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September  2020, 13(9): 2561-2573. doi: 10.3934/dcdss.2020138

Controllability analysis of nonlinear fractional order differential systems with state delay and non-instantaneous impulsive effects

Baskar Sundaravadivoo ,

Department of Mathematics, Alagappa University, Karaikudi-630 004, India

* Corresponding author

Received  November 2018 Revised  April 2019 Published  September 2020 Early access  November 2019

This manuscript prospects the controllability analysis of non-instantaneous impulsive Volterra type fractional differential systems with state delay. By enroling an appropriate Grammian matrix with the assistance of Laplace transform, the conditions to obtain the necessary and sufficiency for the controllability of non-instantaneous impulsive Volterra-type fractional differential equations are derived using algebraic approach and Cayley-Hamilton theorem. A distinctive approach presents in the manuscript, i have taken non-instantaneous impulses into the fractional order dynamical system with state delay and studied the controllability analysis, since this not exists in the available source of literature. Inclusively, i have provided two illustrative examples with the existence of non-instantaneous impulse into the fractional dynamical system. So this demonstrates the validity and efficacy of our obtained criteria of the main section.

Keywords: Fractional differential equations, Caputo fractional derivative, non-instantaneous impulses, Laplace transformation, Mittag-Leffler function.
Mathematics Subject Classification: 93B05, 34A08.
Citation: Baskar Sundaravadivoo. Controllability analysis of nonlinear fractional order differential systems with state delay and non-instantaneous impulsive effects. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2561-2573. doi: 10.3934/dcdss.2020138
References:
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R. Agarwal, M. Benchohra and B. Slimani, Existence results for differential equations with fractional order and impulses, Mem. Differential Equations Math. Phys., 44 (2008), 1-21.   Google Scholar

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R. Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109 (2010), 973-1033.  doi: 10.1007/s10440-008-9356-6.  Google Scholar

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R. Agarwal, S. Hristova and D. O. Regan, Noninstantaneous impulses in Caputo fractional differential equations and practical stability via Lyapunov functions, J. Franklin Inst., 354 (2017), 3097-3119.  doi: 10.1016/j.jfranklin.2017.02.002.  Google Scholar

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R. Agarwal, S. Hristova and D. O. Regan, p-Moment exponential stability of Caputo fractional differential equations with noninstantaneous random impulses, Journal of Applied Mathematics and Computing, 2016 (2016), 1-26.   Google Scholar

[5]

R. Agarwal, S. Hristova and D. O. Regan, Stability of Solutions to Impulsive Caputo Fractinal Differential Equations, Electron. J. Differential Equations, 2016.  Google Scholar

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R. Agarwal, S. Hristova and D. O. Regan, A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations, Fract. Calc. Appl. Anal., 19 (2016), 290-318.  doi: 10.1515/fca-2016-0017.  Google Scholar

[7]

R. Agarwal, D. O. Regan and S. Hristova, Monotone iterative technique for the initial value problem for differential equations with noninstantaneous impulses, Appl. Math. Comput., 298 (2017), 45-56.  doi: 10.1016/j.amc.2016.10.009.  Google Scholar

[8]

R. Agarwal, D. O. Regan and S. Hristova, Stability by Lyapunov like functions of nonlinear differential equations with noninstantaneous impulses, J. Appl. Math. Comput., 53 (2017), 147-168.  doi: 10.1007/s12190-015-0961-z.  Google Scholar

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M. Benchohra and D. Seba, Impulsive Fractional Differential Equations in Banach Spaces, Electron. J. Qual. Theory Differ. Equ., Special Edition I, 2009. doi: 10.14232/ejqtde.2009.4.8.  Google Scholar

[10]

G. Bonanno, R. Rodriquez-Lopez and S. Tersian, Existence of solutions to boundary value problem for impulsive fractional differential equation, Fract. Calc. Appl. Anal., 17 (2014), 717-744.  doi: 10.2478/s13540-014-0196-y.  Google Scholar

[11]

J. Cao and H. Chen, Some results on impulsive boundary valueproblem for fractional differential inclusions, Electron. J. Qual. Theory Differ. Equ., 11 (2011), 1-24.  doi: 10.14232/ejqtde.2011.1.11.  Google Scholar

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M. Feckan, J. R. Wang and Y. Zhou, Periodic solutions for nonlinear evolution equations with non-istantaneous impulses, Nonauton. Dyn. Syst., 1 (2014), 93-101.   Google Scholar

[13]

