A class of fractional evolution equations and optimal controls. (English) Zbl 1214.34010

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.


34A08 Fractional ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
49J27 Existence theories for problems in abstract spaces
Full Text: DOI


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