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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

\[ \begin{cases}\text{}^C\!D_t^qx(t)+Ax(t)=f\left(t,x_t,\displaystyle\int_0^tg(t,s,x_s)ds\right)+B(t)u(t),\,\,\,0<t\leq T,\\ x(t)=\varphi(t),\,\,\,-r\leq t\leq 0,\end{cases}.\tag{1} \]

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.

\[ \begin{cases}\text{}^C\!D_t^qx(t)+Ax(t)=f\left(t,x_t,\displaystyle\int_0^tg(t,s,x_s)ds\right)+B(t)u(t),\,\,\,0<t\leq T,\\ x(t)=\varphi(t),\,\,\,-r\leq t\leq 0,\end{cases}.\tag{1} \]

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.

Reviewer: Rodica Luca Tudorache (Iaşi)

### 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 |

### Keywords:

fractional delay integrodifferential equation; mild solutions; fractional calculus; optimal controls; analytic semigroup
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\textit{J. Wang} et al., Commun. Nonlinear Sci. Numer. Simul. 16, No. 10, 4049--4059 (2011; Zbl 1223.45007)

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