Dynamical systems with controllable singularities: multi-scale and limit representations and optimal control

A new class of systems, the dynamical systems with controllable singularities, is considered. This class refers to systems that admit introduction of the impulsive control actions during singular phases of their motion, such as changes in dimension, discontinuities in the state, and other nonsmooth types of motion. A well-posed representation of the discontinuous in the limit behavior of these systems is given in terms of differential equations with measure and the corresponding generalized (discontinuous) solution of the new type is introduced. This representation, which admits discontinuity of the entire state, is put into correspondence with the detailed multi-scale system description via a space-time transformation followed by a limit procedure. Finally, using the framework developed, an approach to constructive optimal controller synthesis for this class of systems is presented.