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## Archive of Issues

Russia Izhevsk
Year
2015
Volume
25
Issue
1
Pages
51-59
 Section Mathematics Title Lyapunov functions and comparison theorems for control systems with impulsive actions Author(-s) Larina Ya.Yu.a Affiliations Udmurt State Universitya Abstract We extend the results of E.L. Tonkov and E.A. Panasenko to differential equations and control systems with impulsive actions. In terms of Lyapunov functions and the Clarke derivative we obtain comparison theorems for systems with impulsive effect. We consider the set $\mathfrak M\doteq\bigl\{(t,x)\in[t_0,+\infty)\times\mathbb{R}^n: x\in M(t)\bigr\},$ defined by continuous function $t\rightarrow M(t)$, where for every $t \in \mathbb R$ the set $M(t)$ is nonempty and compact. We obtain conditions for the positive invariance of this set, the uniform Lyapunov stability and the uniform asymptotic stability. We make a comparison with the researches of other authors who have considered the zero solution stability for similar systems. Keywords control systems with impulsive actions, Lyapunov function, differential inclusions UDC 517.935, 517.938 MSC 34A60, 37N35, 49J15, 93B03 DOI 10.20537/vm150106 Received 17 February 2015 Language Russian Citation Larina Ya.Yu. Lyapunov functions and comparison theorems for control systems with impulsive actions, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 1, pp. 51-59. References Gurman V.I. Optimal processes of singular control, Automat. Remote Control, 1965, vol. 26, pp. 783-792. Gurman V.I. Vyrozhdennye zadachi optimal'nogo upravleniya (Singular problems of optimal control), Moscow: Nauka, 1977, 304 p. Dykhta V.A., Samsonyuk О.N. Optimal’noe impul’snoe upravlenie s prilozheniyami (Optimal impulse control with applications), Moscow: Fizmatlit, 2000, 256 p. Zavalishchin S.T., Sesekin A.N. Impul'snye protsessy: modeli i prilozheniya (Impulse processes: models and applications), Moscow: Nauka, 1991, 256 p. Miller B.M. Method of discontinuous time change in optimal control pulse and discrete-continuous systems, Automat. Remote Control, 1993, vol. 54, no. 12, pp. 1727-1750. Myshkis A.D. Stability of solutions of differential equations under generalized pulse perturbations, Automat. Remote Control, 2007, vol. 68, no. 10, pp. 1844-1851. Sesekin A.N. On the connectedness of the set of discontinuous solutions of a nonlinear dynamical system with impulse control, Russian Mathematics, 1996, vol. 40, no. 11, pp. 82-89. Samoilenko A.M., Perestyuk N.A. Differentsial'nye uravneniya s impul'snym vozdeistviem (Impulsive differential equations), Kiev: Vishcha shkola, 1987, 287 p. Lakshmikantham V., Leela S., Martynyuk A.A. Ustoichivost' dvizheniya: metod sravneniya (Stability of motion: comparison method), Kiev: Naukova dumka, 1991, 247 p. Panasenko E.A., Tonkov E.L. Invariant and stably invariant sets for differential inclusions, Proceedings of the Steklov Institute of Mathematics, 2008, vol. 262, pp. 194-212. Panasenko E.A., Tonkov E.L. Extension of E.A. Barbashin’s and N.N. Krasovskii’s stability theorems to controlled dynamical systems, Proceedings of the Steklov Institute of Mathematics, 2010, vol. 268, suppl. 1, pp. 204-221. Rodina L.I. Invariant and statistically weakly invariant sets of control systems, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2012, no. 2 (40), pp. 3-164 (in Russian). Rodina L.I. Estimation of statistical characteristics of attainability sets of controllable systems, Russian Mathematics, 2013, vol. 57, no. 11, pp. 17-27. Filippov A.F. Differentsial'nye uravneniya s razryvnoi pravoi chast'yu (Differential equations with discontinuous right-hand side), Мoscow: Nauka, 1985, 223 p. Clarke F. Optimization and nonsmooth analysis, Wiley, 1983. Translated under the title Optimizatsiya i negladkii analiz, Moscow: Nauka, 1988, 300 p. Nemytskii V.V., Stepanov V.V. Qualitative theory of differential equations, New Jersey: Princeton University Press, 1960, 523 p. Federer H. Geometricheskaya teoriya mery (Geometric theory of measure), Moscow: Nauka, 1987, 761 p. Filippov V.V. Prostranstva reshenii obyknovennykh differentsial’nykh uravnenii (Spaces of solutions of ordinary differential equations), Moscow: Moscow State University, 1993, 336 p. Demidovich B.P. Lektsii po matematicheskoi teorii ustoichivosti (Lectures on the mathematical theory of stability), Moscow: Nauka, 1967, 472 p. Chaplygin S.A. Novyi metod priblizhennogo integrirovaniya differentsial'nykh uravnenii (A new method of approximate integration of differential equations), Moscow-Leningrad: Gostekhizdat, 1950, 102 p. Full text