Schauder's fixed-point theorem in approximate controllability problems

• Autorzy: Artur Babiarz , Jerzy Klamka , Michał Niezabitowski
• Języki publikacji: EN

Abstrakty

EN

The main objective of this article is to present the state of the art concerning approximate controllability of dynamic systems in infinite-dimensional spaces. The presented investigation focuses on obtaining sufficient conditions for approximate controllability of various types of dynamic systems using Schauder's fixed-point theorem. We describe the results of approximate controllability for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions, impulsive neutral functional evolution integro-differential systems, stochastic impulsive systems with control-dependent coefficients, nonlinear impulsive differential systems, and evolution systems with nonlocal conditions and semilinear evolution equation.

Słowa kluczowe

EN

approximate controllability; Banach space; Hilbert space; Schauder's fixed-point theorem; infinite-dimensional space

• Wydawca: University of Zielona Gora Press
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2016
• Tom: 26
• Numer: 2
• Strony: 263-275

Daty

wydano

2016

otrzymano

2015-09-14

poprawiono

2015-12-16

zaakceptowano

2016-01-29

Twórcy

autor

Artur Babiarz

• Institute of Automatic Control, Silesian University of Technology, Akademicka 16, 44-100 Gliwice, Poland

autor

Jerzy Klamka

• Institute of Automatic Control, Silesian University of Technology, Akademicka 16, 44-100 Gliwice, Poland

autor

Michał Niezabitowski

• Institute of Automatic Control, Silesian University of Technology, Akademicka 16, 44-100 Gliwice, Poland

