Relaxation of optimal control problems in \(L^p\)-spaces (Q2701822)

The author considers a semilinear parabolic system NEWLINE\[NEWLINE {\partial y(t, x) \over \partial t} + Ay(t, x) + \Phi(t, x, y(t, x)) = 0 NEWLINE\]NEWLINE in an \(n\)-dimensional domain \(\Omega\) with boundary \(\Gamma,\) control \(v(t, x)\) applied through a Robin boundary condition NEWLINE\[NEWLINE {\partial y \over \partial n_A} + \Psi(t, x, y(t, x), v(t, x)) = 0.NEWLINE\]NEWLINE The control space is a suitable subset of \(L^p((0, T) \times \Gamma).\) Without convexity assumptions, the problem of minimizing a functional \(J(y, v)\) subject to this system with initial condition \(y(0, x) = y_0(x)\) may not have a solution, which makes it necessary to use relaxation. As the control set \(V\) is noncompact, construction of the relaxed controls \((t\)-dependent probability measures \(\mu(t))\) requires imbedding of \(V\) in a compact space, at least if one insists on the measures being \(\sigma\)-additive. The author defines such a compactification and studies various properties of the relaxed system, among them necessary optimality conditions.