M. Feckan, Y. Zhou and J. Wang, On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 3050-3060.  doi: 10.1016/j.cnsns.2011.11.017.  Google Scholar

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J. Henderson and A. Ouahab, Impulsive differential inclusions with fractional order, Comput. Math. Appl., 59 (2010), 1191-1226.  doi: 10.1016/j.camwa.2009.05.011.  Google Scholar

[15]

E. Hernandez and D. O. Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 141 (2013), 1641-1649.  doi: 10.1090/S0002-9939-2012-11613-2.  Google Scholar

[16]

S. Hristova and R. Terzieva, Lipschitz Stability of Differential Equations with Non-Instantaneous Impulses, Adv. Difference Equ., 2016. doi: 10.1186/s13662-016-1045-6.  Google Scholar

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W. Jiang and W. Z. Song, Controllability of singular systems with control delay, Automatica, 37 (2001), 1873-1877.   Google Scholar

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R. E. Kalman, Y. C. Ho and K. S. Narendra, Controllability of linear dynamical systems, Contributions to Differential Equations, 1 (1963), 189-213.   Google Scholar

[19]

T. D. Ke and D. Lan, Decay integral solutions for a class of impulsive fractional differential equations in Banach spaces, Fract. Calc. Appl. Anal., 17 (2014), 96-121.  doi: 10.2478/s13540-014-0157-5.  Google Scholar

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A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.  Google Scholar

[21]

P. Li and Ch. Xu, Boundary Value Problems of Fractional Order Differential Equation with Integral Boundary Conditions and Not Instantaneous Impulses, J. Funct. Spaces, 2015. doi: 10.1155/2015/954925.  Google Scholar

[22]

N. I. Mahmudov, Controllability of Linear Stochastic Systems in Hilbert Spaces, J. Math. Anal. Appl., 259 (2001), 64-82.  doi: 10.1006/jmaa.2000.7386.  Google Scholar

[23]

N. I. Mahmudov, Controllability of Semilinear Stochastic Systems in Hilbert Spaces, J. Math. Anal. Appl., 288 (2003), 197-211.  doi: 10.1016/S0022-247X(03)00592-4.  Google Scholar

[24]

K. S. Miller and B. Ross, An Introduction to The Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1993.  Google Scholar

[25]

D. N. Pandey, S. Das and N. Sukavanam, Existence of solutions for a second order neutral differential equation with state dependent delay and not instantaneous impulses, Int. J. Nonlinear Sci., 18 (2014), 145-155.   Google Scholar

[26]

M. Pierri, D. O. Regan and V. Rolnik, Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses, Appl. Math. Comput., 219 (2013), 6743-6749.  doi: 10.1016/j.amc.2012.12.084.  Google Scholar

[27]

I. Podlubny, Fractional Differential Equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[28]

R. Rodriquez-Lopez and S. Tersian, Multiple solutions to boundary value problm for impulsive fractional differential equations, Fract. Calc. Appl. Anal., 17 (2014), 1016-1038.  doi: 10.2478/s13540-014-0212-2.  Google Scholar

[29]

X. B. Shu, Y. Lai and Y. Chen, The existence of mild solutions for impulsive fractional partial differential equations, Nonlinear Anal., 74 (2011), 2003-2011.  doi: 10.1016/j.na.2010.11.007.  Google Scholar

[30]

A. Sood and S. K. Srivastava, On Stability of Differential Systems with Noninstantaneous Impulses, Math. Probl. Eng., 2015. doi: 10.1155/2015/691687.  Google Scholar

[31]

C. Tunc, A note on the qualitative behaviors of non-linear Volterra integro-differential equation, J. Egyptian Math. Soc., 24 (2016), 187-192.  doi: 10.1016/j.joems.2014.12.010.  Google Scholar

[32]

C. Tunc and O. Tunc, New qualitative criteria for solutions of Volterra integro-differential equations, Arab Journal of Basic and Applied Sciences, 25 (2018), 158-165.   Google Scholar

[33]

J. R. Wang, M. Feckan and Y. Zhou, Relaxed controls for nonlinear fractional impulsive evolution equations, J. Optim. Theory Appl., 156 (2013), 13-32.  doi: 10.1007/s10957-012-0170-y.  Google Scholar

[34]

J. Wang and Z. Lin, A class of impulsive nonautonomous differential equations and Ulam - Hyers-Rassias stability, Math. Methods Appl. Sci., 38 (2015), 868-880.  doi: 10.1002/mma.3113.  Google Scholar