Bibliografia

• Arapostathis, A., George, R.K. and Ghosh, M.K. (2001). On the controllability of a class of nonlinear stochastic systems, Systems & Control Letters 44(1): 25-34.
• Babiarz, A., Czornik, A., Klamka, J. and Niezabitowski, M. (2015a). Controllability of discrete-time linear switched systems with constrains on switching signal, in N.T. Nguyen et al. (Eds.), Intelligent Information and Database Systems, Lecture Notes in Computer Science, Vol. 9011, Springer International Publishing, Berlin, pp. 304-312.
• Babiarz, A., Czornik, A., Klamka, J. and Niezabitowski, M. (2015b). The selected problems of controllability of discrete-time switched linear systems with constrained switching rule, Bulletin of the Polish Academy of Sciences: Technical Sciences 63(3): 657-666.
• Babiarz, A., Czornik, A. and Niezabitowski, M. (2016). Output controllability of the discrete-time linear switched systems, Nonlinear Analysis: Hybrid Systems 21: 1-10.
• Bader, R., Gabor, G. and Kryszewski, W. (1996). On the extension of approximations for set-valued maps and the repulsive fixed points, Bollettino della Unione Matematica Italiana B 10(2): 399-416.
• Bader, R. and Kryszewski, W. (1994). Fixed-point index for compositions of set-valued maps with proximally ∞-connected values on arbitrary ANR's, Set-Valued Analysis 2(3): 459-480.
• Balachandran, K. and Dauer, J. (2002). Controllability of nonlinear systems in Banach spaces: A survey, Journal of Optimization Theory and Applications 115(1): 7-28.
• Balachandran, K. and Sakthivel, R. (2001). Controllability of integrodifferential systems in Banach spaces, Applied Mathematics and Computation 118(1): 63-71.
• Bashirov, A.E. and Kerimov, K.R. (1997). On controllability conception for stochastic systems, SIAM Journal on Control and Optimization 35(2): 384-398.
• Bashirov, A.E. and Mahmudov, N.I. (1999). On concepts of controllability for deterministic and stochastic systems, SIAM Journal on Control and Optimization 37(6): 1808-1821.
• Benchohra, M., Gorniewicz, L., Ntouyas, S. and Ouahab, A. (2004). Controllability results for impulsive functional differential inclusions, Reports on Mathematical Physics 54(2): 211-228.
• Benchohra, M. and Ouahab, A. (2005). Controllability results for functional semilinear differential inclusions in Fréchet spaces, Nonlinear Analysis: Theory, Methods & Applications 61(3): 405-423.
• Bian, W. (1999). Constrained controllability of some nonlinear systems, Applicable Analysis 72(1-2): 57-73.
• Chang, Y.-K. (2007). Controllability of impulsive functional differential systems with infinite delay in Banach spaces, Chaos, Solitons & Fractals 33(5): 1601-1609.
• Curtain, R.F. and Zwart, H. (1995). An Introduction to Infinitedimensional Linear Systems Theory, Springer Science & Business Media, Berlin.
• Czornik, A. and Świerniak, A. (2001). On controllability with respect to the expectation of discrete time jump linear systems, Journal of the Franklin Institute 338(4): 443-453.
• Czornik, A. and Świerniak, A. (2004). On direct controllability of discrete time jump linear system, Journal of the Franklin Institute 341(6): 491-503.
• Czornik, A. and Świerniak, A. (2005). Controllability of discrete time jump linear systems, Dynamics of Continuous Discrete and Impulsive Systems B: Applications & Algorithms 12(2): 165-189.
• Dauer, J. and Mahmudov, N. (2002). Approximate controllability of semilinear functional equations in Hilbert spaces, Journal of Mathematical Analysis and Applications 273(2): 310-327.
• Dauer, J., Mahmudov, N. and Matar, M. (2006). Approximate controllability of backward stochastic evolution equations in Hilbert spaces, Journal of Mathematical Analysis and Applications 323(1): 42-56.
• Do, V. (1989). A note on approximate controllability of semilinear systems, Systems & Control Letters 12(4): 365-371.
• Dubov, M. and Mordukhovich, B. (1978). Theory of controllability of linear stochastic systems, Differential Equations 14: 1609-1612.
• George, R.K. (1995). Approximate controllability of nonautonomous semilinear systems, Nonlinear Analysis: Theory, Methods & Applications 24(9): 1377-1393.
• Gorniewicz, L., Granas, A. and Kryszewski, W. (1991). On the homotopy method in the fixed point index theory of multi-valued mappings of compact absolute neighborhood retracts, Journal of Mathematical Analysis and Applications 161(2): 457-473.
• Gorniewicz, L., Ntouyas, S. and O'Regan, D. (2005). Controllability of semilinear differential equations and inclusions via semigroup theory in Banach spaces, Reports on Mathematical Physics 56(3): 437-470.
• Henríquez, H.R. (2008). Approximate controllability of linear distributed control systems, Applied Mathematics Letters 21(10): 1041-1045.
• Hong, X.Z. (1982). A note on approximate controllability for semilinear one-dimensional heat equation, Applied Mathematics and Optimization 8(1): 275-285.
• Jeong, J.-M. and Roh, H.-H. (2006). Approximate controllability for semilinear retarded systems, Journal of Mathematical Analysis and Applications 321(2): 961-975.
• Klamka, J. (2000). Constrained approximate controllability, IEEE Transactions on Automatic Control 45(9): 1745-1749.
• Kryszewski, W. and Zezza, P. (1994). Remarks on the relay controllability of control systems, Journal of Mathematical Analysis and Applications 188(1): 45-65.
• Kumlin, P. (2004). A note on fixed point theory, Functional Analysis Lecture, Mathematics, Chalmers & GU, Gothenburg.
• Lasiecka, I. and Triggiani, R. (1991). Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems, Applied Mathematics and Optimization 23(1): 109-154.
• Li, M., Wang, M. and Zhang, F. (2006). Controllability of impulsive functional differential systems in Banach spaces, Chaos, Solitons & Fractals 29(1): 175-181.
• Mahmudov, N.I. (2001a). Controllability of linear stochastic systems, IEEE Transactions on Automatic Control 46(5): 724-731.
• Mahmudov, N.I. (2001b). Controllability of linear stochastic systems in Hilbert spaces, Journal of Mathematical Analysis and Applications 259(1): 64-82.
• Mahmudov, N.I. (2002). On controllability of semilinear stochastic systems in Hilbert spaces, IMA Journal of Mathematical Control and Information 19(4): 363-376.
• Mahmudov, N.I. (2003). Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM Journal on Control and Optimization 42(5): 1604-1622.
• Mahmudov, N.I. (2008). Approximate controllability of evolution systems with nonlocal conditions, Nonlinear Analysis: Theory, Methods & Applications 68(3): 536-546.
• Mahmudov, N. and Zorlu, S. (2003). Controllability of non-linear stochastic systems, International Journal of Control 76(2): 95-104.
• Naito, K. (1987). Controllability of semilinear control systems dominated by the linear part, SIAM Journal on Control and Optimization 25(3): 715-722.
• Naito, K. (1989). Approximate controllability for trajectories of semilinear control systems, Journal of Optimization Theory and Applications 60(1): 57-65.
• Narayanamoorthy, S. and Sowmiya, S. (2015). Approximate controllability result for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions, Advances in Difference Equations 2015(1): 1-16.
• Pazy, A. (2012). Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Science & Business Media, Berlin.
• Przeradzki, B. (2012). A generalization of Krasnoselskii fixed point theorem for sums of compact and contractible maps with application, Open Mathematics 10(6): 2012-2018.
• Radhakrishnan, B. and Balachandran, K. (2011). Controllability of impulsive neutral functional evolution integrodifferential systems with infinite delay, Nonlinear Analysis: Hybrid Systems 5(4): 655-670.
• Sakthivel, R., Mahmudov, N. and Kim, J. (2007). Approximate controllability of nonlinear impulsive differential systems, Reports on Mathematical Physics 60(1): 85-96.
• Sakthivel, R., Nieto, J.J. and Mahmudov, N. (2010). Approximate controllability of nonlinear deterministic and stochastic systems with unbounded delay, Taiwanese Journal of Mathematics 14(5): 1777-1797.
• Shen, L. and Sun, J. (2011). Approximate controllability of stochastic impulsive systems with control-dependent coefficients, IET Control Theory & Applications 5(16): 1889-1894.
• Sikora, B. and Klamka, J. (2012). On constrained stochastic controllability of dynamical systems with multiple delays in control, Bulletin of the Polish Academy of Sciences: Technical Sciences 60(2): 301-305.
• Sırbu, M. and Tessitore, G. (2001). Null controllability of an infinite dimensional SDE with state-and control-dependent noise, Systems & Control Letters 44(5): 385-394.
• Wang, L. (2006). Approximate controllability and approximate null controllability of semilinear systems, Communications on Pure and Applied Analysis 5(4): 953-962.
• Zang, Y. and Li, J. (2013). Approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions, Boundary Value Problems 2013(1): 1-13.
• Zhou, H.X. (1983). Approximate controllability for a class of semilinear abstract equations, SIAM Journal on Control and Optimization 21(4): 551-565.

Identyfikatory

DOI

10.1515/amcs-2016-0018