Schauder’s fixed-point theorem in approximate controllability problems

Babiarz, Artur; Klamka, Jerzy; Niezabitowski, Michał

doi:10.1515/amcs-2016-0018

Abstract

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.

References

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): 25-34.10.1016/S0167-6911(01)00123-2
Search in Google Scholar
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.
Search in Google Scholar
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): 657-666.10.1515/bpasts-2015-0077
Search in Google Scholar
Babiarz, A., Czornik, A. and Niezabitowski, M. (2016). Output controllability of the discrete-time linear switched systems, Nonlinear Analysis: Hybrid Systems 21: 1-10.10.1016/j.nahs.2015.12.004
Search in Google Scholar
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.
Search in Google Scholar
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): 459-480.10.1007/BF01026835
Search in Google Scholar
Balachandran, K. and Dauer, J. (2002). Controllability of nonlinear systems in Banach spaces: A survey, Journal of Optimization Theory and Applications 115(1): 7-28.10.1023/A:1019668728098
Search in Google Scholar
Balachandran, K. and Sakthivel, R. (2001). Controllability of integrodifferential systems in Banach spaces, Applied Mathematics and Computation 118(1): 63-71.10.1016/S0096-3003(00)00040-0
Search in Google Scholar
Bashirov, A.E. and Kerimov, K.R. (1997). On controllability conception for stochastic systems, SIAM Journal on Control and Optimization 35(2): 384-398.10.1137/S0363012994260970
Search in Google Scholar
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): 1808-1821.10.1137/S036301299732184X
Search in Google Scholar
Benchohra, M., Gorniewicz, L., Ntouyas, S. and Ouahab, A. (2004). Controllability results for impulsive functional differential inclusions, Reports on Mathematical Physics 54(2): 211-228.10.1016/S0034-4877(04)80015-6
Search in Google Scholar
Benchohra, M. and Ouahab, A. (2005). Controllability results for functional semilinear differential inclusions in Fr´echet spaces, Nonlinear Analysis: Theory, Methods & Applications 61(3): 405-423.10.1016/j.na.2004.12.002
Search in Google Scholar
Bian, W. (1999). Constrained controllability of some nonlinear systems, Applicable Analysis 72(1-2): 57-73.10.1080/00036819908840730
Search in Google Scholar
Chang, Y.-K. (2007). Controllability of impulsive functional differential systems with infinite delay in Banach spaces, Chaos, Solitons & Fractals 33(5): 1601-1609.10.1016/j.chaos.2006.03.006
Search in Google Scholar
Curtain, R.F. and Zwart, H. (1995). An Introduction to Infinitedimensional Linear Systems Theory, Springer Science & Business Media, Berlin.10.1007/978-1-4612-4224-6
Search in Google Scholar
Czornik, A. and ´Swierniak, A. (2001). On controllability with respect to the expectation of discrete time jump linear systems, Journal of the Franklin Institute 338(4): 443-453.10.1016/S0016-0032(01)00014-X
Search in Google Scholar
Czornik, A. and ´Swierniak, A. (2004). On direct controllability of discrete time jump linear system, Journal of the Franklin Institute 341(6): 491-503.10.1016/j.jfranklin.2004.05.005
Search in Google Scholar
Czornik, A. and ´Swierniak, A. (2005). Controllability of discrete time jump linear systems, Dynamics of Continuous Discrete and Impulsive Systems B: Applications & Algorithms 12(2): 165-189.
Search in Google Scholar
Dauer, J. and Mahmudov, N. (2002). Approximate controllability of semilinear functional equations in Hilbert spaces, Journal of Mathematical Analysis and Applications 273(2): 310-327.10.1016/S0022-247X(02)00225-1
Search in Google Scholar
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): 42-56.10.1016/j.jmaa.2005.09.089
Search in Google Scholar
Do, V. (1989). A note on approximate controllability of semilinear systems, Systems & Control Letters 12(4): 365-371.10.1016/0167-6911(89)90047-9
Search in Google Scholar
Dubov, M. and Mordukhovich, B. (1978). Theory of controllability of linear stochastic systems, Differential Equations 14: 1609-1612.
Search in Google Scholar
George, R.K. (1995). Approximate controllability of nonautonomous semilinear systems, Nonlinear Analysis: Theory, Methods & Applications 24(9): 1377-1393.10.1016/0362-546X(94)E0082-R
Search in Google Scholar
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): 457-473.10.1016/0022-247X(91)90345-Z
Search in Google Scholar
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): 437-470.10.1016/S0034-4877(05)80096-5
Search in Google Scholar
Henríquez, H.R. (2008). Approximate controllability of linear distributed control systems, Applied Mathematics Letters 21(10): 1041-1045.10.1016/j.aml.2007.10.024
Search in Google Scholar
Hong, X.Z. (1982). A note on approximate controllability for semilinear one-dimensional heat equation, Applied Mathematics and Optimization 8(1): 275-285.10.1007/BF01447763
