Admissibility and optimal control for difference equations (Q2785867)

The author considers an optimal control problem of the form \(x_{n+1} = A_nx_n + B_nu_n, \;n \geq 0,\;u = (u(0),u(1),....) \in K \subset l^2, \;\sum_{i\geq 0}g_ix^2(i) +\langle Qu,u\rangle \to \min\) with the partial initial condition \(Px(0) = \eta \in R^n\). Here \(K\) is a closed convex set and the linear operator \(Q: l^2 \to l^2\) is positively definite and self-adjoint. Assuming, in particular, admissibility of the pair \((l^2, l^{\infty})\) for the equation \(x_{n+1} = A_nx_n + f_n\), the author proves the existence of a unique minimizing control.