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Zbl 1223.45007
Wang, Jinrong; Zhou, Yong; Wei, W.
A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces. (English)
[J] Commun. Nonlinear Sci. Numer. Simul. 16, No. 10, 4049-4059 (2011). ISSN 1007-5704

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.
[Rodica Luca Tudorache (Iaşi)]

MSC 2000:: *45J05 Integro-ordinary differential equations
26A33 Fractional derivatives and integrals (real functions)
49J21
93C30 Control systems governed by other functional relations
45G10 Nonsingular nonlinear integral equations

Keywords: fractional delay integrodifferential equation; mild solutions; fractional calculus; optimal controls; analytic semigroup

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Zentralblatt MATH Berlin [Germany]