Controllability results for weakly blowing up reaction-diffusion system ∗

In this paper, we consider the controllability of a general reaction-diffusion system with homogeneous Dirichlet boundary conditions. We prove the exact controllability to the trajectories and the approximate controllability of the system which contains certain superlinear nonlinearities. The Kakutani fixed point theorem, global Carleman estimates, and the regularity argument of the parabolic system are used.

Theorem 1.1 Let (H1) and (H2) be satisfied.Then for any f, g The result above may not be new, but it is difficult to find its proof.For the completeness of the text, we will give the proof in the Appendix at the end of this paper.
Let us now analyze the controllability property.Consider the solution of the problem (without control functions) with the initial and boundary conditions Let (u * , v * ) be an arbitrary bounded trajectory of (1.10)-(1.12)globally defined on [0, T ], T < T * , where T * ∈ (0, ∞] is the maximal existence time, corresponding to the data 3), we obtain where Then the system (1.13)-(1.15)can be rewritten as follows: where In this paper, we assume that (H3) 3) is said to be exactly controllable to the trajectories at time T < T * if for any initial data 3) is also defined on [0, T ] and satisfies (1.20) Remark 1.1 Clearly, the exact controllability to the trajectories of (1.1)-(1.3) is equivalent to the exact null controllability of (1.16)- (1.18).Therefore, we only need to prove the exact null controllability of (1.16)-(1.18).
In recent years, the controllabilities of the nonlinear parabolic systems have been studied by many authors (see [3]- [9] and the references therein).For reaction-diffusion systems, Anita and Barbu [3] have considered the local null controllability with f i (x, u, v) = α i a(x)uv, i = 1, 2, where the α i are the positive constants and a is a function in L ∞ (Ω) such that a ≥ a 0 > 0 a.e. in Ω, where a 0 is a constant.Wang and Zhang [6] extended that result to the systems with only one control force.In [7], F. Ammar Khodja, A. Benabdallah and C. Dupaix obtained local null controllability of a general reaction-diffusion system.There seems to have been relatively little work devoted EJQTDE, 2012 No. 11, p. 4 to the global exact controllability and the approximate controllability of the systems which contains certain superlinear nonlinearities.This is a precisely problem which we consider in this paper.It is worth mentioning that these nonlinearities may lead to the state of the systems blow-up without imposing any control functions (see Section 2).It is well-known that the blow-up phenomena is usually adverse to us in reality.Therefore, we want to prevent the blow-up phenomena happening.Our method is that we introduce some control functions into the systems to change the dynamics of the systems.The advantage of this method not only avoids the occurrence of the blow-up phenomena, but also leads the state to the ideal targets at the given time.In other words, the systems achieve the controllability.Since the control functions act on a subset of the domain where the state work on, the method is realizable in the point view of practice.
Motivated by the article [1], our main results are stated as follows: Suppose that (H1) and (H3) hold.Then the system 3) is exactly controllable to the trajectories at time T .
As a consequence of Theorem 1.2, we have the approximate controllability result.
Our results rest on a generalized fixed point theorem of Kakutani (see [10], p.7) which has been used in a variety of areas in differential equations and control theory (for instance see [1], [4]).
Theorem 1.4 (Kakutani) Let K be a compact convex subset of a Banach space X and let T : K → 2 X be an upper semicontinuous mapping with convex values T (x) such that T (x) ⊂ K, ∀x ∈ K. Then there is at least one x ∈ K such that x ∈ T (x).
The rest of this paper is organized as follows.In Section 2, we give some blow-up and global existence results to a special case of (1.1)-(1.3),which point out that blowup may occur.The exact controllability and the approximate controllability results are proved in the Section 3 and the Section 4 respectively.In Appendix at the end of the paper, we give the proof of Theorem 1.1 for the sake of completeness.