[35]

J. Wang, Y. Zhou and Z. Lin, On a new class of impulsive fractional differential equations, Appl. Math. Comput., 242 (2014), 649-657.  doi: 10.1016/j.amc.2014.06.002.  Google Scholar

[36]

R. Wang, M. Feckan and Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations, Dyn. Partial Differ. Equ., 8 (2011), 345-361.  doi: 10.4310/DPDE.2011.v8.n4.a3.  Google Scholar

[37]

X. Zhang, X. Huang and Z. Liu, The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay, Nonlinear Anal. Hybrid Syst., 4 (2010), 775-781.  doi: 10.1016/j.nahs.2010.05.007.  Google Scholar

show all references

References:
[1]

R. Agarwal, M. Benchohra and B. Slimani, Existence results for differential equations with fractional order and impulses, Mem. Differential Equations Math. Phys., 44 (2008), 1-21.   Google Scholar

[2]

R. Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109 (2010), 973-1033.  doi: 10.1007/s10440-008-9356-6.  Google Scholar

[3]

R. Agarwal, S. Hristova and D. O. Regan, Noninstantaneous impulses in Caputo fractional differential equations and practical stability via Lyapunov functions, J. Franklin Inst., 354 (2017), 3097-3119.  doi: 10.1016/j.jfranklin.2017.02.002.  Google Scholar

[4]

R. Agarwal, S. Hristova and D. O. Regan, p-Moment exponential stability of Caputo fractional differential equations with noninstantaneous random impulses, Journal of Applied Mathematics and Computing, 2016 (2016), 1-26.   Google Scholar

[5]

R. Agarwal, S. Hristova and D. O. Regan, Stability of Solutions to Impulsive Caputo Fractinal Differential Equations, Electron. J. Differential Equations, 2016.  Google Scholar

[6]

R. Agarwal, S. Hristova and D. O. Regan, A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations, Fract. Calc. Appl. Anal., 19 (2016), 290-318.  doi: 10.1515/fca-2016-0017.  Google Scholar

[7]

R. Agarwal, D. O. Regan and S. Hristova, Monotone iterative technique for the initial value problem for differential equations with noninstantaneous impulses, Appl. Math. Comput., 298 (2017), 45-56.  doi: 10.1016/j.amc.2016.10.009.  Google Scholar

[8]

R. Agarwal, D. O. Regan and S. Hristova, Stability by Lyapunov like functions of nonlinear differential equations with noninstantaneous impulses, J. Appl. Math. Comput., 53 (2017), 147-168.  doi: 10.1007/s12190-015-0961-z.  Google Scholar

[9]

M. Benchohra and D. Seba, Impulsive Fractional Differential Equations in Banach Spaces, Electron. J. Qual. Theory Differ. Equ., Special Edition I, 2009. doi: 10.14232/ejqtde.2009.4.8.  Google Scholar

[10]

G. Bonanno, R. Rodriquez-Lopez and S. Tersian, Existence of solutions to boundary value problem for impulsive fractional differential equation, Fract. Calc. Appl. Anal., 17 (2014), 717-744.  doi: 10.2478/s13540-014-0196-y.  Google Scholar

[11]

J. Cao and H. Chen, Some results on impulsive boundary valueproblem for fractional differential inclusions, Electron. J. Qual. Theory Differ. Equ., 11 (2011), 1-24.  doi: 10.14232/ejqtde.2011.1.11.  Google Scholar

[12]

M. Feckan, J. R. Wang and Y. Zhou, Periodic solutions for nonlinear evolution equations with non-istantaneous impulses, Nonauton. Dyn. Syst., 1 (2014), 93-101.   Google Scholar

[13]

M. Feckan, Y. Zhou and J. Wang, On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 3050-3060.  doi: 10.1016/j.cnsns.2011.11.017.  Google Scholar

[14]

J. Henderson and A. Ouahab, Impulsive differential inclusions with fractional order, Comput. Math. Appl., 59 (2010), 1191-1226.  doi: 10.1016/j.camwa.2009.05.011.  Google Scholar

[15]

E. Hernandez and D. O. Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 141 (2013), 1641-1649.  doi: 10.1090/S0002-9939-2012-11613-2.  Google Scholar

[16]

S. Hristova and R. Terzieva, Lipschitz Stability of Differential Equations with Non-Instantaneous Impulses, Adv. Difference Equ., 2016. doi: 10.1186/s13662-016-1045-6.  Google Scholar

