## A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces.(English)Zbl 1223.45007

This work deals with the fractional delay nonlinear integrodifferential controlled system
$\begin{cases}\text{}^C\!D_t^qx(t)+Ax(t)=f\left(t,x_t,\displaystyle\int_0^tg(t,s,x_s)ds\right)+B(t)u(t),\,\,\,0<t\leq T,\\ x(t)=\varphi(t),\,\,\,-r\leq t\leq 0,\end{cases}.\tag{1}$
where $$\text{}^C\!D_t^q$$ denotes the Caputo fractional derivative of order $$q\in (0,1)$$, $$-A:D(A)\to X$$ is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators $$\{S(t),\,\,t\geq 0\}$$ on a separable reflexive Banach space $$X$$, $$f$$ is $$X$$-value function and $$g$$ is $$X_{\alpha}$$-value function. Here $$X_{\alpha}=D(A^{\alpha})$$ is a Banach space with the norm $$\|x\|_{\alpha}=\|A^{\alpha}x\|$$ for $$x\in X_{\alpha}$$, $$u$$ takes values from another separable reflexive Banach space $$Y$$, $$B$$ is a linear operator from $$Y$$ into $$X$$, and $$x_t:[-r,0]\to X_{\alpha},\,\,t\geq 0$$ represents the history of the state from time $$t-r$$ up to the present time $$t$$, defined by $$x_t=\{x(t+s),\,\,\,s\in [-r,0]\}$$. The authors prove the existence and uniqueness of $$\alpha$$-mild solutions for $$(1)$$, and the continuous dependence result of these solutions. The Lagrange problem of system $$(1)$$ is also formulated and an existence result of optimal controls is presented. To illustrate the obtained results, an example is finally addressed.

### MSC:

 45J05 Integro-ordinary differential equations 26A33 Fractional derivatives and integrals 49J21 Existence theories for optimal control problems involving relations other than differential equations 93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) 45G10 Other nonlinear integral equations
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