THREE-DIMENSIONAL

. This paper is concerned with optimal boundary control of a three dimensional reaction-diﬀusion system in a more general form than what has been presented in the literature. The state equations are analyzed and the optimal control problem is investigated. Necessary and suﬃcient optimality conditions are derived. The model is widely applicable due to its generality. Some examples in applications are discussed.


1.
Introduction.We investigate a three-dimensional semi-linear parabolic reactiondiffusion system.The model may arise in a chemical or biological process where the species involved are subject to diffusion and reaction among each other.As an example, we consider the reaction A + B → C which obeys the law of mass action.To simplify the discussion, we assume that the backward reaction C → A + B is negligible and that the forward reaction proceeds with a constant (e.g., not temperaturedependent) rate.This leads to a coupled semilinear parabolic system in three spatial dimensions for the respective concentrations (see,e.g., [3,8]) as the following: x ∈ Ω, 326 W. YANG, J. SUN AND S. ZHANG w(x, 0) = w 0 (x) x ∈ Ω.
The first boundary equation ∂ n u = c(t) describes the boundary flux of the first substance A by means of a given shape function c(t), while c(t) denotes the control intensity at time t, which is to be determined.The remaining homogeneous Neumann boundary conditions simply correspond to a "no-outflow" condition of the substances through the boundary of the reaction vessel Ω.Our goal is to drive the above reaction-diffusion system from the given state to a desired state with minimal cost min J(u, v, w, c) , where the real constants α u , α v , α w , α uT , α vT , α wT , and α c are nonnegative.The given desired terminal states u Q , v Q , w Q , u Ω , v Ω , w Ω are elements of L 2 (Q), L 2 (Ω), respectively.This problem belongs to the class of optimization problems with PDF constraints [11], in particular, it is an optimal control problem with semilinear parabolic equation system, to which quite a number of publications were devoted, see e.g., [3,12].We note that, for instance, in [15] a nonlinear boundary condition of Stefan-Boltzmann type was considered.The papers [4,9,10] studied a nonlinear phase field model, and the papers [2,14] discussed on the Pontryagin principle for parabolic control problems.Further references on the control of nonlinear parabolic equations can be found in the monographs [3,11,16].
The present paper studies the optimal boundary control problems governed by a system of semilinear parabolic PDEs in a very general form.Similar, but more specific optimal control problems were discussed first in the PhD thesis [6] by Griesse and later extended by Griesse and Volkwein [7,8].In contrast, we became interested in dealing with more general nonlinearities.Specifically, we consider the optimal control problems (2) subject to the following class of three-dimensional reactiondiffusion systems with general right-hand terms together with the Neumann boundary conditions and the box constraint on control c Here, ε is a nonnegative constant.The functions c a , c b are given in L ∞ (Σ) such that c a ≤ c b holds almost everywhere in Σ.The control c(t) ∈ C ab enters the righthand side of (4), which is the first one in the inhomogeneous Neumann condition.
In [7,8] Griesse and Volkwein studied system (3) with the following form of and added a penalized integral constraint.The cost function J(u, v, w, c) takes the following form Under a coercivity condition on the Hessian of the Lagrange function, the optimal value of problem ( 8) with ( 7) is shown to be a directionally differentiable function of the perturbation parameters such as the reaction and diffusion constants and initial states.The derivative, termed parametric sensitivity, is characterized as the solution of an auxiliary linear-quadratic optimal control problem.
In [1], Barthel, John, and Tröltzsch considered a class of boundary control problems which are governed by a semi-linear two-dimension reaction-diffusion systems with point-wise control constraints with the following form of and, after discussing existence and uniqueness of the state equation with both linear and nonlinear boundary conditions, the existence of an optimal solution is shown.Necessary and sufficient optimality conditions are then derived.
Compared with [1] and [7,8], we consider more general f 1 , f 2 , and f 3 , which allow wider applications of the model.We show the existence and uniqueness of a solution to the system of the state equations ( 3)-( 6) with a general nonlinear form.The existence of a solution to the optimal control problem and necessary and sufficient optimality conditions are also derived.
The article is organized as follows.In Section 2, the state equations are analyzed and the optimal control problem is investigated.Section 3 is devoted to the necessary and sufficient optimality conditions.Some application examples are discussed in Section 4.
2. The existence of the solution for state systems and optimal control problem.In this section, we consider the systems (3)-( 6) with nonlinear boundary conditions.To show an existence and uniqueness theorem for the nonlinear system, we invoke the method of ordered upper and lower solutions.This method was also used in [1,6,13] for the similar problem with different boundary conditions mentioned in the introduction.
We make the following blanket assumptions.
(H) 3 : The function b satisfies b(x, t, 0) ≤ c a (x, t) for all (x, t) ∈ Σ and We give the definition of a weak solution of system (3)-(5).
We next present sufficient conditions so that the L 2 -norm of the sum of concentrations u, v, w does not increase with time.
The proof is similar to the proof of theorem 2.1.We omit it.The following theorem guarantees that problem (2)-( 6) has a solution.
Proof.The claim follows by standard arguments.Let be a minimizing sequence in C ad for the non-negative cost J.Since J is radially unbounded, it follows from Theorem 2.1 that this sequence is bounded in X := (W (0, T )) 3 × L 2 (0, T ).
3. Necessary and sufficient optimality conditions.Let S : c → (u, v, w) be the control-to-state operator with S : L r (Σ) → Y 3 , where we fix r > N +1 throughout the following.Since the cost functional is quadratic, we obtain the next lemma by standard arguments.
