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December  2017, 6(4): 599-613. doi: 10.3934/eect.2017030

Exact controllability results for a class of abstract nonlocal Cauchy problem with impulsive conditions

Poongodi Rathinasamy 1,, , Murugesu Rangasamy 2, and  Nirmalkumar Rajendran 2,

1. 

Department of Mathematics, Kongunadu Arts and Science College, Coimbatore -641 029, Tamil Nadu, India
2. 

Department of Mathematics, SRMV College of Arts and Science, Coimbatore -641 020, Tamil Nadu, India

* Corresponding author: R. Poongodi

Received  June 2016 Revised  May 2017 Published  September 2017

This paper deals with exact controllability of a class of abstract nonlocal Cauchy problem with impulsive conditions in Banach spaces. By using Sadovskii fixed point theorem and Mönch fixed point theorem, exact controllability results are obtained without assuming the compactness and Lipschitz conditions for nonlocal functions. An example is given to illustrate the main results.

Keywords: Exact controllability, abstract Cauchy problem, the measure of noncompactness, impulsive systems, fixed point theorem.
Mathematics Subject Classification: Primary: 34A60, 34A37; Secondary: 34K30.
Citation: Poongodi Rathinasamy, Murugesu Rangasamy, Nirmalkumar Rajendran. Exact controllability results for a class of abstract nonlocal Cauchy problem with impulsive conditions. Evolution Equations & Control Theory, 2017, 6 (4) : 599-613. doi: 10.3934/eect.2017030
References:
[1]

A. Anguraj and M. Mallika Arjunan, Existence and uniqueness of mild and classical solutions of impulsive evolution equations, Electronic Journal of Differential Equations, 111 (2005), 1-8.   Google Scholar

[2]

A. Anguraj and M. Mallika Arjunan, Existence results for an impulsive neutral integro-differential equations in Banach spaces, Nonlinear Studies, 16 (2009), 33-48.   Google Scholar

[3]

A. Anguraj and K. Karthikeyan, Existence of solutions for impulsive neutral functional differential equations with nonlocal conditions, Nonlinear Analysis, 70 (2009), 2717-2721.  doi: 10.1016/j.na.2008.03.059.  Google Scholar

[4]

K. Balachandran, J. Y. Park and S. H. Park, Controllability of nonlocal impulsive quasilinear integrodifferential systems in Banach spaces, Reports on Mathematical Physics, 65 (2010), 247-257.  doi: 10.1016/S0034-4877(10)80019-9.  Google Scholar

[5]

J. Banas and K. Goebel, Measure of Noncompactness in Banach Space, in: Lecture Notes in Pure and Applied Matyenath, , Dekker, New York, 1980.  Google Scholar

[6]

I. Benedetti, V. Obukhovskii and P. Zecca, Controllability for impulsive semilinear functional differential inclusions with a non-compact evolution operator, Discussiones Mathematicae Differential Inclusions, Control and Optimization, 31 (2011), 39-69.  doi: 10.7151/dmdico.1127.  Google Scholar

[7]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional System, Birkh" auser Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4581-6.  Google Scholar

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L. Byszewski, Theorems about existence and uniqueness of solutions of a semi-linear evolution nonlocal Cauchy problem, Journal of Mathematical Analysis and Applications, 162 (1991), 494-505.  doi: 10.1016/0022-247X(91)90164-U.  Google Scholar

[9]

L. Byszewski, Existence and uniqueness of classical solutions to a functional differential abstract nonlocal Cauchy problem, Journal of Applied Mathematics and Stochastic Analysis, 12 (1999), 91-97.  doi: 10.1155/S1048953399000088.  Google Scholar

[10]

Y. K. Chang, A. Anguraj and M. Mallika Arjunan, Existence results for impulsive neutral functional differential equations with infinite delay, Nonlinear Analysis: Hybrid Systems, 2 (2008), 209-218.  doi: 10.1016/j.nahs.2007.10.001.  Google Scholar

[11]

Y. K. Chang, J. J. Nieto and W. S. Li, Controllability of semilinear differential systems with nonlocal initial conditions in Banach spaces, Journal of Optimization Theory and Applications, 142 (2009), 267-273.  doi: 10.1007/s10957-009-9535-2.  Google Scholar

[12]

P. Y. Chen and Y. X. Li, Existence and uniqueness of strong solutions for nonlocal evolution equations, Electronic Journal of Differential Equations, 18 (2014), 1-9.   Google Scholar

[13]

C. Cuevas, E. Hernandez and M. Rabello, The existence of solutions for impulsive neutral functional differential equations, Computers and Mathematics with Applications, 58 (2009), 744-757.  doi: 10.1016/j.camwa.2009.04.008.  Google Scholar

