# American Institute of Mathematical Sciences

July  2017, 22(5): 1977-1986. doi: 10.3934/dcdsb.2017116

## On practical stability of differential inclusions using Lyapunov functions

 Taras Shevchenko National University of Kyiv, Department of Computer Science and Cybernetics, Volodymyrska Str. 60,01033, Kyiv, Ukraine

Received  January 2016 Revised  February 2016 Published  March 2017

In this paper we consider the problem of practical stability for differential inclusions. We prove the necessary and sufficient conditions using Lyapunov functions. Then we solve the practical stability problem of linear differential inclusion with ellipsoidal righthand part and ellipsoidal initial data set. In the last section we apply the main result of this paper to the problem of practical stabilization.

Citation: Volodymyr Pichkur. On practical stability of differential inclusions using Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1977-1986. doi: 10.3934/dcdsb.2017116
##### References:
 [1] D. Angeli, B. Ingalls, E. D. Sontag and Y. Wang, Uniform global asymptotic stability of differential inclusions, Journal of Dynamical and Control Systems, 10 (2004), 391-412. doi: 10.1023/B:JODS.0000034437.54937.7f. [2] E. Arzarello and A. Bacciotti, On stability and boundedness for lipschitzian differential inclusions: The converse of Lyapunov's theorems, Set-Valued Analysis, 5 (1997), 377-390. doi: 10.1023/A:1008603707291. [3] J. P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory Berlin-Heidelberg-New York-Tokyo, Springer-Verlag, 1984. [4] J. P. Aubin and H. Frankowska, Set-valued Analysis Boston, Birkhäuser, 2009. [5] A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory Berlin -Heidelberg -New York, Springer, 2005. [6] O. M. Bashnyakov, F. G. Garashchenko and V. V. Pichkur, Practical Stability, Estimations and Optimization, Kyiv : Taras Shevchenko National University of Kyiv, 2008. [7] A. N. Bashnyakov, V. V. Pichkur and I. V. Hitko, On Maximal Initial Data Set in Problems of Practical Stability of Discrete System, J. Automat. Inf. Scien., 43 (2011), 1-8. doi: 10.1615/JAutomatInfScien.v43.i3.10. [8] B. N. Bublik, F. G. Garashchenko and N. F. Kirichenko, Structural -Parametric Optimization and Stability of Bunch Dynamics, Kyiv: Naukova dumka, 1985. [9] N. G. Chetaev, On certain questions related to the problem of the stability of unsteady motion, J. Appl. Math. Mech., 24 (1960), 6-19. doi: 10.1016/0021-8928(60)90135-0. [10] K. Deimling, Multivalued Differential Equations Berlin-New York: Walter de Gruyter, 1992. [11] R. Gama and G. Smirnov, Stability and optimality of solutions to differential inclusions via averaging method, Set-Valued and Variational Analysis, 22 (2014), 349-374. doi: 10.1007/s11228-013-0261-4. [12] F. G. Garashchenko and V. V. Pichkur, Garashchenko and V. V. Pichkur, Properties of optimal sets of practical stability of differential inclusions. Part Ⅰ. Part Ⅱ, (Russian), Problemy Upravlen. Inform., (2006), 163-170. [13] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides Dordrecht-Boston-London: Kluwer Academic, 1988. [14] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides and Differential Inclusions, in Nonlinear Analysis and Nonlinear Differential Equations (eds. V. A. Trenogin and A. F. Filippov), Moscow: FIZMATLIT, (2003), 265-288. [15] N. F. Kirichenko, Introduction to the Stability Theory, Kyiv: Vyshcha Shkola, 1978. [16] V. Lakshmikantham, S. Leela and A. A. Martynyuk, Practical Stability of Nonlinear Systems Singapore : World Scientific, 1990. [17] J. Lasalle and S. Lefshetz, Stability by Lyapunov Direct Method and Application, Academic Press, New York:, 1961. [18] A. Michel, K. Wang and B. Hu, Qualitative Theory of Dynamical Systems. The Role of Stability-Preserving Mappings, Marcel Dekker, Inc. , New York, 1995. [19] V. V. Pichkur and M. S. Sasonkina, Maximum set of initial conditions for the problem of weak practical stability of a discrete inclusion, J. Math. Sci., 194 (2013), 414-425. doi: 10.1007/s10958-013-1537-9. [20] G. Smirnov, Introduction to the Theory of Differential Inclusions, American Mathematical Society, 2002. [21] V. Veliov, Stability-like properties of differential inclusions, Set-Valued Analysis, 5 (1997), 73-88. doi: 10.1023/A:1008683223676.

