Stability of elliptic optimal control problems (Q5948785)

The paper considers a family \(\{\text{P}_h\}\) of minimization problems (depending upon a functional parameter \(h\)) NEWLINE\[NEWLINE\begin{aligned} &I_h= \int_\Omega F(x,z_u(x),\nabla z_u(x), u(x), h(x)) dx\to \min\\ & (u,z_u)\in U\times H^1_0(\Omega, \mathbb{R}^m),\end{aligned}\tag{\(\text{P}_h\)}NEWLINE\]NEWLINE where \(U\) is the set of admissible controls and \(z_u\) is the minimizer of the functional (for a chosen \(u\) and a fixed \(h\)) NEWLINE\[NEWLINEJ_h= \int_\Omega f(x, z(x),\nabla z(x), u(x), h(x)) dx.NEWLINE\]NEWLINE Here \(\Omega\subset \mathbb{R}^n\) is a bounded Lipschitz domain, the integrand \(f\) is convex in \((z,\nabla z)\), quadratic in \(\nabla z\) and affine in \(u\), the integrand \(F\) is convex in \((\nabla z,u)\). Under some continuity and growth properties of \(f\) and \(F\) the authors give continuity properties of the sets of solutions of \((\text{P}_h)\) with respect to \(h\).