Ensemble Control of Dynamic Systems with Square Impulses

Abstract

In the article, the basic mathematical concepts of quadratic-impulse control are extended to dynamical systems in the space of probability measures (local continuity equations) representing dynamic ensembles of (infinitely many) homotypic individuals. Starting from a model with quadratic dependence on an unbounded control, we constructively design its impulsive trajectory extension, which, as a byproduct, ensures also the well-posedness of an associated optimal control problem. The main attention is paid to the characteristic control ODE whose extension combines a discontinuous time change with a convexification of the velocity set via generalized controls. Finally, we show that it is possible to exclude the generalized controls and represent extended states as impulsive solutions of a measure-driven ODE. The resulting impulsive continuity equation can be viewed as a measure-driven equation in the space of measures.