Approximate controllability results in \(\alpha \)-norm for some partial functional integrodifferential equations with nonlocal initial conditions in Banach spaces. (English) Zbl 1521.93017
In this work the authors have discussed the approximate controllability of some nonlinear partial functional integrodifferential equations with nonlocal initial condition in Hilbert spaces under the assumption that the corresponding linear part is approximately controllable. The results are obtained by using the fractional power theory and \(\alpha\)-norm, the measure of noncompactness and the Mönch fixed-point theorem, and the theory of analytic resolvent operators for integral equations. This paper is a generalization of the work of N. I. Mahmudov [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 68, No. 3, 536–546 (2008; Zbl 1129.93004)] and established without the assumption of compactness of the resolvent operator. An example is provided to illustrate the main results.
Reviewer: Krishnan Balachandran (Coimbatore)
MSC:
| 93B05 | Controllability |
| 93C20 | Control/observation systems governed by partial differential equations |
| 35R10 | Partial functional-differential equations |
| 45K05 | Integro-partial differential equations |
| 93B28 | Operator-theoretic methods |
| 47H10 | Fixed-point theorems |
| 47D06 | One-parameter semigroups and linear evolution equations |
Keywords:
approximate controllability; integro-differential equations; nonlocal initial condition; resolvent operator for integral equations; fractional power; measure of noncompactness; Mönch fixed-point theoremCitations:
Zbl 1129.93004References:
| [1] | R. Atmania and S. Mazouzi, Controllability of semilinear integrodifferential equations with nonlocal conditions, Electron. J. Differential Equations 2005 (2005), Paper No. 75. · Zbl 1075.34081 |
| [2] | J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes Pure Appl. Math. 60, Marcel Dekker, New York, 1980. · Zbl 0441.47056 |
| [3] | D. N. Chalishajar and A. Kumar, Total controllability of the second order semi-linear differential equation with infinite delay and non-instantaneous impulses, Math. Comput. Appl. 23 (2018), no. 3, Paper No. 32. |
| [4] | Y. K. Chang, J. J. Nieto and W. S. Li, Controllability of semilinear differential systems with nonlocal initial conditions in Banach spaces, J. Optim. Theory Appl. 142 (2009), no. 2, 267-273. · Zbl 1178.93029 |
| [5] | R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts Appl. Math. 21, Springer, New York, 1995. · Zbl 0839.93001 |
| [6] | W. Desch, R. Grimmer and W. Schappacher, Well-posedness and wave propagation for a class of integrodifferential equations in Banach space, J. Differential Equations 74 (1988), no. 2, 391-411. · Zbl 0663.45008 |
| [7] | K. Ezzinbi, G. Degla and P. Ndambomve, Controllability for some partial functional integrodifferential equations with nonlocal conditions in Banach spaces, Discuss. Math. Differ. Incl. Control Optim. 35 (2015), no. 1, 25-46. · Zbl 1513.93005 |
| [8] | K. Ezzinbi, H. Toure and I. Zabsonre, Existence and regularity of solutions for some partial functional integrodifferential equations in Banach spaces, Nonlinear Anal. 70 (2009), no. 7, 2761-2771. · Zbl 1176.45013 |
| [9] | X. Fu and R. Huang, Existence of solutions for neutral integro-differential equations with state-dependent delay, Appl. Math. Comput. 224 (2013), 743-759. · Zbl 1334.34143 |
| [10] | Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math. 27 (1990), no. 2, 309-321. · Zbl 0790.45009 |
| [11] | R. C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc. 273 (1982), no. 1, 333-349. · Zbl 0493.45015 |
| [12] | R. C. Grimmer and A. J. Pritchard, Analytic resolvent operators for integral equations in Banach space, J. Differential Equations 50 (1983), no. 2, 234-259. · Zbl 0519.45011 |
| [13] | A. Kumar, R. K. Vats and A. Kumar, Approximate controllability of second-order non-autonomous system with finite delay, J. Dyn. Control Syst. 26 (2020), no. 4, 611-627. · Zbl 1489.93010 |
| [14] | N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim. 42 (2003), no. 5, 1604-1622. · Zbl 1084.93006 |
| [15] | N. I. Mahmudov, Approximate controllability of evolution systems with nonlocal conditions, Nonlinear Anal. 68 (2008), no. 3, 536-546. · Zbl 1129.93004 |
| [16] | N. I. Mahmudov and S. Zorlu, Approximate controllability of fractional integro-differential equations involving nonlocal initial conditions, Bound. Value Probl. 2013 (2013), Article ID 118. · Zbl 1295.93013 |
| [17] | F. Z. Mokkedem and X. Fu, Approximate controllability of semi-linear neutral integro-differential systems with finite delay, Appl. Math. Comput. 242 (2014), 202-215. · Zbl 1334.93033 |
| [18] | H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal. 4 (1980), no. 5, 985-999. · Zbl 0462.34041 |
| [19] | A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer, New York, 1983. · Zbl 0516.47023 |
| [20] | R. Sakthivel, Y. Ren and N. I. Mahmudov, On the approximate controllability of semilinear fractional differential systems, Comput. Math. Appl. 62 (2011), no. 3, 1451-1459. · Zbl 1228.34093 |
| [21] | S. Selvi and M. Mallika Arjunan, Controllability results for impulsive differential systems with finite delay, J. Nonlinear Sci. Appl. 5 (2012), no. 3, Special issue, 206-219. · Zbl 1293.93107 |
| [22] | J. Wang, Z. Fan and Y. Zhou, Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces, J. Optim. Theory Appl. 154 (2012), no. 1, 292-302. · Zbl 1252.93028 |
| [23] | J. Wang and W. Wei, Controllability of integrodifferential systems with nonlocal initial conditions in Banach spaces, J. Math. Sci. (N. Y.) 177 (2011), no. 3, 459-465. · Zbl 1290.93025 |
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