Optimal impulse equation with applications (Q2723296)

The monograph deals with nonlinear control problems in which control domains are unbounded. The authors give a clear, thorough and original exposition of many investigations in cases when optimal controls are generalized functions of impulse type and trajectories of optimal dynamical processes are discontinues. Relaying on ideas developed by \textit{B. S. Goh} [SIAM J. Control 4, 309--325 (1966; Zbl 0146.11906); J. Math. Anal. Appl. 20, 534--539 (1967; Zbl 0155.15604)] and by \textit{V. I. Gurman} [``Degenerate optimal control problems'' (1977; Zbl 0463.49002); ``The extension principle in control problems'' (1985; Zbl 0900.49024)], the authors present the constructive methods in the theory of extension of optimal problems for systems linear in control in order to transform the initial optimal impulse control problem to a classical one. The maximum principle is obtained in this way as a necessary condition for the optimality of the first order for impulse and singular processes. A special attention is turned to a substantiation of the maximum principle in nonsmooth problems of impulse control with trajectories of bounded variation under the presence of limitations on controls and under suitable assumptions. These results belong to the authors and generalize investigations in the theory of impulse control and are based on: the modified nonlinear transformation of B. S. Goh, principle of \textit{I. Ekeland} [Bull. Am. Math. Soc., New Ser. 1, 443--474 (1979; Zbl 0441.49011)], the notion of a generalized solution derived by \textit{S. T. Zavalishchin} and \textit{A. N. Sesekin} [``Impulsive processes. Models and applications'' (1991; Zbl 0745.47042)] and \textit{B. M. Miller} [Autom. Remote Control 50, No. 6, 733--742 (1989); translation from Avtom. Telemekh. 1989, No. 6, 23--34 (1989; Zbl 0716.49002)], and the maximum principle for nonsmooth problems obtained by \textit{F. H. Clarke} and \textit{R. B. Vinter} [SIAM J. Control Optimization 27, No. 5, 1048--1071 (1989; Zbl 0695.49014); ibid. 1072--1091 (1989; Zbl 0684.49007)]. Various meaningful modeling examples from mechanics, physics, robotics, medicine, ecology and economics are investigated through the book to illustrate applications of the theory and the efficiency of necessary conditions obtained. Interesting results which have substantial significance for applications are obtained by the authors. This monograph is a good foundation for both study and research in the theory of impulse and singular controls as well as in their applications.