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K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

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X. Fu, Controllability of abstract neutral functional differential systems with unbounded delay, Applied Mathematics and Computation Archive, 151 (2004), 299-314.  doi: 10.1016/S0096-3003(03)00342-4.  Google Scholar

[16]

M. Guo, X. Xue and R. Li, Controllability of impulsive evolution inclusions with nonlocal conditions, Journal of Optimization Theory and Applications, 120 (2004), 355-374.  doi: 10.1023/B:JOTA.0000015688.53162.eb.  Google Scholar

[17]

S. Ji, G. Li and M. Wang, Controllability of impulsive differential systems with nonlocal conditions, Applied Mathematics and Computation, 217 (2011), 6981-6989.  doi: 10.1016/j.amc.2011.01.107.  Google Scholar

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M. Kamenskii, P. Obukhovskii and Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter, 2001. doi: 10.1515/9783110870893.  Google Scholar

[19]

V. Kavitha, M. M. Arjunan and C. Ravichandran, Existence results for impulsive systems with nonlocal conditions in Banach spaces, The Journal of Nonlinear Sciences and Applications, 4 (2011), 138-151.   Google Scholar

[20]

J. Liang, H. J. Liu and T. Xiao, Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Mathematical and Computer Modelling, 49 (2009), 798-804.  doi: 10.1016/j.mcm.2008.05.046.  Google Scholar

[21]

J. Liang and H. Yang, Controllability of fractional integro-differential evolution equations with nonlocal conditions, Applied Mathematics and Computation, 254 (2015), 20-29.  doi: 10.1016/j.cam.2015.03.017.  Google Scholar

[22]

J. A. Machado, C. Ravichandran, M. Rivero and J. J. Trujillo, Controllability results for impulsive mixed-type functional integro-differential evolution equations with nonlocal conditions, Fixed Point Theory and Applications, 2013 (2013), 1-16.  doi: 10.1186/1687-1812-2013-66.  Google Scholar

[23]

N. I. Mahmudov and A. Denker, On controllability of linear stochastic system, International Journal of Control, 73 (2000), 144-151.  doi: 10.1080/002071700219849.  Google Scholar

[24]

H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Analysis, 4 (1980), 985-999.  doi: 10.1016/0362-546X(80)90010-3.  Google Scholar

[25]

V. Obukhovski and P. Zecca, Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup, Nonlinear Analysis, 70 (2009), 3424-3436.  doi: 10.1016/j.na.2008.05.009.  Google Scholar

[26]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[27]

B. Radhakrishnan, A. Mohanraj and V. Vinoba, Existence of solutions for nonlinear impulsive neutral integro-differential equations of Sobolev type with nonlocal conditions in Banach spaces, Electronic Journal of Differential Equations, 18 (2013), 1-13.   Google Scholar

[28]

S. Sivasankaran, M. Mallika Arjunan and V. Vijayakumar, Existence of global solutions for impulsive functional differential equations with nonlocal conditions, The Journal of Nonlinear Sciences and its Applications, 4 (2011), 102-114.   Google Scholar

[29]

S. Sivasankaran, V. Vijayakumar and M. Mallika Arjunan, Existence of global solutions for impulsive abstract partial neutral functional differential equations, International Journal of Nonlinear Science, 11 (2011), 412-426.   Google Scholar

[30]

Z. Tai, Controllability of fractional impulsive neutral integrodifferential systems with a nonlocal Cauchy condition in Banach spaces, Applied Mathematics Letters, 24 (2011), 2158-2161.  doi: 10.1016/j.aml.2011.06.018.  Google Scholar

[31]

C. C. Travis and G. F. Webb, Partial functional differential equations with deviating arguments in time variables, Journal of Mathematical Analysis and Applications, 56 (1976), 397-409.  doi: 10.1016/0022-247X(76)90052-4.  Google Scholar

[32]

V. Vijayakumar, C. Ravichandran and R. Murugesu, Nonlocal controllability of mixed Volterra-Fredholm type fractional semilinear integro-differential inclusions in banach spaces, Dynamics of Continuous, Discrete and Impulsive Systems: Series B, 20 (2013), 485-502.   Google Scholar

[33]

V. Vijayakumar, A. Selvakumar and R. Murugesu, Controllability for a class of fractional neutral integro-differential equations with unbounded delay, Applied Mathematics and Computation, 232 (2014), 303-312.  doi: 10.1016/j.amc.2014.01.029.  Google Scholar

[34]

