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. AngeliB. IngallsE. 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. BashnyakovV. 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.

show all references

References:
[1]

D. AngeliB. IngallsE. 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. BashnyakovV. 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.

[1]

Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115

[2]

Xing Wang, Nan-Jing Huang. Stability analysis for set-valued vector mixed variational inequalities in real reflexive Banach spaces. Journal of Industrial & Management Optimization, 2013, 9 (1) : 57-74. doi: 10.3934/jimo.2013.9.57

[3]

Zhenhua Peng, Zhongping Wan, Weizhi Xiong. Sensitivity analysis in set-valued optimization under strictly minimal efficiency. Evolution Equations & Control Theory, 2017, 6 (3) : 427-436. doi: 10.3934/eect.2017022

[4]

Robert Baier, Thuy T. T. Le. Construction of the minimum time function for linear systems via higher-order set-valued methods. Mathematical Control & Related Fields, 2019, 9 (2) : 223-255. doi: 10.3934/mcrf.2019012

[5]

Yihong Xu, Zhenhua Peng. Higher-order sensitivity analysis in set-valued optimization under Henig efficiency. Journal of Industrial & Management Optimization, 2017, 13 (1) : 313-327. doi: 10.3934/jimo.2016019

[6]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461

[7]

Geng-Hua Li, Sheng-Jie Li. Unified optimality conditions for set-valued optimizations. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1101-1116. doi: 10.3934/jimo.2018087

[8]

Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327

[9]

Qingbang Zhang, Caozong Cheng, Xuanxuan Li. Generalized minimax theorems for two set-valued mappings. Journal of Industrial & Management Optimization, 2013, 9 (1) : 1-12. doi: 10.3934/jimo.2013.9.1

[10]

Sina Greenwood, Rolf Suabedissen. 2-manifolds and inverse limits of set-valued functions on intervals. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5693-5706. doi: 10.3934/dcds.2017246

[11]

Mariusz Michta. Stochastic inclusions with non-continuous set-valued operators. Conference Publications, 2009, 2009 (Special) : 548-557. doi: 10.3934/proc.2009.2009.548

[12]

Guolin Yu. Topological properties of Henig globally efficient solutions of set-valued problems. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 309-316. doi: 10.3934/naco.2014.4.309

[13]

Zengjing Chen, Yuting Lan, Gaofeng Zong. Strong law of large numbers for upper set-valued and fuzzy-set valued probability. Mathematical Control & Related Fields, 2015, 5 (3) : 435-452. doi: 10.3934/mcrf.2015.5.435

[14]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519

[15]

Jiawei Chen, Zhongping Wan, Liuyang Yuan. Existence of solutions and $\alpha$-well-posedness for a system of constrained set-valued variational inequalities. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 567-581. doi: 10.3934/naco.2013.3.567

[16]

Guolin Yu. Global proper efficiency and vector optimization with cone-arcwise connected set-valued maps. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 35-44. doi: 10.3934/naco.2016.6.35

[17]

Benjamin Seibold, Morris R. Flynn, Aslan R. Kasimov, Rodolfo R. Rosales. Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models. Networks & Heterogeneous Media, 2013, 8 (3) : 745-772. doi: 10.3934/nhm.2013.8.745

[18]

Shay Kels, Nira Dyn. Bernstein-type approximation of set-valued functions in the symmetric difference metric. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1041-1060. doi: 10.3934/dcds.2014.34.1041

[19]

Ying Gao, Xinmin Yang, Jin Yang, Hong Yan. Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps. Journal of Industrial & Management Optimization, 2015, 11 (2) : 673-683. doi: 10.3934/jimo.2015.11.673

[20]

Zhiang Zhou, Xinmin Yang, Kequan Zhao. $E$-super efficiency of set-valued optimization problems involving improvement sets. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1031-1039. doi: 10.3934/jimo.2016.12.1031

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (6)
  • HTML views (1)
  • Cited by (0)

Other articles
by authors

[Back to Top]