A study on analytic resolvent semilinear integro-differential equations with control functions

Document Type : Research Paper

Authors

Rajkiya Engineering College Kannauj, India

Abstract

The goal of this research is to look at some of the sufficient conditions for approximate controllability in nonlinear resolvent integro-differential evolution control systems. We have considered that nonlinear term is satisfying Lipschitz continuity. To show the key results, we employ Gronwall's inequality, semigroup theory, and the resolvent operators. The main results have been discussed under two sets of assumptions. Application of common fixed point theorems such as Banach, Schauder, Sadovskii, etc. is avoided as discussed earlier by several researchers in the available literature. Finally, one case study based on the proposed problem is discussed in order to verify the theoretical findings.

Keywords

[1] P. Balasubramaniam and P. Tamilalagan, Approximate controllability of a class of fractional neutral stochastic
integro-differential inclusions with infinite delay by using Mainardi’s function, Appl. Math. Comput. 256 (2015),
232–246.
[2] W. Desch, R. Grimmer and W. Schappacher, Some considerations for Linear integrodifferential equations, J.
Math. Anal. Appl. 104 (1984), no. 1, 219–234.
[3] C. Dineshkumar, R. Udhayakumar, V. Vijayakumar and K.S. Nisar, A discussion on the approximate controllability of Hilfer fractional neutral stochastic integro-differential systems, Chaos Solitons Fractals 142 (2021),
110472[4] C. Dineshkumar, R. Udhayakumar, V. Vijayakumar, K.S. Nisar and A. Shukla, A note on the approximate
controllability of Sobolev type fractional stochastic integro-differential delay inclusions with order 1 < r < 2,
Math. Comput. Simul. 190 (2021), 1003–1026.
[5] J.P.C. dos Santos, M. Mallika Arjunan and C. Cuevas, Existence results for fractional neutral integro-differential
equations with state-dependent delay, Comput. Math. Appl. 62 (2011), 1275–1283.
[6] J.P.C. dos Santos, V. Vijayakumar and R. Murugesu, Existence of mild solutions for nonlocal Cauchy problem
for fractional neutral integro-differential equation with unbounded delay, Commun. Math. Anal. 14 (2013), no.
1, 59–71.
[7] R. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc. 273 (1982),
333–349.
[8] R. Grimmer and R. Miller, Existence, uniqueness, and continuity for integral equations in a Banach space, J.
Math. Anal. Appl. 57 (1977), no. 2, 429–447.
[9] R. Grimmer and A.J. Pritchard, Analytic resolvent operators for integral equations in a Banach space, J. Differ.
Equ. 50 (1983), 234–259.
[10] R. Grimmer and J. Pr¨uss, On linear Volterra equations in Banach spaces, Comput. Math. Appl. 11 (1985), no.
1-3, 189–205.
[11] K. Kavitha, V. Vijayakumar, A. Shukla, K.S. Nisar and R. Udhayakumar, Results on approximate controllability
of Sobolev-type fractional neutral differential inclusions of Clarke subdifferential type, Chaos Solitons Fractals 151
(2021), Paper No. 111264, 1–8.
[12] K. Kavitha, V. Vijayakumar, R. Udhayakumar and C. Ravichandran, Results on controllability of Hilfer fractional
differential equations with infinite delay via measures of noncompactness, Asian J. Control 24 (2022), no. 3, 1406–
1415
[13] N.I. Mahmudov, Approximate controllability of some nonlinear systems in Banach spaces, Boundary Value Prob.
2013 (2013), no. 1, 1–13.
[14] N.I. Mahmudov and A. Denker, On controllability of linear stochastic systems, Int. J. Control 73 (2000), 144–151.
[15] N.I. Mahmudov, R. Murugesu, C. Ravichandran and V. Vijayakumar, Approximate controllability results for
fractional semilinear integro-differential inclusions in Hilbert spaces, Results Math. 71 (2017), 45–61.
[16] N.I. Mahmudov, V. Vijayakumar and R. Murugesu, Approximate controllability of second-order evolution differential inclusions in Hilbert spaces, Mediterr. J. Math. 13 (2016), no. 3, 3433–3454.
