Null controllability for a parabolic equation with nonlocal nonlinearities

doi:10.1016/j.sysconle.2011.09.017

Systems & Control Letters

Volume 61, Issue 1, January 2012, Pages 107-111

https://doi.org/10.1016/j.sysconle.2011.09.017 Get rights and content

Abstract

In this paper, we establish a local null controllability result for a nonlinear parabolic PDE with nonlocal nonlinearities. The result relies on the (global) null controllability of similar linear equations and a fixed point argument. We also analyze other similar controllability problems and we present several open questions.

Section snippets

Introduction and main result

Let $Ω$ be a bounded domain of $R^{N}, N \geq 1$ , with boundary $Γ = \partial Ω$ of class $C^{2}$ . We fix $T > 0$ and we consider the cylinder $Q = Ω \times (0, T)$ of $R^{N + 1}$ , with lateral boundary $Σ = Γ \times (0, T)$ . We also consider a non-empty (small) open set of $ω \subset Ω$ ; as usual, $1_{ω}$ denotes the characteristic function of $ω$ .

Throughout this paper, $C$ (and sometimes $C_{0}, K, K_{0}, \dots$ ) will denote various positive constants. Frequently, we will emphasize the fact that $C$ depends on (say) $f$ by writing $C = C (f)$ . The inner product and norm in $L^{2} (Ω)$ will be denoted,

Analysis of the controllability of the linearized system

Let us fix $T > 0$ and let us assume that the functions $B_{i j}$ satisfy (1.5), (1.6), (1.7).

In this section we will prove that, for any $z \in Z$ , the linear system ${\begin{matrix} u_{t} - B (t; z) u = v 1_{ω} in Q, \\ u (x, t) = 0 on Σ, \\ u (x, 0) = u^{0} (x) in Ω, \end{matrix}$ is null controllable and, also, that we can find control-state pairs $(v, u)$ satisfying (1.3), (1.4) and, moreover, ${‖ (v, u) ‖}_{W} \leq C ({‖ z ‖}_{Z}) {| u^{0} |}_{L^{2}} .$

Here, $B (t; z) u ≔ \sum_{i, j = 1}^{N} B_{i j} (z (\cdot, t), t) \frac{\partial^{2} u}{\partial x_{i} \partial x_{j}}$ and the spaces $Z$ and $W$ are given by: $Z = {z \in L^{1} (0, T; L^{2} (Ω)) : z_{t} \in L^{\infty} (0, T; L^{2} (Ω))},$ $W = {(v, u) : v \in C^{κ, κ / 2} (\bar{ω} \times [0, T]), u \in C^{2 + κ, 1 + κ / 2} (\bar{Q}$

The local null controllability of the nonlinear problem

This section is devoted to prove the main result in this paper, namely Theorem 1.1. It will be a consequence of the results in Section 2 and Kakutani’s Theorem.

Let us fix $R > 0$ and let us denote by $B_{R}$ the closed ball in $Z$ of center 0 and radius $R$ . Remember that $Z = {z \in L^{1} (0, T; L^{2} (Ω)) : z_{t} \in L^{\infty} (0, T; L^{2} (Ω))}$ .

For each fixed $z \in Z$ , we consider the null controllability problem for the associated linear system (2.1). In view of Theorem 2.3, there exist control-states $(v, u) \in W$ satisfying (2.17) and, consequently, $‖ ($

Further remarks and open questions

This section is devoted to present several additional remarks and open problems concerning controllability problems of the kind (1.1), (1.2), (1.3).

First, we point out that, from Theorem 1.1 we can deduce the null controllability of (1.1) for large $T$ , i.e. the following holds:

Theorem 4.1

Let us assume that, for any $T > 0$ , one has (1.5). Then, for each $u^{0} \in L^{2} (Ω)$ , there exists $T^{*} = T^{*} (u^{0})$ with the following property: if $T > T^{*}$ , there exist controls $v \in L^{2} (ω \times (0, T))$ and associated solutions to (1.1) satisfying (1.3).

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Citation Excerpt :
The local null controllability of (1) can be proved following other strategies. See for instance [8], where a similar result was established using a fixed point method (that is, rewriting the controllability problem as a fixed point equation in an appropriate space). On the other hand, the global null controllability of (1) is an open question.
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Null controllability for a parabolic equation with nonlocal nonlinearities

Abstract

Section snippets

Introduction and main result

Analysis of the controllability of the linearized system

The local null controllability of the nonlinear problem

Further remarks and open questions

Ann. Inst. H. Poincaré Anal. Non Linéaire

Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms

Asymptot. Anal.

Remarks on a nonlocal problem involving the Dirichlet energy

Rend. Semin. Mat. Univ. Padova

On some model diffusion problems with a nonlocal lower order term

Chinese Ann. Math. Ser. B

Vibrations of elastic strings: mathematical aspects—part one

J. Comput. Anal. Appl.

On the controllability of parabolic systems with a nonlinear term involving the state and the gradient

SIAM J. Control Optim.

Approximate controllability of the semilinear heat equation

Proc. Roy. Soc. Edinburgh Sect. A