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##### References:
 [1] D. Angeli, B. Ingalls, E. D. Sontag and Y. Wang, Uniform global asymptotic stability of differential inclusions, Journal of Dynamical and Control Systems, 10 (2004), 391-412. doi: 10.1023/B:JODS.0000034437.54937.7f. [2] E. Arzarello and A. Bacciotti, On stability and boundedness for lipschitzian differential inclusions: The converse of Lyapunov's theorems, Set-Valued Analysis, 5 (1997), 377-390. doi: 10.1023/A:1008603707291. [3] J. P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory Berlin-Heidelberg-New York-Tokyo, Springer-Verlag, 1984. [4] J. P. Aubin and H. Frankowska, Set-valued Analysis Boston, Birkhäuser, 2009. [5] A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory Berlin -Heidelberg -New York, Springer, 2005. [6] O. M. Bashnyakov, F. G. Garashchenko and V. V. Pichkur, Practical Stability, Estimations and Optimization, Kyiv : Taras Shevchenko National University of Kyiv, 2008. [7] A. N. Bashnyakov, V. V. Pichkur and I. V. Hitko, On Maximal Initial Data Set in Problems of Practical Stability of Discrete System, J. Automat. Inf. Scien., 43 (2011), 1-8. doi: 10.1615/JAutomatInfScien.v43.i3.10. [8] B. N. Bublik, F. G. Garashchenko and N. F. Kirichenko, Structural -Parametric Optimization and Stability of Bunch Dynamics, Kyiv: Naukova dumka, 1985. [9] N. G. Chetaev, On certain questions related to the problem of the stability of unsteady motion, J. Appl. Math. Mech., 24 (1960), 6-19. doi: 10.1016/0021-8928(60)90135-0. [10] K. Deimling, Multivalued Differential Equations Berlin-New York: Walter de Gruyter, 1992. [11] R. Gama and G. Smirnov, Stability and optimality of solutions to differential inclusions via averaging method, Set-Valued and Variational Analysis, 22 (2014), 349-374. doi: 10.1007/s11228-013-0261-4. [12] F. G. Garashchenko and V. V. Pichkur, Garashchenko and V. V. Pichkur, Properties of optimal sets of practical stability of differential inclusions. Part Ⅰ. Part Ⅱ, (Russian), Problemy Upravlen. Inform., (2006), 163-170. [13] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides Dordrecht-Boston-London: Kluwer Academic, 1988. [14] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides and Differential Inclusions, in Nonlinear Analysis and Nonlinear Differential Equations (eds. V. A. Trenogin and A. F. Filippov), Moscow: FIZMATLIT, (2003), 265-288. [15] N. F. Kirichenko, Introduction to the Stability Theory, Kyiv: Vyshcha Shkola, 1978. [16] V. Lakshmikantham, S. Leela and A. A. Martynyuk, Practical Stability of Nonlinear Systems Singapore : World Scientific, 1990. [17] J. Lasalle and S. Lefshetz, Stability by Lyapunov Direct Method and Application, Academic Press, New York:, 1961. [18] A. Michel, K. Wang and B. Hu, Qualitative Theory of Dynamical Systems. The Role of Stability-Preserving Mappings, Marcel Dekker, Inc. , New York, 1995. [19] V. V. Pichkur and M. S. Sasonkina, Maximum set of initial conditions for the problem of weak practical stability of a discrete inclusion, J. Math. Sci., 194 (2013), 414-425. doi: 10.1007/s10958-013-1537-9. [20] G. Smirnov, Introduction to the Theory of Differential Inclusions, American Mathematical Society, 2002. [21] V. Veliov, Stability-like properties of differential inclusions, Set-Valued Analysis, 5 (1997), 73-88. doi: 10.1023/A:1008683223676.
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