Blow-up and global existence of solutions for (1.1) -(1.3) in the absence of control functions
In this section, we will give an example to show that the solutions of (1.1)-(1.3)may occur blow-up phenomena, provided that F i (i = 1, 2, 3, 4) satisfy (H3).Our crucial theorem is the following result to be proved later.
In order to prove Theorem 2.1, we merely consider the classical solutions of the following weakly coupled reaction-diffusion system with initial and boundary conditions where u 0 , v 0 ∈ C 2,α 0 (Ω)(0 < α 0 < 1) are nonnegative functions and constants p i , q i ≥ 0, i = 1, 2.
For the local existence of a classical solution for (2.1)-( 2.3) we refer to Chapter 12 in [16].Many authors have considered the global existence and blow-up of solutions for some reaction-diffusion systems (see e.g.[13][14]).As far as we know, there are no the similar results for the reaction functions in (2.1).Therefore, we prove the global existence and the blow-up of the solutions for the system (2.1)-(2.3)first.
Let us begin with a single parabolic equation. where ) is a uniformly positive definite matrix, the coefficients of L are sufficiently smooth in Ω × [0, T ), φ is a Hölder continuous function in Ω, and f (u) is Lipschitz continuous in R. The following blow-up results are basic and well known (see [12]): 3) has a unique solution (u, v) and This lemma is a particular case of Theorem 2.2 in [13].Therefore, we omit the proof here.Now, we are in a position to show the following global existence result for (2.1)-(2.3).
If p i + q i = 1, the solution of the following linear system could be regarded as a upper solution of (2.1)-(2.3),since We can get the conclusion as the solution of linear system is global.Corresponding to Lemma 2.1 for the single equation case, we have the following blow-up result for (2.1)-(2.3).EJQTDE, 2012 No. 11, p. 7 Then the solutions of (2.1)-(2.3)blow-up in a finite time.
Proof.By the conditions, we can get Using Lemma 2.1 and Remark 2.1, we get immediately that u blow-up in a finite time.This implies the same conclusion about v as well.
Proof of Theorem 2.1.We consider the nonnegative solutions of (2.1)-( 2.3) with nonnegative initial data.Obviously, the condition (H3) holds by virtue of γ i < 3  2 .If we choose one of the initial data large enough and use γ i > 1, then all the conditions of Theorem 2.3 are satisfied.Then the blow-up phenomena will occur in the absence of control functions.

Proof of the exact controllability result
In this section, we are devoted to prove Theorem 1.2.The proof is based on the null controllability of the linear parabolic system and the Kakutani fixed point theorem.
EJQTDE, 2012 No. 11, p. 9 Lemma 3.2 (see [7]) Let λ 0 > 1, C being the constant given in Theorem 3.1.Then for any L ∞ (Q T ) and s ≥ −3, the solution (ψ, ζ) of (3.4)-(3.6)satisfies: where δ = τ ρ and I(s, z) From the results above, we can obtain the observability estimate as follows: Lemma 3.3 Under the assumptions of Lemma 3.2, the solution (ψ, ζ) of (3.4)-(3.6)satisfies: ) , where By the definition of function I and Lemma 3.2, × Ω ,we get  We obtain that the function e −2m(T −t) ( ψ(t ) is increasing in t.Then by the monotonicity and the mean value theorem of integral, we can get Our crucial lemma is the following: Suppose that (H1) and (H3) hold.Then the system where C T is defined in Lemma 3.3.
Proof.For any given (y, z) ∈ K R and any ε > 0, we consider the following optimal control problem where (f, g) ∈ L 2 (Q T ) and (u, v) is the solution of (3.1)-(3.3)corresponding to (f, g).

6 Lemma 2 . 2
(1 + u) with p > 1 satisfies the requirements in Lemma 2.1, and the solution of (2.4)-(2.6)may blow-up in a finite time.Since the reaction functions in (2.1) are quasi-monotone increasing, we have the following existence-comparison theorem.EJQTDE, 2012 No. 11, p.Let ( u, v) and ( u, v) be a pair of upper and lower solutions ([13]) of equations (2.1)-