state constraints and obstacles, which can be stated as follows.
(i) For the definition of a set to be finite codimensional in Z and for related results, one can refer to [1,17].(ii) Some basic examples of the F, g, h are: F (u) = u(x, T ), g(t, u) = α|u| 2  2 and h(w) = |w| 2  2 , where α > 0, one can see [18] for more details.pagebreak Nonlinear Anal.Model.Control, 21(1):18-39 We define then S is a convex and closed subset with finite codimensionality in R N .Consider a state constraint of the form A ad , which satisfies that L(u * , v * , w * ) = min L(u, v, w).(vi) The relations ( 5), (6) 2 form the adjoint system, (p * , q * ) is called the adjoint state and it represents a Lagrange multiplier associated with the state constraint.Equation ( 6) 1 expresses the maximum principle.
The rest of this paper is organized as follows.In Section 2, we consider the approximation problem (P ε ) of problem (P).After showing the solvability of (P ε ), we obtain the relationship between the optimal control problem (P) and its approximation problem (P ε ).In Section 3, we derive a priori estimates for the optimal pair (u ε , v ε , w ε ) of (P ε ) and then use a passage-to-limit procedure with ε 0 to get the optimality conditions for (P).

The approximation problem
This section is to show the existence of the optimal control of the approximation problem corresponding to Lengyel-Epstein model.Firstly, we show some technical lemmas and the existence of problem (2), which is presented below for the sake of completeness and easy reference.Next, we prove the existence of the control optimal problem (P ε ), which is the approximation of problem (P).In order to approximate the where [•] + denotes the positive part.Then We fix a primitive βε of β ε such that Now, we consider the following approximating system of (2) where C > 0 is a constant independent of u, v and ε. (ii) Let w n ∈ L 2 (0, T ; U ), w n → u weakly in L 2 (0, T ; U ) and (u, v), (u n , v n ) be the solutions of (9) corresponding to w and w n , respectively.Then on some subsequence of (u n , v n ), still denoted by itself, we have Proof.The existence of weak solution is proved by the standard Galerkin method.Indeed, let Then A is a linear, self-adjoint operator in H with D(A) dense in H. Therefore, we can define the powers A s of s, s ∈ R, and V is nothing other than D(A 1/2 ).Thus, there exists an orthonormal family ψ j (j ∈ N) of H and a sequence η j (j ∈ N) such that For n ∈ N, we define the discrete ansatz space by and require that u n,0 (x) → u 0 in H. Performing the Galerkin procedure for system (9), According to the ODE theory, there is a unique solution to (15) in the interval [0, T n ), where T n → T is a consequence of the following a priori estimates.Multiplying (15) 1 by u n and (15) 2 by v n and integrating them, respectively, we derive and Here we have use the fact that where ξ locates between 0 and u n .Observing that Therefore, from ( 16), Young's inequality and Hölder's inequality it follows that http://www.mii.lt/NA Here and throughout the proof of Lemma 1, we shall denote by C several positive constants independent of u n , v n and ε.With similar arguments in the above, we show that which, together with (18), implies that which, combined with (8) and the Gronwall's inequality, yields Here we have use the fact that βε (0) = 0 and βε (r) 0 for any r ∈ R.
On the other hand, testing (15) 1 by −∆u n and (15) 2 by −∆v n , respectively, and integrating the resulting equations over Ω, we derive and Notice that Nonlinear Anal.Model.Control, 21(1):  and Inserting ( 25) and ( 26) into ( 23) and ( 24), respectively, we derive and which, combined with ( 7), ( 21) and the Gronwall's inequality, implies that and Now, multiplying (15) 1 by β ε , integrating over [0, T ] and invoking the Young's inequality, we derive Thanks to (7), (21) and the Gronwall's inequality, we derive Finally, multiplying (15) 1 and (15) 2 by u n,t and v n,t , respectively, after some basic calculation, we end up with http://www.mii.lt/NABy ( 29), ( 30) and ( 32)-(33) and applying the rather standard argument, we can conclude that there exist a function (u, v) and a subsequence of (u n , v n ), still denoted by themselves, such that and (u, v) is the solution of problem (9).The uniqueness of the solution to problem ( 9) can be got easily, we omit it.Now, we prove the w-dependence of this lemma.To this end, replacing (u, v) and w by (u n , v n ) and w n in (9), respectively, we obtain By the above analysis, we have and where C > 0 is a constant independent of n and ε.By (36)-(37) and using Ascoli-Arzela theorem and compactness lemma, we infer that there exists a subsequence of (u n , v n ), still denoted by itself, such that as n → ∞.The proof is completed.
