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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 $$\cases\text{}^C\!D_t^qx(t)+Ax(t)=f\left(t,x_t,\displaystyle\int_0^tg(t,s,x_s)ds\right)+B(t)u(t),\,\,\,0<t\le T,\\ x(t)=\varphi(t),\,\,\,-r\le t\le 0,\endcases.\tag1$$ 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 $\{S(t),\,\,t\ge 0\}$ 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 $x_t:[-r,0]\to X_{\alpha},\,\,t\ge 0$ 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 (real functions) 49J21 Optimal control problems involving relations other than differential equations 93C30 Control systems governed by other functional relations 45G10 Nonsingular nonlinear integral equations
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