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On the approximate controllability of some semilinear parabolic boundary-value problems. (English) Zbl 0936.93010

The paper studies the ${L}^{p}$-approximate controllability problem in time $T$ for a class of semilinear parabolic systems in a bounded domain ${\Omega }$ in ${ℝ}^{n}$. Considered are two types of problems: $\left({𝒫}_{D}\right)$ and $\left({𝒫}_{N}\right)$. In each problem, the control functions are fairly general, given by the form $v=v\left(x,t\right)$. In $\left({𝒫}_{D}\right)$, the control $v\left(x,t\right)$ and the nonlinear term $f\left(y\right)$, $y$ being the state of the system, enter the equation under a homogeneous boundary condition of the Dirichlet type, and $v$, supposed to be nonnegative on ${\Omega }×\left(0,T\right)$, belongs to a dense subset $𝒰$ of ${L}_{+}^{p}\left({\Omega }×\left(0,T\right)\right)$. In $\left({𝒫}_{N}\right)$, $v$ and $f$ appear on the boundary condition of the Neumann type, and $v$ belong to a dense subset $𝒰$ of ${L}^{p}\left(\partial {\Omega }×\left(0,T\right)\right)$. For example, $\left({𝒫}_{N}\right)$ is written as

$\frac{\partial y}{\partial t}-{\Delta }y=0,\phantom{\rule{1.em}{0ex}}\frac{\partial y}{\partial \nu }+f\left(y\right){|}_{\partial {\Omega }}=v,\phantom{\rule{1.em}{0ex}}y\left(·,0\right)={y}_{0}·$

In both cases, the problem in the linear case where $f\left(y\right)=0$ is first solved. Based on this result, the semilinear problem is solved as a perturbation of a linear problem cancelling the nonlinear term. The problem is also studied via the Kakutani fixed-point theorem for other semilinear parabolic systems including multivalued coefficients (a case of flux boundary controls).

Reviewer: T.Nambu (Kobe)
##### MSC:
 93B05 Controllability 93C20 Control systems governed by PDE 35B37 PDE in connection with control problems (MSC2000) 35K55 Nonlinear parabolic equations