[17]

W. Jiang and W. Z. Song, Controllability of singular systems with control delay, Automatica, 37 (2001), 1873-1877.   Google Scholar

[18]

R. E. Kalman, Y. C. Ho and K. S. Narendra, Controllability of linear dynamical systems, Contributions to Differential Equations, 1 (1963), 189-213.   Google Scholar

[19]

T. D. Ke and D. Lan, Decay integral solutions for a class of impulsive fractional differential equations in Banach spaces, Fract. Calc. Appl. Anal., 17 (2014), 96-121.  doi: 10.2478/s13540-014-0157-5.  Google Scholar

[20]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.  Google Scholar

[21]

P. Li and Ch. Xu, Boundary Value Problems of Fractional Order Differential Equation with Integral Boundary Conditions and Not Instantaneous Impulses, J. Funct. Spaces, 2015. doi: 10.1155/2015/954925.  Google Scholar

[22]

N. I. Mahmudov, Controllability of Linear Stochastic Systems in Hilbert Spaces, J. Math. Anal. Appl., 259 (2001), 64-82.  doi: 10.1006/jmaa.2000.7386.  Google Scholar

[23]

N. I. Mahmudov, Controllability of Semilinear Stochastic Systems in Hilbert Spaces, J. Math. Anal. Appl., 288 (2003), 197-211.  doi: 10.1016/S0022-247X(03)00592-4.  Google Scholar

[24]

K. S. Miller and B. Ross, An Introduction to The Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1993.  Google Scholar

[25]

D. N. Pandey, S. Das and N. Sukavanam, Existence of solutions for a second order neutral differential equation with state dependent delay and not instantaneous impulses, Int. J. Nonlinear Sci., 18 (2014), 145-155.   Google Scholar

[26]

M. Pierri, D. O. Regan and V. Rolnik, Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses, Appl. Math. Comput., 219 (2013), 6743-6749.  doi: 10.1016/j.amc.2012.12.084.  Google Scholar

[27]

I. Podlubny, Fractional Differential Equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[28]

R. Rodriquez-Lopez and S. Tersian, Multiple solutions to boundary value problm for impulsive fractional differential equations, Fract. Calc. Appl. Anal., 17 (2014), 1016-1038.  doi: 10.2478/s13540-014-0212-2.  Google Scholar

[29]

X. B. Shu, Y. Lai and Y. Chen, The existence of mild solutions for impulsive fractional partial differential equations, Nonlinear Anal., 74 (2011), 2003-2011.  doi: 10.1016/j.na.2010.11.007.  Google Scholar

[30]

A. Sood and S. K. Srivastava, On Stability of Differential Systems with Noninstantaneous Impulses, Math. Probl. Eng., 2015. doi: 10.1155/2015/691687.  Google Scholar

[31]

C. Tunc, A note on the qualitative behaviors of non-linear Volterra integro-differential equation, J. Egyptian Math. Soc., 24 (2016), 187-192.  doi: 10.1016/j.joems.2014.12.010.  Google Scholar

[32]

C. Tunc and O. Tunc, New qualitative criteria for solutions of Volterra integro-differential equations, Arab Journal of Basic and Applied Sciences, 25 (2018), 158-165.   Google Scholar

[33]

J. R. Wang, M. Feckan and Y. Zhou, Relaxed controls for nonlinear fractional impulsive evolution equations, J. Optim. Theory Appl., 156 (2013), 13-32.  doi: 10.1007/s10957-012-0170-y.  Google Scholar

[34]

J. Wang and Z. Lin, A class of impulsive nonautonomous differential equations and Ulam - Hyers-Rassias stability, Math. Methods Appl. Sci., 38 (2015), 868-880.  doi: 10.1002/mma.3113.  Google Scholar

[35]

J. Wang, Y. Zhou and Z. Lin, On a new class of impulsive fractional differential equations, Appl. Math. Comput., 242 (2014), 649-657.  doi: 10.1016/j.amc.2014.06.002.  Google Scholar

[36]

R. Wang, M. Feckan and Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations, Dyn. Partial Differ. Equ., 8 (2011), 345-361.  doi: 10.4310/DPDE.2011.v8.n4.a3.  Google Scholar

[37]

X. Zhang, X. Huang and Z. Liu, The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay, Nonlinear Anal. Hybrid Syst., 4 (2010), 775-781.  doi: 10.1016/j.nahs.2010.05.007.  Google Scholar

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