*(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 ${\mathbb{R}}^{n}$. Considered are two types of problems: $\left({\mathcal{P}}_{D}\right)$ and $\left({\mathcal{P}}_{N}\right)$. In each problem, the control functions are fairly general, given by the form $v=v(x,t)$. In $\left({\mathcal{P}}_{D}\right)$, the control $v(x,t)$ 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 ${\Omega}\times (0,T)$, belongs to a dense subset $\mathcal{U}$ of ${L}_{+}^{p}({\Omega}\times (0,T))$. In $\left({\mathcal{P}}_{N}\right)$, $v$ and $f$ appear on the boundary condition of the Neumann type, and $v$ belong to a dense subset $\mathcal{U}$ of ${L}^{p}(\partial {\Omega}\times (0,T))$. For example, $\left({\mathcal{P}}_{N}\right)$ is written as

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).

##### MSC:

93B05 | Controllability |

93C20 | Control systems governed by PDE |

35B37 | PDE in connection with control problems (MSC2000) |

35K55 | Nonlinear parabolic equations |