Asymptotic analysis of some control problems (Q2730764)

The system is described by a semilinear parabolic equation with distributed and (Dirichlet) boundary control, NEWLINE\[NEWLINE{\partial y(t, x) \over \partial t} + Ay(t, x) + \Phi(t, x, y(t, x)) = u(t, x) \quad (0 \leq t \leq T, \;x \in \Omega), NEWLINE\]NEWLINE NEWLINE\[NEWLINEy(t, x) = v(t, x) \quad (0 \leq t \leq T, \;x \in \Gamma), \quad y(0, x) = y_0(x) \quad (x \in \Omega)NEWLINE\]NEWLINE in a bounded domain \(\Omega\) with boundary \(\Gamma.\) The distributed control \(u(t, x)\) belongs to a closed convex subset \(U_{ad} \subset L^q((0, T) \times \Omega)\) and the boundary control to a bounded closed subset \(V_{ad} \subset L^\infty((0, T) \times \Gamma).\) The problem is to minimize an integral cost functional under the control constraints plus a state constraint of the form \(g(y) \in {\mathcal C},\) where \(g\) is a continuous map \(g : C(K) \to C(K),\) \(K\) a compact subset of \((0, T] \times \Omega .\) NEWLINENEWLINENEWLINETo bypass the complications associated with a nonsmooth Dirichlet boundary condition, the authors approximate the problem replacing the boundary condition by NEWLINE\[NEWLINE {\partial y(t, x) \over \partial n} + \alpha y(t, x) = \alpha v(t, x) NEWLINE\]NEWLINE \((\partial / \partial n\) the conormal derivative, \(\alpha\) a penalty parameter \(\to \infty).\) If the state constraint is let unmodified the minimum of the functional for the approximate problem may not approach the minimum for the original problem as \(\alpha \to \infty.\) This difficulty is overcome by using the approximate state constraint NEWLINE\[NEWLINE \inf_{z \in {\mathcal C}}\|z - g(y)\|_{C(K)} \leq \varepsilon \to 0.NEWLINE\]