## List of issues > Series «Mathematics». 2014. Vol. 7

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Monotonicity of Lyapunov Type Functions for Impulsive Control Systems

The paper is devoted to the study of impulsive dynamical systems with trajectories of bounded variation and impulsive controls (regular vector measures). A new concept of solutions for these systems is introduced. According to this concept, the solution is an upper semicontinuous set-valued mapping. The relationship between the new solution concept and conventional one is established. We prove that the set of solutions is a closure of the set of the absolutely continuous solutions. Here, the closure is understood in the sense of the convergence in Hausdorff metric for graphs of the supplemented absolutely continuous trajectories. In this paper, we focus mainly on the study of some monotonicity properties of a continuous function with respect to a nonlinear impulsive control system with trajectories of bounded variation. Definitions of strong and weak monotonicity and V-monotonicity are proposed and discussed. The set of conventional variables t and x of Lyapunov type functions is now supplemented with the variable V, which, on the one hand, is responsible for the impulsive dynamics of the system and has the property of the time variable and, on the other hand, characterizes some resource of the impulsive control. We show that such double interpretation of variable V leads to different definitions of monotonicity, which are called monotonicity and V-monotonicity. For smooth Lyapunov type functions, infinitesimal conditions of monotonicity in the form of Hamilton-Jacobi differential inequalities are presented.

1. Aubin J.-P., Cellina A. Differential Inclusions. Berlin, Springer-Verlag, 1984. 342 p.

2. Clarke F.H., Ledyaev Yu.S., Stern R.J., Wolenski P.R. Nonsmooth Analysis and Control Theory. New York, Springer-Verlag, Grad. Texts in Math. Vol. 178, 1998, 276 p.

3. Vinter R.B. Optimal Control. Birkhauser, Boston, 2000.

4. Pereira F.L., Silva G.N. Stability for Impulsive Control Systems. Dynamical Systems, 2002, vol. 17, no. 4, pp. 421-434.

5. Dykhta V.A., Samsonyuk O.N. Hamilton-Jacobi Inequalities in Control Problems for Impulsive Dynamical Systems. Proc. of the Steklov Institute of Mathematics, 2010, vol. 271, pp. 86-102.

6. Samsonyuk O.N. Strong and Weak Monotonicity Conditions of Lyapunov Type Functions for nonlinear impulsive system [Usloviya sil'noj i slaboj monotonnosti funkcij tipa Lyapunova dlya nelinejnyh impul'snyh upravlyaemyh sistem]. Book of Abstracts of the XI International Conference «Stability and Oscillations in Nonlinear Control Systems» (Pyatnitskii Conference), June 1-4, 2010, pp. 347-349.

7. Samsonyuk O. Lyapunov Type Functions for Nonlinear Impulsive Control Systems: Monotonicity Conditions and Applications. Book of Abstracts of the 5th International Conference on Physics and Control (PhysCon 2011), September 5-8, 2011, pp. 87.

8. Dykhta V., Samsonyuk O. Some Applications of Hamilton-Jacobi Inequalities for Classical and Impulsive Optimal Control Problems. European Journal of Control, 2011, vol. 17, pp. 55-69.

9. Dykhta V.A., Samsonyuk O.N. Optimal'noe impul'snoe upravlenie s prilozheniyami [Optimal Impulsive Control with Applications]. Moscow, Fizmatlit, 2000.

10. Zavalishchin S.T., Sesekin A.N. Impul'snye processy: modeli i prilozhenija [Impulse Processes: Models and Applications]. Moscow, Nauka, 1991.

11. Miller B.M. Method of Discontinuous Time Change in Problems of Control of Impulse and Discrete-Continuous Systems. Autom. Remote Control, 1993, vol. 54, no. 12, part 1, pp. 1727-1750.

12. Miller B.M., Rubinovich E.Ya. Optimizatsiya dinamicheskikh sistem s impul'snymi upravleniyami [Optimization of Dynamic Systems with Impulsive Controls]. Moscow, Nauka, 2005.

13. Gurman V.I. Printsip rasshireniya v zadachakh optimal'nogo upravleniya [The Extension Principle in Optimal Control Problems]. Moscow, Nauka, 1997.

14. Baturin V.A., Dykhta V.A., Moskalenko A.I., et al. Metody resheniya zadach teroii upravleniya na osnove principa rasshireniya [Methods for Solving Problems in Control Theory on the Basis of an Extension Principle]. Novosibirsk, Nauka, 1990.

15. Dykhta V.A. Variatsionnyi printsip maksimuma i kvadratichnye usloviya optimal'nosti impul'snykh protsessov [The Variational Maximum Principle and Quadratic Optimality Conditions for Pulse Processes]. Irkutsk, IGEA, 1995.

16. Miller, B.M. Optimality Condition in the Control Problem for a System Described by a Measure Differential Equation. Autom. Remote Control, 1982, vol. 43, no. 6, part 1, pp. 752-761.

17. Miller, B.M. Conditions for the Optimality in Problems of Generalized Control. I, II, Autom. Remote Control, 1992, vol. 53, no. 3, part 1, pp. 362-370 no. 4, pp. 505-513.

18. Arutyunov A.V., Karamzin D.Yu., Pereira F.L. On Constrained Impulsive Control Problems. J. Math. Sci., 2010, vol. 165, no. 6, pp. 654-688.

19. Motta M., Rampazzo F. Space-time Trajectories of Nonlinear Systems Driven by Ordinary and Impulsive Controls. Differential Integral Equations. 1995, vol. 8, pp. 269-288.

20. Silva G.N., Vinter R.B. Measure Differential Inclusions. J. of Mathematical Analysis and Applications. 1996, vol. 202, pp. 727-746.

21. Miller B.M. The Generalized Solutions of Nonlinear Optimization Problems with Impulse Control. SIAM J. Control Optim. 1996, vol. 34, pp. 1420-1440.

22. Sesekin A.N. On the Set of Discontinuous Solutions of Nonlinear Differential Equations. Izv. Vyssh. Uchebn. Zaved., Mat. , 1994, vol. 38, no. 6, pp. 83-89.

23. Sesekin A.N. Dynamical Systems with Nonlinear Impulsive structure [Dinamicheskie sistemy s nelinejnoj impul'snoj strukturoj]. Trudy Inst. Mat. Mekh. UrO RAN, Ekaterinburg, 2000, vol. 6, pp. 497-510.

24. Samsonyuk O.N. Invariant Sets for Nonlinear Impulsive Control Systems. Autom. Remote Control, 2014. (To appear).

25. Samsonyuk O.N. Strong and Weak Invariance for Nonlinear Impulsive Control Systems. Book ofAbstracts ofIFAC WC 2011, 2011, pp. 3480-3485.

26. Samsonyuk O.N. Compound Lyapunov Type Functions in Control Problems of Impulsive Dynamical Systems [Sostavnye funkcii tipa Lyapunova v zadachah upravleniya impul'snymi dinamicheskimi sistemami]. Trudy Inst. Mat. Mekh. UrORAN, Ekaterinburg, vol. 16, pp. 170-178.