Optimal programmed controls: Existence and approximation (Q2784552)

The paper is concerned with the optimal control problem \(g(x(1)) \to \inf\), subject to \({\dot x}(t)=\varphi(t, x(t), u(t), \omega)\), \(u(t) \in U_t\) for a.e. \(t \in (0,1)\), \(x(0)=x_0\), and \(x(\cdot) \in X\) a.s., where \(\varphi(t,x,u,\omega)\) is an \(\mathbb{R}^n\)-valued function on \((0,1) \times \mathbb{R}^n \times U \times \Omega\), the parameter \(\omega \in \Omega\) and the initial value \(x_0\) are assumed to be random, \(U_t \subset U\) with \(U\) compact and \(X \subset C(0,1;\mathbb{R}^n)\). The author proves the existence of solution of the problem. For obtaining suboptimal controls a method is considered that uses the discretization of the random parameters. It is shown that the results can be extended to stochastic optimal control problems with distributed parameters.