Local Exact Controllability to Stationary Solutions of a Semilinear Parabolic Equation

TL;DR

This paper proves the local exact controllability of a quasilinear porous media equation with Dirichlet boundary conditions, showing that small initial deviations can be driven to stationary solutions within finite time.

Contribution

It establishes the first local exact controllability result for the quasilinear porous media equation with specific boundary conditions and regularity assumptions.

Findings

Controllability holds for small initial deviations from stationary solutions.

Finite time controllability is achieved under regularity conditions on the stationary solution.

The results apply to equations with smooth nonlinearities and bounded domains with smooth boundaries.

Abstract

This paper establishes the local exact controllability of the quasilinear porous media equation with Dirichlet boundary condition.\\ Consider the equation $$\begin{aligned} &y_t - \Delta a(y) = mu+f \text{ on } Q\\ &y(0)=y_0,\ y \mid_{\Sigma} = 0 \end{aligned}$$ on the $n+1$-dimensional cylinder $Q = \Omega \times (0, T)$ with lateral boundary $\Sigma = \partial \Omega \times (0, T)$. The exact controllability in finite time is proved when $\|y_0 - y_s\|_{W^{1, n}_0(\Omega) \cap C(\overline \Omega)}$ is sufficiently small, $n > 1$, for every stationary solution $y_s$ such that $a(y_s) \in W^{2, q}(\Omega)$, where $q>n$. It is assumed that $\Omega$ is a bounded open set with $C^2$ boundary and that $a \in C^2(\mathbb R)$, $a'>0$.