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

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
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