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## Archive of Issues

Russia Izhevsk
Year
2016
Volume
26
Issue
1
Pages
68-78
 Section Mathematics Title Weak asymptotic stability of control systems with impulsive actions Author(-s) Larina Ya.Yu.a Affiliations Udmurt State Universitya Abstract We continue investigating the conditions of positive invariance and asymptotic stability of a given set relative to a control system with impulsive actions. We consider the set $\mathfrak M \doteq \bigl\{ (t,x) \in[t_0,+\infty) \times \mathbb{R}^n: x\in M(t)\bigr\}$, given by a function $t\rightarrow M(t)$ that is continuous in the Hausdorff metric, where the set $M(t)$ is nonempty and compact for each $t \in \mathbb R$. In terms of the Lyapunov functions and the Clarke derivative, we obtain conditions for weak positive invariance, weak uniform Lyapunov stability and weak asymptotic stability of the set $\mathfrak M$. Also we prove a comparison theorem for solutions of systems and equations with impulses the consequence of which is the conditions for existence of solutions of the system that asymptotically tends to zero. The obtained results are illustrated by the example of model for competition of two species exposed to impulse control at given times. Keywords control systems with impulsive actions, Lyapunov function, weak asymptotic stability UDC 517.935, 517.938 MSC 34A60, 37N35, 49J15, 93B03 DOI 10.20537/vm160106 Received 17 January 2016 Language Russian Citation Larina Ya.Yu. Weak asymptotic stability of control systems with impulsive actions, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 1, pp. 68-78. References Larina Ya.Yu. Lyapunov functions and comparison theorems for control systems with impulsive actions, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2015, vol. 25, no. 1, pp. 51-59 (in Russian). Samoilenko A.M., Perestyuk N.A. Differentsial'nye uravneniya s impul'snym vozdeistviem (Impulsive differential equations), Kiev: Vishcha shkola, 1987, 287 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. Rodina L.I., Tonkov E.L. Statistical characteristics of attainable set of controllable system, non-wandering, and minimal attraction center, Nelin. Dinam., 2009, vol. 5, no. 2, pp. 265-288 (in Russian). 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. Blagodatskikh V.I., Filippov A.F. Differential inclusions and optimal control, Proc. Steklov Inst. Math., 1986, vol. 169, pp. 199-259. Demidovich B.P. Lektsii po matematicheskoi teorii ustoichivosti (Lectures on the mathematical stability theory), 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. Riznichenko G.Yu. Lektsii po matematicheskim modelyam v biologii. Chast' 1 (Lectures on the mathematical models in biology. Part 1), Izhevsk: Regular and Chaotic Dynamics, 2002, 236 p. Kuzenkov O.A., Ryabova E.A. Matematicheskoe modelirovanie protsessov otbora (Mathematical modeling of processes of selection), Nizhnii Novgorod: Nizhnii Novgorod State University, 2007, 324 p. Full text