Minimum period control problem for an infinite dimensional system (Q1389923)

Let \(X\) be a separable reflexive Banach space, \(S\) a bounded closed sphere in \(X\), and \(e^A\) be a compact \(c_0\)-semigroup on \(X\). Moreover, let \(U\) be a bounded subset of some Banach space \(Z\), and let \({\mathcal U}_0= \{u:u: \mathbb{R}_+\to U\) is measurable\}. The authors are interested in minimum period control problems governed by the following law \[ x(t; x_0,u(0))= e^{tA}x_0+ \int^t_0 e^{(t- \tau)A}B(u(\tau))d\tau,\quad t\in\mathbb{R}_+, \] with \(x_0\in S\), \(u\in{\mathcal U}_0\) and \(B: Z\to X\) is a continuous mapping. First, the authors study an infinite time optimal control problem with mixed type target set. To the latter problem complete results are established. Moreover, these results are applied to derive existence results for minimal period control systems.