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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 q x(t)=-Ax(t)+f(t,x(t)),tJ=[0,T],q(0,1),
x(t 0 )=x 0 ·

They introduce a notion of α-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.

MSC:
34A08Fractional differential equations
34G20Nonlinear ODE in abstract spaces
49J27Optimal control problems in abstract spaces (existence)
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