V. Vijayakumar, C. Ravichandran, R. Murugesu and J. J. Trujillo, Controllability results for a class of fractional semilinear integro-differential inclusions via resolvent operators, Applied Mathematics and Computation, 247 (2014), 152-161.  doi: 10.1016/j.amc.2014.08.080.  Google Scholar

[35]

J. Wang and W. Wei, Controllability of integrodifferential systems with nonlocal initial conditions in Banach spaces, Journal of Mathematical Sciences, 177 (2011), 459-465.  doi: 10.1007/s10958-011-0471-y.  Google Scholar

[36]

Y. Zhou, V. Vijayakumar and R. Murugesu, Controllability for fractional evolution inclusions without compactness, Evolution Equations and Control Theory, 4 (2015), 507-524.  doi: 10.3934/eect.2015.4.507.  Google Scholar

show all references

References:
[1]

A. Anguraj and M. Mallika Arjunan, Existence and uniqueness of mild and classical solutions of impulsive evolution equations, Electronic Journal of Differential Equations, 111 (2005), 1-8.   Google Scholar

[2]

A. Anguraj and M. Mallika Arjunan, Existence results for an impulsive neutral integro-differential equations in Banach spaces, Nonlinear Studies, 16 (2009), 33-48.   Google Scholar

[3]

A. Anguraj and K. Karthikeyan, Existence of solutions for impulsive neutral functional differential equations with nonlocal conditions, Nonlinear Analysis, 70 (2009), 2717-2721.  doi: 10.1016/j.na.2008.03.059.  Google Scholar

[4]

K. Balachandran, J. Y. Park and S. H. Park, Controllability of nonlocal impulsive quasilinear integrodifferential systems in Banach spaces, Reports on Mathematical Physics, 65 (2010), 247-257.  doi: 10.1016/S0034-4877(10)80019-9.  Google Scholar

[5]

J. Banas and K. Goebel, Measure of Noncompactness in Banach Space, in: Lecture Notes in Pure and Applied Matyenath, , Dekker, New York, 1980.  Google Scholar

[6]

I. Benedetti, V. Obukhovskii and P. Zecca, Controllability for impulsive semilinear functional differential inclusions with a non-compact evolution operator, Discussiones Mathematicae Differential Inclusions, Control and Optimization, 31 (2011), 39-69.  doi: 10.7151/dmdico.1127.  Google Scholar

[7]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional System, Birkh" auser Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4581-6.  Google Scholar

[8]

L. Byszewski, Theorems about existence and uniqueness of solutions of a semi-linear evolution nonlocal Cauchy problem, Journal of Mathematical Analysis and Applications, 162 (1991), 494-505.  doi: 10.1016/0022-247X(91)90164-U.  Google Scholar

[9]

L. Byszewski, Existence and uniqueness of classical solutions to a functional differential abstract nonlocal Cauchy problem, Journal of Applied Mathematics and Stochastic Analysis, 12 (1999), 91-97.  doi: 10.1155/S1048953399000088.  Google Scholar

[10]

Y. K. Chang, A. Anguraj and M. Mallika Arjunan, Existence results for impulsive neutral functional differential equations with infinite delay, Nonlinear Analysis: Hybrid Systems, 2 (2008), 209-218.  doi: 10.1016/j.nahs.2007.10.001.  Google Scholar

[11]

Y. K. Chang, J. J. Nieto and W. S. Li, Controllability of semilinear differential systems with nonlocal initial conditions in Banach spaces, Journal of Optimization Theory and Applications, 142 (2009), 267-273.  doi: 10.1007/s10957-009-9535-2.  Google Scholar

[12]

P. Y. Chen and Y. X. Li, Existence and uniqueness of strong solutions for nonlocal evolution equations, Electronic Journal of Differential Equations, 18 (2014), 1-9.   Google Scholar

[13]

C. Cuevas, E. Hernandez and M. Rabello, The existence of solutions for impulsive neutral functional differential equations, Computers and Mathematics with Applications, 58 (2009), 744-757.  doi: 10.1016/j.camwa.2009.04.008.  Google Scholar

[14]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[15]

X. Fu, Controllability of abstract neutral functional differential systems with unbounded delay, Applied Mathematics and Computation Archive, 151 (2004), 299-314.  doi: 10.1016/S0096-3003(03)00342-4.  Google Scholar

[16]

M. Guo, X. Xue and R. Li, Controllability of impulsive evolution inclusions with nonlocal conditions, Journal of Optimization Theory and Applications, 120 (2004), 355-374.  doi: 10.1023/B:JOTA.0000015688.53162.eb.  Google Scholar

[17]

S. Ji, G. Li and M. Wang, Controllability of impulsive differential systems with nonlocal conditions, Applied Mathematics and Computation, 217 (2011), 6981-6989.  doi: 10.1016/j.amc.2011.01.107.  Google Scholar

