Schauder’s fixed-point theorem in approximate controllability problems

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.

[1]  V. N. Do.  A note on approximate controllability of semilinear systems , 1989 .

[2]  Irena Lasiecka,et al.  Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems , 1989 .

[3]  Hernán R. Henríquez,et al.  Approximate controllability of linear distributed control systems , 2008, Appl. Math. Lett..

[4]  Abdelghani Ouahab,et al.  Controllability results for impulsive functional differential inclusions , 2004 .

[5]  Hong Xing Zhou,et al.  Approximate Controllability for a Class of Semilinear Abstract Equations , 1983 .

[6]  Mihai Sîrbu,et al.  Null controllability of an infinite dimensional SDE with state- and control-dependent noise , 2001, Syst. Control. Lett..

[7]  Nazim I. Mahmudov,et al.  Approximate controllability of evolution systems with nonlocal conditions , 2008 .

[8]  T. L Hicks,et al.  A note on fixed-point theorems , 1982 .

[9]  Yong-Kui Chang,et al.  Controllability of impulsive functional differential systems with infinite delay in Banach spaces , 2007 .

[10]  Jitao Sun,et al.  Approximate controllability of stochastic impulsive systems with control-dependent coefficients , 2011 .

[11]  K. Naito.  Approximate controllability for trajectories of semilinear control systems , 1989 .

[12]  Mrinal K. Ghosh,et al.  On the controllability of a class of nonlinear stochastic systems , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[13]  A. Świerniak,et al.  On Controllability of Discrete Time Jump Linear System , 2000 .

[14]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[15]  Adam Czornik,et al.  On direct controllability of discrete time jump linear system , 2004, J. Frankl. Inst..

[16]  Nazim I. Mahmudov,et al.  Controllability of non-linear stochastic systems , 2003 .

[17]  Junping Li,et al.  Approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions , 2013, Boundary Value Problems.

[18]  Rathinasamy Sakthivel,et al.  Controllability of integrodifferential systems in Banach spaces , 2001, Appl. Math. Comput..

[19]  Artur Babiarz,et al.  Output controllability of the discrete-time linear switched systems , 2016 .

[20]  Donal O'Regan,et al.  Controllability of semilinear differential equations and inclusions via semigroup theory in banach spaces , 2005 .

[21]  P. Zezza,et al.  Remarks on the Relay Controllability of Control Systems , 1994 .

[22]  Hans Zwart,et al.  An Introduction to Infinite-Dimensional Linear Systems Theory , 1995, Texts in Applied Mathematics.

[23]  Nazim I. Mahmudov,et al.  Controllability of Linear Stochastic Systems in Hilbert Spaces , 2001 .

[24]  Jerzy Klamka,et al.  Controllability of Discrete-Time Linear Switched Systems with Constrains on Switching Signal , 2015, ACIIDS.

[25]  J. P. Dauer,et al.  Controllability of Nonlinear Systems in Banach Spaces: A Survey , 2002 .

[26]  Nazim I. Mahmudov.  Controllability of linear stochastic systems , 2001, IEEE Trans. Autom. Control..

[27]  Nazim I. Mahmudov,et al.  On controllability of semilinear stochastic systems in Hilbert spaces , 2002 .

[28]  Jerzy Klamka,et al.  Constrained approximate controllability , 2000, IEEE Trans. Autom. Control..

[29]  J. Klamka,et al.  On constrained stochastic controllability of dynamical systems with multiple delays in control , 2012 .

[30]  Koichiro Naito,et al.  Controllability of semilinear control systems dominated by the linear part , 1987 .

[31]  J. Goldstein.  Semigroups of Linear Operators and Applications , 1985 .

[32]  Krishnan Balachandran,et al.  Controllability of impulsive neutral functional evolution integrodifferential systems with infinite delay , 2011 .

[33]  Nazim I. Mahmudov,et al.  Controllability of semilinear stochastic systems in Hilbert spaces , 2003 .

[34]  Nazim I. Mahmudov,et al.  Approximate controllability of semilinear functional equations in Hilbert spaces , 2002 .

[35]  S. Narayanamoorthy,et al.  Approximate controllability result for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions , 2015 .

[36]  R. Sakthivel,et al.  APPROXIMATE CONTROLLABILITY OF NONLINEAR DETERMINISTIC AND STOCHASTIC SYSTEMS WITH UNBOUNDED DELAY , 2010 .

[37]  Jerzy Klamka,et al.  The selected problems of controllability of discrete-time switched linear systems with constrained switching rule , 2015 .

[38]  Nazim I. Mahmudov,et al.  Approximate controllability of nonlinear impulsive differential systems , 2007 .

[39]  Lianwen Wang,et al.  Approximate controllability and approximate null controllability of semilinear systems , 2006 .

[40]  Lech Górniewicz,et al.  On the homotopy method in the fixed point index theory of multi-valued mappings of compact absolute neighborhood retracts , 1991 .

[41]  Ari Arapostathis,et al.  On the controllability of a class of nonlinear stochastic systems , 2001, Syst. Control. Lett..

[42]  Adam Czornik,et al.  On controllability with respect to the expectation of discrete time jump linear systems , 2001, J. Frankl. Inst..

[43]  Fengqin Zhang,et al.  Controllability of impulsive functional differential systems in Banach spaces , 2006 .

[44]  W. M. Bian.  Constrained controllability of some nonlinear systems , 1999 .

[45]  Xing Zhou Hong.  A note on approximate controllability for semilinear one-dimensional heat equation , 1982 .

[46]  Nazim I. Mahmudov,et al.  Approximate controllability of backward stochastic evolution equations in Hilbert spaces , 2006 .

[47]  Stojan Radenović,et al.  A note on fixed point theory for cyclic φ-contractions , 2015, Fixed Point Theory and Applications.

[48]  Raju K. George,et al.  Approximate controllability of nonautonomous semilinear systems , 1995 .

[49]  Bogdan Przeradzki,et al.  A generalization of Krasnosel’skii fixed point theorem for sums of compact and contractible maps with application , 2012 .

[50]  Agamirza E. Bashirov,et al.  On Controllability Conception for Stochastic Systems , 1997 .

[51]  Abdelghani Ouahab,et al.  Controllability results for functional semilinear differential inclusions in Fréchet spaces , 2005 .

[52]  Nazim I. Mahmudov,et al.  Approximate Controllability of Semilinear Deterministic and Stochastic Evolution Equations in Abstract Spaces , 2003, SIAM J. Control. Optim..

[53]  Jin-Mun Jeong,et al.  Approximate controllability for semilinear retarded systems , 2006 .

[54]  R. Bader,et al.  Fixed-point index for compositions of set-valued maps with proximally ∞-connected values on arbitrary ANR's , 1994 .

[55]  Nazim I. Mahmudov,et al.  On Concepts of Controllability for Deterministic and Stochastic Systems , 1999 .