Exact controllability to the trajectories for parabolic PDEs with nonlocal nonlinearities

Abstract

This paper deals with the analysis of the internal control of a parabolic PDE with nonlinear diffusion, nonlocal in space. In our main result, we prove the local exact controllability to the trajectories with distributed controls, locally supported in space. The main ingredients of the proof are a compactness–uniqueness argument and Kakutani’s fixed-point theorem in a suitable functional setting. Some possible extensions and open problems concerning other nonlocal systems are presented.

Appendix: proof of Lemma 2.2

Let $w \in Z$ be given, and let us set

$$\begin{aligned} \displaystyle \alpha (t):=a\Bigl (\int _\Omega w(x',t)\,\hbox {d}x'\Bigr ). \end{aligned}$$

Let us assume that $\varphi$ satisfies

$$\begin{aligned} \left\{ \begin{array}{lllll} -\varphi _{t} - \alpha (t) \Delta \varphi = 0 &{}\quad \text{ in } &{} \, \,Q,\\ \varphi =0 &{}\quad \text{ on } &{}\,\, \Sigma ,\\ \varphi (x,T)=f(x) &{}\quad \text{ in } &{}\,\, \Omega . \end{array}\right. \end{aligned}$$

From the standard Carleman estimates for $\varphi$, one has

$$\begin{aligned} \iint _Q e^{-C_0/(T-t)} |\varphi |^2\,\hbox {d}x\,\hbox {d}t \le \tilde{C}_0 \iint _{\omega \times (0,T)}e^{-2s\sigma } \xi ^{3}|\varphi |^2\,\hbox {d}x\,\hbox {d}t, \end{aligned}$$

(5.1)

where $C_0$ only depends on $\Omega$, $\omega$, T, $a_0$, $a_1$. M and $\Vert w\Vert _{Z}$ and $\tilde{C}_0$ only depends on $\Omega$, $\omega$, $a_0$, $a_1$, M and $\Vert w\Vert _{Z}$; see [7, 9]. Taking into account that

$$\begin{aligned} \varphi (x,t)=\sum _{j \ge 1} e^{-\lambda _j \int _t^T \alpha (s)\,\hbox {d}s} (f,\phi _j) \phi _j \end{aligned}$$

and

$$\begin{aligned} \Vert \varphi \Vert ^2 = \sum _{j \ge 1} e^{-2\lambda _j \int _t^T \alpha (s)\,\hbox {d}s} |(f,\phi _j)|^2, \end{aligned}$$

we see from (5.1) that

$$\begin{aligned} \int _0^T\sum _{j \ge 1} e^{-2\lambda _j \int _t^T \alpha (s)\,\hbox {d}s-C_0/(T-t)} |(f,\phi _j)|^2\,\hbox {d}t \le \tilde{C}_0 \iint _{\omega \times (0,T)}e^{-2s\sigma } \xi ^{3}|\varphi |^2\,\hbox {d}x\,\hbox {d}t, \end{aligned}$$

and consequently,

$$\begin{aligned} \sum _{j \ge 1} \left( \int _0^T e^{-2\lambda _j a_1(T-t)-C_0/(T-t)}\,\hbox {d}t\ \right) |(f,\phi _j)|^2 \le \tilde{C}_0 \iint _{\omega \times (0,T)} e^{-2s\sigma }\xi ^{3}|\varphi |^2\,\hbox {d}x\,\hbox {d}t. \end{aligned}$$

The asymptotic behavior of the integrals in the left hand side is well known. Indeed, one has

$$\begin{aligned} \int _0^T e^{-2\lambda a_1(T-t)-C_0/(T-t)}\hbox {d}t \thicksim \left( \dfrac{\pi ^2 C_0}{4(\lambda a_1)^3}\right) ^{1/4} e^{-4 \sqrt{C_0 a_1 \lambda }}, \,\,\, \text{ as }\,\, \lambda a_1\rightarrow \infty \end{aligned}$$

(see, for instance, [8]). Thus, there exists $\tilde{C}_1$, again depending on $\Omega$, $\omega$, $a_0$, $a_1$, M and $\Vert w\Vert _{Z}$, such that

$$\begin{aligned} \int _0^T e^{-2\lambda _j a_1(T-t)-C_0/(T-t)}\,\hbox {d}t \ge \tilde{C}_1 e^{-2 R_0 \sqrt{\lambda _j}}\,\,\,\ \forall \, j\ge 1. \end{aligned}$$

Obviously, this leads to (2.1) and ends the proof.