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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 and Continuous Dynamical Systems - S, 2020, 13 (9) : 2561-2573. doi: 10.3934/dcdss.2020138
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. 

[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.

[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.

[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. 

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[30]

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

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

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. 

[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.

[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.

[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. 

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[30]

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

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

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