[17] N.I. Mahmudov and S. Zorlu, On the approximate controllability of fractional evolution equations with compact
analytic semigroup, J. Comput. Appl. Math. 259 (2014), 194–204.
[18] M. Mohan Raja, V. Vijayakumar, A. Shukla, K.S. Nisar and S. Rezapour, New discussion on nonlocal controllability for fractional evolution system of order 1 < r < 2, Adv. Difference Equ. 2021 (2021), no. 1, Paper No. 481,
1–19.
[19] M. Mohan Raja, V. Vijayakumar, R. Udhayakumar, Y. Zhou, A new approach on the approximate controllability
of fractional differential evolution equations of order 1 < r < 2 in Hilbert spaces, Chaos, Solitons Fractals 141
(2020), 1–10.
[20] K. Naito, Controllability of semilinear control systems dominated by the linear part, SIAM J. Control Optim. 25
(1987), no. 3, 715–722.
[21] R. Patel, A. Shukla and S. Singh Jadon, Existence and optimal control problem for semilinear fractional order
(1,2] control system, Math. Meth. Appl. Sci. 2020 (2020), no. Special issue, 1- 12.
[22] A. Shukla, N. Sukavanam and D.N. Pandey, Controllability of semilinear stochastic system with multiple delays
in control, IFAC Proc. 47 (2014), 306–312.
[23] A. Shukla and R. Patel, Existence and optimal control results for second-order semilinear system in Hilbert spaces,
Circuits Syst. Signal Process 40 (2021), 4246–4258.
[24] S. Rezapour, H.R. Henr´─▒quez, V. Vijayakumar, K.S. Nisar and A. Shukla, A note on existence of mild solutionsfor second-order neutral integro-differential evolution equations with state-dependent delay, Fractal Fractional 5
(2021), no. 3, 126.
[25] R. Ravi Kumar, Nonlocal Cauchy problem for analytic resolvent operator integrodifferential equations in Banach
spaces, Appl. Math. Comput. 204 (2008), 352–362.
[26] R. Ravi Kumar, Regularity of solutions of evolution integrodifferential equations with deviating argument, Appl.
Math. Comput. 217 (2011), 9111-9121.
[27] R. Sakthivel, R. Ganesh and S.M. Anthoni, Approximate controllability of fractional nonlinear differential inclusions, Appl. Math. Comput. 225 (2013), 708–717.
[28] A. Singh, A. Shukla, V. Vijayakumar and R. Udhayakumar, Asymptotic stability of fractional order (1, 2]
stochastic delay differential equations in Banach spaces, Chaos, Solitons Fractals 150 (2021), 111095.
[29] N. Sukavanam and S. Kumar, Approximate controllability of fractional order semilinear delay systems, J. Optim.
Theory Appl. 151 (2011), no. 2, 373–384.
[30] A. Shukla, N. Sukavanam and D.N. Pandey, Approximate Controllability of Semilinear Fractional Control Systems
of Order (1,2], Proc. Conf. Control Appl. Soc. Ind. Appl. Math., 2015, pp. 175-180
[31] A. Shukla, N. Sukavanam and D.N. Pandey, Complete controllability of semi-linear stochastic system with delay,
Rend. Circ. Mat. Palermo 64 (2015), 209–220.
[32] A. Shukla, N. Sukavanam and D.N. Pandey, Approximate controllability of semilinear fractional stochastic control
system, Asian-Eur. J. Math. 11 (2018), no. 6, 1850088, 1–15.
[33] V. Vijayakumar, Approximate controllability results for analytic resolvent integro-differential inclusions in Hilbert
spaces, Int. J. Control 91 (2018), no. 1, 204–214.
[34] V. Vijayakumar, C. Ravichandran and R. Murugesu and J.J. Trujillo, Controllability results for a class of fractional
semilinear integro-differential inclusions via resolvent operators, Appl. Math. Comput. 247 (2014), 152–161.
[35] V. Vijayakumar, A. Selvakumar and R. Murugesu, Controllability for a class of fractional neutral integrodifferential equations with unbounded delay, Appl. Math. Comput. 232 (2014), 303–312.
[36] Z. Yan, Approximate controllability of partial neutral functional differential systems of fractional order with statedependent delay, Int. J. Control 85 (2012), no. 8, 1051–1062.
Volume 13, Issue 2
July 2022
Pages 1685-1692
  • Receive Date: 28 December 2021
  • Revise Date: 20 March 2022
  • Accept Date: 23 March 2022