An initial-boundary-value problem for a system of degenerate parabolic integro-differential equations is considered. The sufficient conditions for the existence and uniqueness of its generalized solution and for the existence of at least one optimal control for a given performance functional are obtained. A stable numerical solution to the initial-boundary-value problem is derived for a locally one-dimensional case and conditions are formulated for constructing a stable numerical algorithm of the optimal control problem on a class of piecewise-smooth control functions.
In this paper the following two connected problems are discussed. The problem of the existence of a stationary solution for the abstract equation ϵx″(t)+x′(t)=Ax(t)+∫−∞tE(t−s)x(s)ds+ξ(t),t∈R containing a small parameter ϵ in Banach space B is considered. Here A∈ℒ(B) is a fixed operator, E∈C([0,+∞),ℒ(B)) and ξ is a stationary process. The asymptotic expansion of the stationary solution for equation (1) in the series on degrees of e is given
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.