Melnik, R.V.N.
Non-smooth dynamic systems appear routinely in many applications of control and game theory. By constructing mathematical models in these fields we often have to deal with evolutionary partial differential equations (PDEs), solution of which are not smooth enough to satisfy these equations in the classical sense. A classical example is provided by Hamilton-Jacobi-Bellman (HJB) equations that approximately describe dynamics of value/cost functions which may not be differentiable everywhere. These equations can be formally derived by applying Bellman's dynamic programming approach in a heuristic manner, and much research in control theory has been devoted to the clarification of the connection between this approach and the Pontryagin maximum principle in the non-smooth and stochastic cases. This paper is devoted to the analysis of this connection under relaxed smoothness requirements in non-reflexive Banach spaces and to the approximation of HJB equations in these spaces.
The author proposes a unified treatment of non-smooth and stochastic control problems using mathematical models of HJB-type obtained with Steklov's operator technique. Until recently, the analysis of such models has been predominantly conducted in reflexive Banach spaces. The situation has changes since the introduction of the L1 control theory. Due to new and emerging applications of control systems in engineering, non-reflexive Banach spaces become the most reasonable functional classes for perturbations in such problems where control systems have to work in a dynamic and uncertain environment. This paper contributes to the construction and the analysis of approximate PDE models for the description of control systems in such situations.
Key words: evolutionary problems; non-smooth dynamic systems; control and game theory problems; non-reflexive Banach spaces; Hamilton-Jacobi-Bellman equations; Steklov's operator technique; HJB approximations; Bellman's dynamic programming approach; perturbed Hamiltonians; Pontryagin maximum principle; control systems in dynamic and uncertain environments.