Completion by gamma-convergence for optimal control problems (Q1327528)

In many optimal control problems one is led to solve \(\min\{J(u, y): u\in U,\;y\in \arg\min G(u,\cdot)\}\), where \(y\) is the state variable varying in a metric space \(Y\), \(u\) is the control variable varying in a set \(U, J: U\times Y\to \overline{{\mathbf R}}\) is the cost functional and \(G: U\times Y\to \overline{{\mathbf R}}\) is the so-called state functional. In the paper, in order to apply the standard direct methods of calculus of variations, a topology on \(U\) is first introduced and then, under some additional assumptions, the existence of an optimal pair is proved. Such topology on \(U\) is related to the \(\Gamma\)-convergence of the mappings \(G(u,\cdot)\). A relaxed formulation of the above minimum problem is proposed when no lower semicontinuity hypotheses on \(J\) are assumed; this is done by considering a relaxed function of \(J\) and relaxed controls in the completion of \(U\) in the introduced topology. Finally, some examples are discussed: the first one dealing with shape optimization problems in which the control set \(U\) is the class of all domains contained in a given open subset \(\Omega\) of \({\mathbf R}^ n\); the others with problems where the control is on the coefficients of the differential state equation.