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**Approximate controllability for abstract semilinear impulsive functional differential inclusions based on Hausdorff product measures.**
*(English)*
Zbl 1414.34061

Summary: A second order semilinear impulsive functional differential inclusion in a separable Hilbert space is considered. Without imposing hypotheses of the compactness on the cosine families of operators, some sufficient conditions of approximate controllability are formulated in the case where the multivalued nonlinearity of the inclusion is a completely continuous map dominated by a function. By the use of resolvents of controllability Gramian operators and developing appropriate computing techniques for the Hausdorff product measures of noncompactness, the results of approximate controllability for position and velocity are derived. An example is also given to illustrate the application of the obtained results.

### MSC:

34K35 | Control problems for functional-differential equations |

34K45 | Functional-differential equations with impulses |

34K09 | Functional-differential inclusions |

34K30 | Functional-differential equations in abstract spaces |

93B05 | Controllability |

### Keywords:

approximate controllability; impulsive system; second order semilinear differentail inclusion; cosine family of operators; fixed point for multivalued mapping
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\textit{J.-Z. Xiao} and \textit{X.-H. Zhu}, Topol. Methods Nonlinear Anal. 52, No. 2, 353--372 (2018; Zbl 1414.34061)

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