Exact controllability of mild solutions for a class of semilinear differential equations

Abstract

In this paper, by using the noncompact measure, we study the exact controllability of mild solutions for semilinear differential equation when the nonlocal term is Lipschitz continuous in general Banach space. Here, the regularity conditions of noncompact measure assumption in previous literature are not used, but instead we adopt the estimate of noncompact measure in different topological spaces, give the proper control function by the property of quotient space, and obtain the exact controllability of the differential equation. Therefore, our results can be regarded as a Generalization of the controllability field.

Keywords

Exact controllability / noncompact measure / semilinear differential equation / nonlocal condition

References

[1] BanaśJGoebelK. Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, Vol 60. New York: Marcel Dekker, 1980

[2] Byszewski L. Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J Math Anal Appl 1991; 162(2): 494–505

[3] Chang Y K, Nieto J J, Li W S. Controllability of semilinear differential systems with nonlocal initial conditions in Banach spaces. J Optim Theory Appl 2009; 142(2): 267–273

[4] CurtainR FZwartH. An Introduction to Infinite-dimensional Linear Systems Theory. Texts in Applied Mathematics, Vol 21. New York: Springer-Verlag, 1995

[5] DeimlingK. Nonlinear Functional Analysis. Berlin: Springer-Verlag, 1985

[6] Fan Z B. Approximate controllability of fractional differential equations via resolvent operators. Adv Difference Equ 2014; 2014: 54

[7] Fan Z B, Dong Q X, Li G. Approximate controllability for semilinear composite fractional relaxation equations. Fract Calc Appl Anal 2016; 19(1): 267–284

[8] Ji S C, Li G, Wang M. Controllability of impulsive differential systems with nonlocal conditions. Appl Math Comput 2011; 217(16): 6981–6989

[9] Lian T T, Fan Z B, Li G. Approximate controllability of semilinear fractional differential systems of order 1<q<2 via resolvent operators. Filomat 2017; 31(18): 5769–5781

[10] Liu X H, Wang J R, O’Regan D. On the approximate controllability for fractional evolution inclusions of Sobolev and Clarke subdifferential type. IMA J Math Control Inform 2019; 36(1): 1–17

[11] Lü J F, Yang X Y. Approximate controllability of Hilfer fractional integro-differential equations using sequence method. Acta Math Sci Ser A (Chinese Ed) 2020; 40(5): 1282–1294

[12] Obukhovski V, Zecca P. Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup. Nonlinear Anal 2009; 70(9): 3424–3436

[13] Pei Y T, Chang Y K. Approximate controllability for stochastic fractional hemivariational inequalities of degenerate type. Fract Calc Appl Anal 2020; 23(5): 1506–1531

[14] Si Y C, Wang J R, Fečkan M. Controllability of linear and nonlinear systems governed by Stieltjes differential equations. Appl Math Comput 2020; 376: 125139

[15] Wang J R, Zhou Y. Complete controllability of fractional evolution systems. Commun Nonlinear Sci Numer Simul 2012; 17(11): 4346–4355

RIGHTS & PERMISSIONS

Higher Education Press 2024