[18]

M. Kamenskii, P. Obukhovskii and Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter, 2001. doi: 10.1515/9783110870893.  Google Scholar

[19]

V. Kavitha, M. M. Arjunan and C. Ravichandran, Existence results for impulsive systems with nonlocal conditions in Banach spaces, The Journal of Nonlinear Sciences and Applications, 4 (2011), 138-151.   Google Scholar

[20]

J. Liang, H. J. Liu and T. Xiao, Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Mathematical and Computer Modelling, 49 (2009), 798-804.  doi: 10.1016/j.mcm.2008.05.046.  Google Scholar

[21]

J. Liang and H. Yang, Controllability of fractional integro-differential evolution equations with nonlocal conditions, Applied Mathematics and Computation, 254 (2015), 20-29.  doi: 10.1016/j.cam.2015.03.017.  Google Scholar

[22]

J. A. Machado, C. Ravichandran, M. Rivero and J. J. Trujillo, Controllability results for impulsive mixed-type functional integro-differential evolution equations with nonlocal conditions, Fixed Point Theory and Applications, 2013 (2013), 1-16.  doi: 10.1186/1687-1812-2013-66.  Google Scholar

[23]

N. I. Mahmudov and A. Denker, On controllability of linear stochastic system, International Journal of Control, 73 (2000), 144-151.  doi: 10.1080/002071700219849.  Google Scholar

[24]

H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Analysis, 4 (1980), 985-999.  doi: 10.1016/0362-546X(80)90010-3.  Google Scholar

[25]

V. Obukhovski and P. Zecca, Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup, Nonlinear Analysis, 70 (2009), 3424-3436.  doi: 10.1016/j.na.2008.05.009.  Google Scholar

[26]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[27]

B. Radhakrishnan, A. Mohanraj and V. Vinoba, Existence of solutions for nonlinear impulsive neutral integro-differential equations of Sobolev type with nonlocal conditions in Banach spaces, Electronic Journal of Differential Equations, 18 (2013), 1-13.   Google Scholar

[28]

S. Sivasankaran, M. Mallika Arjunan and V. Vijayakumar, Existence of global solutions for impulsive functional differential equations with nonlocal conditions, The Journal of Nonlinear Sciences and its Applications, 4 (2011), 102-114.   Google Scholar

[29]

S. Sivasankaran, V. Vijayakumar and M. Mallika Arjunan, Existence of global solutions for impulsive abstract partial neutral functional differential equations, International Journal of Nonlinear Science, 11 (2011), 412-426.   Google Scholar

[30]

Z. Tai, Controllability of fractional impulsive neutral integrodifferential systems with a nonlocal Cauchy condition in Banach spaces, Applied Mathematics Letters, 24 (2011), 2158-2161.  doi: 10.1016/j.aml.2011.06.018.  Google Scholar

[31]

C. C. Travis and G. F. Webb, Partial functional differential equations with deviating arguments in time variables, Journal of Mathematical Analysis and Applications, 56 (1976), 397-409.  doi: 10.1016/0022-247X(76)90052-4.  Google Scholar

[32]

V. Vijayakumar, C. Ravichandran and R. Murugesu, Nonlocal controllability of mixed Volterra-Fredholm type fractional semilinear integro-differential inclusions in banach spaces, Dynamics of Continuous, Discrete and Impulsive Systems: Series B, 20 (2013), 485-502.   Google Scholar

[33]

V. Vijayakumar, A. Selvakumar and R. Murugesu, Controllability for a class of fractional neutral integro-differential equations with unbounded delay, Applied Mathematics and Computation, 232 (2014), 303-312.  doi: 10.1016/j.amc.2014.01.029.  Google Scholar

[34]

V. Vijayakumar, C. Ravichandran, R. Murugesu and J. J. Trujillo, Controllability results for a class of fractional semilinear integro-differential inclusions via resolvent operators, Applied Mathematics and Computation, 247 (2014), 152-161.  doi: 10.1016/j.amc.2014.08.080.  Google Scholar

[35]

J. Wang and W. Wei, Controllability of integrodifferential systems with nonlocal initial conditions in Banach spaces, Journal of Mathematical Sciences, 177 (2011), 459-465.  doi: 10.1007/s10958-011-0471-y.  Google Scholar

[36]

Y. Zhou, V. Vijayakumar and R. Murugesu, Controllability for fractional evolution inclusions without compactness, Evolution Equations and Control Theory, 4 (2015), 507-524.  doi: 10.3934/eect.2015.4.507.  Google Scholar

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