Proof.Rewrite (9) as following: Employing almost exactly the same arguments as in the proof of Lemma 1, we conclude that the results (42)-(44).Furthermore, by a standard argument in [2], we get η ∈ ∂I [σ * ,σ * ] (u) a.e. in L 2 (0, T ; H).This completes the proof.Now, we let (u * , v * , w * ) be optimal for problem (P).For each ε > 0, assume Now, for each ε > 0, the approximating optimal control problems (P ε ) is as follows: Minimize L ε (w) over w ∈ L 2 (0, T ; U ), http://www.mii.lt/NAwhere L ε : L 2 (0, T ; U ) → R by and (u ε , v ε ) is the solution of ( 9).Here, d S (F (u)) denotes the distance between F (u) and S, is the approximations of g [1], where n = [1/ε], ρ n is a mollifier in R n and P n : H → X n is the projection of H on X n , which is the finite dimensional space generated by In this case, one can transform the original state constrained optimal control problem (P) into non-constrained optimal control problem (P ε ) and use the method [3] to obtain the optimality condition for problem (P) by a passage-to-limit procedure for ε 0. First of all, we show the existence of the optimal solutions for (P ε ).
Lemma 3. (P ε ) has at least one optimal solution.
Proof.Let ε > 0 be fixed.It is clear that inf L ε (w) > −∞.Let d ε = inf{L ε (w): w ∈ L 2 (0, T ; U )} and w n be a minimizing sequence such that which, together with (H2), (H3) and (50), implies that w n is bounded in L 2 (0, T ; U ). Without loss of generality, we may assume that w n → w weakly in L 2 (0, T ; U ).Let (u n , v n ) and (ũ, ṽ) be the solutions of ( 9) corresponding to w n and w, respectively.It follows from Lemma 1 that on some subsequence of (u n , v n ), still denoted by itself, With the help of (H2), ( 51) and ( 53), we also obtain On the other hand, due to (53) and (H1), we have and therefore, Finally, ( 50) and ( 54)-( 56) imply that (ũ, ṽ, w) is the optimal for problem (P ε ).This concludes the proof of the Lemma 3.
Lemma 4. Let w ε be optimal for problem (P ε ) and (u ε , v ε ) be the solution of (9) corresponding to w ε .Then on some subsequence ε n , Proof.Since w ε is solution to (P ε ), we have which, together with (49), implies that which, combined with (59), implies that which implies that w ε is bounded in L 2 (0, T ; U ). Without loss of generality, we may assume that w ε → w weakly in L 2 (0, T ; U ), which, together with Lemma 2, implies that there exists a sequence of ε n such that On the other hand, (50) and (61) imply that and thus, lim http://www.mii.lt/NAThus, we conclude from (50), ( 62) and (64) that Hence, ũ = u * , ṽ = v * , w = w * and Finally, it follows from Lemma 2 that This completes the proof.
3 The optimality condition for (P ε ) and (P) In the following, we derive the optimality condition for problem (P) by showing the relation between approximation problem (P ε ) and problem (P).We start this section with the necessary conditions for (u ε , v ε , w ε ) to be optimal for (P ε ).
Lemma 5. Suppose that β ε satisfies (7)-( 8) and (H1)-(H3) hold.Let (u ε , v ε , w ε ) be optimal for problem (P ε ).Then there exists a tetrad Proof.Let w ε be optimal for problem (P ε ) and (u ε , v ε ) be the solution of (9) corresponding to w ε .Set J. Zheng Now, owing to w ε is the optimal for problem (P ε ), we have (L ε (w χ ε )−L ε (w ε ))/χ 0 for all χ > 0. Hence, employing the same arguments as in the proof of [1], we conclude that where (y ε , ȳε ) is the solution of ∇g ε (t, u ε ) denotes the gradient of g ε to the second variable at u ε , ∇h(w ε ) denotes the gradient of h at w ε and Thanks to S is convex and closed, we may also infer that Let and (p ε , q ε ) be the solution of (68).Due to [1,Thm. 1.14], the boundary value problem (68) has a unique solution It follows from (68), ( 71) and (72) that which implies (69).This completes the proof.
On the other hand, by the same argument in [1], we obtain that on a subsequence, still denoted by ε, and Now, we will prove that In fact, let ψ : R → R be a smooth, bounded and monotone approximation of the signum function such that ψ(0) = 0 (see [1,Lemma 3.5]).Now, multiplying (68) 1 by ψ(p ε ) and integrating the resulting equations over [0, T ], we get where C 1 > 0 is independent of ε and γ(c, θ, b) is positive constant depending on c, θ and b.Here we have use the fact that and ψ ∈ L 2 (0, T ; V ) (see [1,Lemma 5.3]).Here and throughout the proof of Theorem 1, we shall denote by C i (i ∈ N) several positive constants independent of ε.Therefore, (85) implies that Nonlinear Anal.Model.Control, 21(1):18-39 Hence, by the above inequality, we infer that there exists η ∈ (L ∞ (Q T )) * such that Thus, (84) holds.
On the other hand, it follows from (74) and ( 75) that Therefore, there exist two generalized subsequences of µ ε and ζ ε such that Here we use the fact that µ ε and ζ ε are bounded on R and Z * , respectively.Using Lemma 4, we may pass to the limit in (69) and derive (6) 1 .