On the twice continuously Fréchet-differentiability of the operator S, we have that with the boundary conditions, ∀(x, t) ∈ Σ and the initial conditions, ∀x ∈ Ω in the following sense.
We consider these operators with image in C( Q) and reformulate the nonlinear system of (3) as Since S uQ , S uΣ , S u0 , S vQ , S v0 , S wQ , and S w0 are linear and continuous and To use the implicit function theorem, we have to show the boundedness and continuous invertibility of the partial Fréchet derivative F (u, v, w, c) = (F 1 , F 2 , F 3 ) T .To verify these properties, we first mention that the equation Since the mapping z → y is not smooth, we substitute r i = z i − y i , i = 1, 2, to obtain the equivalent system together with the Neumann boundary conditions and the initial conditions For every z ∈ (C( Q)) 3 , this linear boundary value problem has a unique solution r ∈ Y 3 cf.e.g., Theorem 5.5 in [1].The mapping z → r is continuous, so is the mapping z → y.Therefore, we can invoke the implicit function theorem and obtain that the control-to-state operator S is twice continuously Fréchet differentiable.
Corollary 1.The derivative of the control-to-state operator S at c in direction c is given by where (u, v, w) is the weak solution of the linearized equation obtained by linearizing system (4) at S(c) = (ū, v, w) together with the Neumann boundary conditions and the initial conditions Proof.Let us briefly sketch the proof and omit the details.The system (4) is of the form where Therefore there holds where Similarly, we can obtain the second-order derivative of S. Let f (c) := J(u(c), v(c), w(c), c) = J(S(c), c).
Corollary 2. The second-order derivative of S at c in direction (ĉ, ĉ) is given by where (u, v, w) is the weak solution of the system (where together with the Neumann boundary conditions (where (x, t) ∈ Σ) and the initial conditions (where x ∈ Ω) Proof.Differentiating (74) with respect to c in direction c 2 yields where Notice that where and From ( 70), (72), and (78), we have that The proof is completed.
To formulate necessary optimality conditions, let c be an optimal control of ( 2)-( 6) with states (ū, v, w).Then we have (u, v, w) = S(c).Let us write for short y = (u, v, w), ȳ = (ū, v, w).We can obtain the following standard result.Lemma 3.3.(cf.[16]).Every locally optimal control function c of ( 2)-( 6) satisfies the variational inequality We determine f by the chain rule and obtain for the direction c In the next, we eliminate the states u and v in (82) by adjoint states (ϕ 1 , ϕ 2 , ϕ 3 ), defined as the solutions of the adjoint system with the boundary conditions and the initial conditions Lemma 3.4.If (u, v, w) is the weak solution of the linearized system (64)-( 66) and (ϕ 1 , ϕ 2 , ϕ 3 ) is the solution of the adjoint system (83)-( 85), then it holds for all From this lemma and (82), we can obtain that It follows that Theorem 3.5.Every locally optimal solution c of ( 2)-( 6) satisfies, together with the adjoint states (ϕ 1 , ϕ 2 , ϕ 3 ) of the adjoint system (83)-( 85), the variational inequality An equivalent pointwise expression of this variational inequality is which leads in a standard way to the projection formula for almost all (x, t) ∈ Σ, where P Next, we consider the sufficient second order optimality conditions for (2)- (6) .By the chain rule, we derive for y = (u, v, w) that (93) Therefore, we have by using the adjoint states (ϕ 1 , ϕ 2 , ϕ 3 ).We then deduce To formulate our sufficient optimality conditions in a more convenient form, we introduce the Lagrange function £(u, v, w, c, ϕ 1 , ϕ 2 , ϕ 3 ) = J(u, v, w, c)+ In view of (93), we obtain To obtain the second-order sufficient conditions, ∀τ > 0, we define A τ (c) := {(x, t) ∈ Σ : |ϕ 1 + α c c| > τ } as the set of strongly active restrictions for c.The τ -critical cone C τ (c) is made up of all c ∈ L ∞ (Σ) with c(x, t) = 0, (x, t) ∈ A τ (c), c(x, t) ≥ 0, for c(x, t) = c a , and (x, t) ∈ A τ (c), c(x, t) ≤ 0, for c(x, t) = c b , and (x, t) ∈ A τ (c).
We have the following second-order sufficient conditions.Theorem 3.6.(Second-order sufficient conditions).Suppose that the control function c satisfies the first-order necessary optimality conditions of Theorem 3.5.If there exist positive constants δ and τ such that holds for all c ∈ C τ (c) and all (u, v, w) ∈ Y 3 satisfying the linearized equation ( 64)-(66), then we find positive constants ε and σ such that the quadratic growth condition holds for all c ∈ C ab with c − c L ∞ (Σ) .Therefore, the control function c is locally optimal in the sense of L ∞ (Σ).
Then, based on the results in Sections 2 and 3, we have the following results.
We introduce the adjoint system as following, ∀(x, t) with the boundary conditions ∀ (x, t) ∈ Σ and the initial conditions ∀x ∈ Ω   (116) Then, the conditions (H) 1 , (H) 2 , and (H) 3 are satisfied, and similar results to Theorems 4.4-4.6 can be obtained.Therefore we can extend the classical results in the literature to the general form where interactions among u, v, and w arise in the right-hand side of system (2).We omit the details.