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Zbl 1214.34010
Wang, Jinrong; Zhou, Yong
A class of fractional evolution equations and optimal controls. (English)
[J] Nonlinear Anal., Real World Appl. 12, No. 1, 262-272 (2011). ISSN 1468-1218

The authors, using the techniques of fractional calculus, a singular version of Gronwall's inequality and the Leray-Schauder fixed point theorem for compact maps, study the existence of solutions of a fractional evolution equation of the type $$D^qx(t)=-Ax(t)+f(t,x(t)),\qquad t \in J=[0,T],~q \in (0,1),$$ $$x(t_0)=x_0.$$ They introduce a notion of $\alpha$-mild solution which is associated with a probability density function and a semigroup operator. Further, the existence of an optimal control for a Lagrange problem is proved. An example is given to demonstrate their results.
[J. Vasundhara Devi (Visakhapatnam)]

MSC 2000:: *34A08
34G20 Nonlinear ODE in abstract spaces
49J27 Optimal control problems in abstract spaces (existence)

Keywords: fractional evolution equations; analytic compact semigroup; fractional power space; $\alpha$-mild solutions; optimal controls

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