Degenerate distributed control systems with fractional time derivative.

*(English)*Zbl 1424.49007Summary: The existence of a unique strong solution for the Cauchy problem to semilinear nondegenerate fractional differential equation and for the generalized Showalter-Sidorov problem to semilinear fractional differential equation with degenerate operator at the Caputo derivative in Banach spaces is proved. These results are used for search of solution existence conditions for a class of optimal control problems to a system described by the degenerate semilinear fractional evolution equation. Abstract results are applied to the research of an optimal control problem solvability for the equations system of Kelvin-Voigt fractional viscoelastic fluids.

##### MSC:

49J20 | Existence theories for optimal control problems involving partial differential equations |

49K20 | Optimality conditions for problems involving partial differential equations |

76A10 | Viscoelastic fluids |

35R11 | Fractional partial differential equations |

35Q35 | PDEs in connection with fluid mechanics |

##### Keywords:

fractional differential calculus; Caputo derivative; Mittag-Leffer function; partial differentialequation; degenerate evolution equation; \((L,p)\)-bounded operator; optimal control; fractional viscoelastic fluid
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\textit{M. V. Plekhanova}, Ural Math. J. 2, No. 2, 58--71 (2016; Zbl 1424.49007)

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