Nonlinear Feedback Types in Impulse and Fast Control

It is well-known that a system with linear structure subjected to bounded control inputs for optimal closed-loop control yields nonlinear feedback of discontinuous bang-bang type. This paper investigates new types of nonlinear feedback in the case of optimal impulsive closed-loop control which may naturally generate discontinuous trajectories. The realization of such feedback under impulsive inputs that are allowed to use δ-functions with their higher derivatives requires physically realizable approximations. Described in this paper is a new class of realizable feedback inputs that also allows to produce smooth approximation of controls. Such approach also applies to problems in micro time scales that require so-called fast or ultra-fast controls.