Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces. (English) Zbl 1252.93028

Wang, JinRong; Fan, Zhenbin; Zhou, Yong

J. Optim. Theory Appl. 154, No. 1, 292-302 (2012).

Summary: In this paper, we establish two sufficient conditions for nonlocal controllability for fractional evolution systems. Since there is no compactness of characteristic solution operators, our theorems guarantee the effectiveness of controllability results under some weakly compactness conditions.

MSC:

93B05	Controllability
34A08	Fractional ordinary differential equations
93C15	Control/observation systems governed by ordinary differential equations

Keywords:

nonlocal controllability; Caputo fractional derivative; equicontinuity; measure of noncompactness; Mönch’s fixed-point theorems

References:

[1]	Byszewski, L.: Theorems about existence and uniqueness of solutions of a semi-linear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162, 494–505 (1991) · Zbl 0748.34040
[2]	Fan, Z., Li, G.: Existence results for semilinear differential equations with nonlocal and impulsive conditions. J. Funct. Anal. 258, 1709–1727 (2010) · Zbl 1193.35099
[3]	Hernández, E., Santos, J.S., Azevedo, K.A.G.: Existence of solutions for a class of abstract differential equations with nonlocal conditions. Nonlinear Anal. 74, 2624–2634 (2011) · Zbl 1221.47079
[4]	Wang, J., Wei, W.: A class of nonlocal impulsive problems for integrodifferential equations in Banach spaces. Results Math. 58, 379–397 (2010) · Zbl 1209.34095
[5]	Hernández, E., O’Regan, D.: Controllability of Volterra–Fredholm type systems in Banach space. J. Franklin Inst. 346, 95–101 (2009) · Zbl 1160.93005
[6]	Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) · Zbl 1092.45003
[7]	Wang, J., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal. 12, 262–272 (2011) · Zbl 1214.34010
[8]	Wang, J., Zhou, Y., Medved’, M.: On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay. J. Optim. Theory Appl. 152, 31–50 (2012) · Zbl 1357.49018
[9]	Balachandran, K., Park, J.Y.: Controllability of fractional integrodifferential systems in Banach spaces. Nonlinear Anal. 3, 363–367 (2009) · Zbl 1175.93028
[10]	Sakthivel, R., Ren, Y., Mahmudov, N.I.: On the approximate controllability of semilinear fractional differential systems. Comput. Math. Appl. 62, 1451–1459 (2011) · Zbl 1228.34093
[11]	Debbouchea, A., Baleanu, D.: Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems. Comput. Math. Appl. 62, 1442–1450 (2011) · Zbl 1228.45013
[12]	Hernández, E., O’Regan, D., Balachandran, K.: On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal. 73, 3462–3471 (2010) · Zbl 1229.34004
[13]	Jaradat, O.K., Al-Omari, A., Momani, S.: Existence of the mild solution for fractional semilinear initial value problems. Nonlinear Anal. 69, 3153–3159 (2008) · Zbl 1160.34300
[14]	Banas, J., Goebel, K.: Measure of Noncompactness in Banach Spaces. Lecture Notes in Pure and Appl. Math., vol. 60. Marcel Dekker, New York (1980) · Zbl 0441.47056
[15]	Kamenskii, M., Obukhovskii, V., Zecca, P.: Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. de Gruyter Ser. Nonlinear Anal. Appl., vol. 7. de Gruyter, Berlin (2001) · Zbl 0988.34001
[16]	Zhou, Y., Jiao, F.: Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59, 1063–1077 (2010) · Zbl 1189.34154
[17]	O’Regan, D., Precup, R.: Existence criteria for integral equations in Banach spaces. J. Inequal. Appl. 6, 77–97 (2001) · Zbl 0993.45011
[18]	Mönch, H.: Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal. 4, 985–999 (1980) · Zbl 0462.34041