Problems of optimal control of mixed states (Q1914136)

The following optimization problem \[ \dot x=f(x,u(t)), \quad t\in [0,T], \qquad x(0)= x_0\in \mathbb{R}^n, \qquad \int_{\mathbb{R}^n} \Phi(x(T)) \rho_0(x_0) dx_0\to \inf_u \] is investigated, where \(u(t)\in U\subset \mathbb{R}^n\) is a piecewise continuous function, \(U\) is a closed bounded set; \(x(T)= x(T; x_0,u)\); the initial states \(x_0\) are not known before but are distributed with a probability density function \(\rho_0(x_0)\). The necessary conditions for optimality are obtained. From the theory of neural networks two model examples are given. Moreover, for the discrete analog of the considered problem a computing algorithm of optimal control is described.