The existence of optimal spaces for linear functional equations (Q1873477)

The author deals with equations for (linear) functionals and not with linear functional equations. The topic of the paper is connected with optimal control problems. Given a metric space \(C_{\text{ad}}\) (of admissible controls), a (real and separable) Hilbert space \(F\) with inner product \([\cdot,\cdot]\) together with a family \(\{F_c\}_{c\in C_{\text{ad}}}\) of closed subspaces of \(F\), the author considers a continuous linear functional \(l: F\to \mathbb{R}\). Let \(u^0_c\in F_c\) be such that \([u^0_c, v]= l(v)\) for all \(v\in F_c\). Then, given some functional \(J: C_{\text{ad}}\times F\to \mathbb{R}\), under some assumptions on \(C_{\text{ad}}\), the family of \(F_c\)'s, and on \(J\), it is shown that 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 \(J\)).