Search in Google Scholar
Jeong, J.-M. and Roh, H.-H. (2006). Approximate controllability for semilinear retarded systems, Journal of Mathematical Analysis and Applications 321(2): 961-975.10.1016/j.jmaa.2005.09.005
Search in Google Scholar
Klamka, J. (2000). Constrained approximate controllability, IEEE Transactions on Automatic Control 45(9): 1745-1749.10.1109/9.880640
Search in Google Scholar
Kryszewski, W. and Zezza, P. (1994). Remarks on the relay controllability of control systems, Journal of Mathematical Analysis and Applications 188(1): 45-65.10.1006/jmaa.1994.1410
Search in Google Scholar
Kumlin, P. (2004). A note on fixed point theory, Functional Analysis Lecture, Mathematics, Chalmers & GU, Gothenburg.
Search in Google Scholar
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): 109-154.10.1007/BF01442394
Search in Google Scholar
Li, M., Wang, M. and Zhang, F. (2006). Controllability of impulsive functional differential systems in Banach spaces, Chaos, Solitons & Fractals 29(1): 175-181.10.1016/j.chaos.2005.08.041
Search in Google Scholar
Mahmudov, N.I. (2001a). Controllability of linear stochastic systems, IEEE Transactions on Automatic Control 46(5): 724-731.10.1109/9.920790
Search in Google Scholar
Mahmudov, N.I. (2001b). Controllability of linear stochastic systems in Hilbert spaces, Journal of Mathematical Analysis and Applications 259(1): 64-82.10.1006/jmaa.2000.7386
Search in Google Scholar
Mahmudov, N.I. (2002). On controllability of semilinear stochastic systems in Hilbert spaces, IMA Journal of Mathematical Control and Information 19(4): 363-376.10.1093/imamci/19.4.363
Search in Google Scholar
Mahmudov, N.I. (2003). Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM Journal on Control and Optimization 42(5): 1604-1622.10.1137/S0363012901391688
Search in Google Scholar
Mahmudov, N.I. (2008). Approximate controllability of evolution systems with nonlocal conditions, Nonlinear Analysis: Theory, Methods & Applications 68(3): 536-546.10.1016/j.na.2006.11.018
Search in Google Scholar
Mahmudov, N. and Zorlu, S. (2003). Controllability of non-linear stochastic systems, International Journal of Control 76(2): 95-104.10.1080/0020717031000065648
Search in Google Scholar
Naito, K. (1987). Controllability of semilinear control systems dominated by the linear part, SIAM Journal on Control and Optimization 25(3): 715-722.10.1137/0325040
Search in Google Scholar
Naito, K. (1989). Approximate controllability for trajectories of semilinear control systems, Journal of Optimization Theory and Applications 60(1): 57-65.10.1007/BF00938799
Search in Google Scholar
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): 1-16.10.1186/s13662-015-0454-2
Search in Google Scholar
Pazy, A. (2012). Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Science & Business Media, Berlin.
Search in Google Scholar
Przeradzki, B. (2012). A generalization of Krasnoselskii fixed point theorem for sums of compact and contractible maps with application, Open Mathematics 10(6): 2012-2018.10.2478/s11533-012-0102-y
Search in Google Scholar
Radhakrishnan, B. and Balachandran, K. (2011). Controllability of impulsive neutral functional evolution integrodifferential systems with infinite delay, Nonlinear Analysis: Hybrid Systems 5(4): 655-670.10.1016/j.nahs.2011.05.001
Search in Google Scholar
Sakthivel, R., Mahmudov, N. and Kim, J. (2007). Approximate controllability of nonlinear impulsive differential systems, Reports on Mathematical Physics 60(1): 85-96.10.1016/S0034-4877(07)80100-5
Search in Google Scholar
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): 1777-1797.10.11650/twjm/1500406016
Search in Google Scholar
Shen, L. and Sun, J. (2011). Approximate controllability of stochastic impulsive systems with control-dependent coefficients, IET Control Theory & Applications 5(16): 1889-1894.10.1049/iet-cta.2010.0422
Search in Google Scholar
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): 301-305.10.2478/v10175-012-0040-7
Search in Google Scholar
Sırbu, M. and Tessitore, G. (2001). Null controllability of an infinite dimensional SDE with state-and control-dependent noise, Systems & Control Letters 44(5): 385-394.10.1016/S0167-6911(01)00158-X
Search in Google Scholar
Wang, L. (2006). Approximate controllability and approximate null controllability of semilinear systems, Communications on Pure and Applied Analysis 5(4): 953-962.10.3934/cpaa.2006.5.953
Search in Google Scholar
Zang, Y. and Li, J. (2013). Approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions, Boundary Value Problems 2013(1): 1-13.10.1186/1687-2770-2013-193
Search in Google Scholar
Zhou, H.X. (1983). Approximate controllability for a class of semilinear abstract equations, SIAM Journal on Control and Optimization 21(4): 551-565. 10.1137/0321033
Search in Google Scholar

Schauder’s fixed-point theorem in approximate controllability problems

Abstract

Paradigm

My account