On the approximate controllability of some semilinear parabolic boundary-value problems
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: \(({\mathcal P}_D)\) and \(({\mathcal P}_N)\). In each problem, the control functions are fairly general, given by the form \(v= v(x,t)\). In \(({\mathcal P}_D)\), the control \(v(x,t)\) and the nonlinear term \(f(y)\), \(y\) 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 \(({\mathcal P}_N)\), \(v\) 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, \(({\mathcal P}_N)\) is written as \[ {\partial y\over\partial t}-\Delta y= 0,\quad {\partial y\over\partial\nu}+ f(y)\bigl|_{\partial\Omega}= v,\quad y(\cdot, 0)= y_0. \] In both cases, the problem in the linear case where \(f(y)= 0\) 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).
scientific article; zbMATH DE number 1129436