On the existence of optimal spaces for nonlinear functional equations (Q1874977)

The author deals with nonlinear equations for (linear) functionals and not with nonlinear functional equations. The topic of the paper is connected with optimal control problems. Let \(C_{\text{ad}}\) be a metric space (of admissible controls), let \(F\) be a (real and separable) Hilbert space with inner product \([\cdot,\cdot]\) and let \(\{F_c\}_{c\in C_{\text{ad}}}\) be a family of closed subspaces of \(F\). Furthermore, let \(W: F\to F\) be a nonlinear operator with continuous inverse \(W^{-1}\) such that for all \(c\) the conditions \(v\in F_c\) and \(W(v)\in F_c\) are equivalent. Given a continuous linear functional \(l: F\to \mathbb{R}\), let \(u^0_c\in F_c\) be such that \([W(u^0_c), v]= l(v)\) for all \(v\in F_c\). Then, given some functional \(J: C_{\text{ad}}\times F\to \mathbb{R}\), the author shows that under some assumptions on \(C_{\text{ad}}\), the family of \(F_c\)'s, and on \(J\), there is (at least) one \(c^*\in C_{\text{ad}}\) such that \(J(c^*, u^0_{c^*})\leq J(c, u^0_c)\) for all \(c\in C_{\text{ad}}\). \(c^*\) is called an optimal control and \(F_{c^*}\) an optimal space (for \(l\) with respect to \(J\)).