Optimal control and minimax synthesis of constrained parabolic systems under uncertain perturbations

Abstract

This paper concerns a minimax control design problem for a class of parabolic PDE systems with nonregular boundary conditions and uncertain distributed perturbations under pointwise control and state constraints. We deal with boundary controllers acting through Dirichlet boundary conditions that are the most challenging for parabolic dynamics. The original motivation for this problem comes from an environmental problem of groundwater control by B.S. Mordukhovich (1986), which has important applications to agriculture, ecology, and other practical areas. The goal of the control is to neutralize the adverse effect of uncertain disturbances (in particular, weather conditions) on the dynamics of the groundwater level. In practice we usually do not have information about the magnitude of the disturbance and neither do we know its probability distribution. The only thing we know about the disturbance is the range of its possible values. Thus the above problem belongs to the class of feedback control problems with the groundwater level as the feedback parameter. Here we study a more general class of multidimensional parabolic control systems that covers a fairly broad range of practical applications.