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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

${D}^{q}x\left(t\right)=-Ax\left(t\right)+f\left(t,x\left(t\right)\right),\phantom{\rule{2.em}{0ex}}t\in J=\left[0,T\right],\phantom{\rule{3.33333pt}{0ex}}q\in \left(0,1\right),$
$x\left({t}_{0}\right)={x}_{0}·$

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.

##### MSC:
 34A08 Fractional differential equations 34G20 Nonlinear ODE in abstract spaces 49J27 Optimal control problems in abstract spaces (existence)
##### References:
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