Stochastic generalized gradient methods in problems of dynamic systems synthesis (Q2366326)

Let be given a dynamic system \(\dot z=F(\xi,\eta,z,u)\) the right-hand part of which contains a control \(u\in U\), a vector of parameters \(\xi\in \Xi\) which should be optimized, and a factor of uncertainty \(\eta\). The factor of uncertainty belongs to \(N(\xi,\omega)\in R^ n\) where \(\omega\) is a stochastic factor. Consider the problem \(K(\xi)=\text{Mathematical expectation}_ \omega\sup_ \eta\inf_{u(.)}J(\xi,\omega,\eta,u)\to \inf_ \xi\) where \(J(\xi,\omega,\eta,u)=\int_{t_ 0}^ t\varphi(\xi,z,u)dt+\varphi^ 0(\xi,\omega,z(t_ 1 ))\), where \(z(t)\) is a trajectory of the dynamic system with \(u=u(t)\), \(t\in [t_ 0,t_ 1]\) and \(z(t_ 0)\) is a function depending on \(\omega,\eta\). In the paper this problem is approximated by a finite dimensional minimax problem. For the minimax problem an algorithm like stochastic method of generalized gradient is considered. For some cases it is proved that this algorithm converges.