Analysis of nonlinear fractional control systems in Banach spaces. (English) Zbl 1223.93059

Wang, JinRong; Zhou, Yong

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 17, 5929-5942 (2011).

Summary: We consider the nonlinear control systems of fractional order and its optimal controls in Banach spaces. Using the fractional calculus, Hölder’s inequality, \(p\)-mean continuity, weakly singular inequality and Leray-Schauder’s fixed-point theorem with compact mapping, a sufficient condition is given for the existence and uniqueness of mild solutions for a broad class of fractional nonlinear infinite dimensional control systems. Utilizing the approximately lower semicontinuity of integral functionals and weakly compactness, we extend the existence result of optimal controls for nonlinear control systems to nonlinear fractional control systems under generally mild conditions. An example is given to illustrate the effectiveness of the results obtained.

MSC:

93C15	Control/observation systems governed by ordinary differential equations
34G10	Linear differential equations in abstract spaces
34G20	Nonlinear differential equations in abstract spaces
34A08	Fractional ordinary differential equations
49J15	Existence theories for optimal control problems involving ordinary differential equations

Keywords:

fractional control systems; mild solutions; weakly singular inequality; optimal controls; weakly compactness

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