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This paper describes the theory of feedback control in the class of inputs which allow delta-functions and their derivatives. It indicates a modification of dynamic programming techniques appropriate for such problems. Introduced are physically realizable bang-bang-type approximations of the “ideal” impulse-type solutions. These may also serve as “fast” feedback controls which solve the terminal control problem in arbitrary small time. Examples of damping high-order oscillations in finite time are presented.

As well known, problems of impulse control had been among the topics of control theory since its creation, where they were mostly treated as those of open-loop control (Krasovski, 1957 and Neustadt, 1964). However, many recent applied motivations (hybrid systems, coordinated control, communication for control, etc.), also require and justify the application of impulsive inputs. But now the request is to deal with closed-loop schemes. Similar mathematical problems, arising in economic models, were indicated by Bensoussan and Lions (1982). In contrast with most previous investigations, this paper deals with the problem of closed-loop impulse control based on generalization of dynamic programming techniques in the form of variational inequalities of the Hamilton–Jacobi–Bellman (HJB) type. Once subjected to closed-loop impulse controls, the originally linear systems treated here, become nonlinear. A special feature, described in this paper, is the application of higher order impulses which are derivatives of delta-functions, introduced for open-loop controls by Kurzhanski and Osipov (1969). Such “ideal” controls allow to transfer a controllable system from one state to another in zero time. They may also serve as virtual controls for system resets in hybrid system models. However, these ideal impulse controls are not physically realizable. In order to ensure their applicability, a scheme for substituting them by realizable approximations is introduced, which leads to the description of “fast” controls that can solve problems of terminal control in arbitrary small time. This paper gives a concise description of related theory with examples on damping high-order oscillating systems to zero in finite time. It is based on two presentations at IFAC Conferences ( Daryin & Kurzhanskii, 2007b; Kurzhanski, 2007), see also Daryin, Kurzhanski, and Seleznev (2005), Kurzhanski (2006), Daryin and Kurzhanskii (2007a).