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**Controllability of second-order semilinear neutral functional differential systems in Banach spaces.**
*(English)*
Zbl 0990.93007

Semilinear second-order neutral functional dynamical systems defined in infinite-dimensional Banach spaces are considered. It is generally assumed that the state equation contains both a linear part without delays and a nonlinear part containing delays. First of all the definition of exact relative controllability in a given finite-time interval is presented. Next, using Leray-Schauder’s fixed-point theorem, sufficient conditions for controllability are formulated and proved. In the proofs of the main results, the semigroup theory of linear operators in Banach spaces is used. Moreover, several remarks and comments concerning controllability problems for infinite-dimensional control systems with delays are presented. The relationship to results existing in the literature are also given. Finally, it should be pointed out that similar controllability problems have been recently considered in the papers [the authors and J. Y. Park, Bull. Korean Math. Soc. 35, No. 1, 1-13 (1999; Zbl 0935.93013)] and [J. Y. Park and H. K. Han, ibid. 34, 411-419 (1997; Zbl 0889.93008)].

Reviewer: J.Klamka (Katowice)

### MSC:

93B05 | Controllability |

93C23 | Control/observation systems governed by functional-differential equations |

34K30 | Functional-differential equations in abstract spaces |

34K35 | Control problems for functional-differential equations |

34K40 | Neutral functional-differential equations |

### Keywords:

semilinear control systems; second-order neutral functional dynamical systems; delays; exact relative controllability; Leray-Schauder fixed-point theorem; semigroup theory of linear operators in Banach spaces
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\textit{K. Balachandran} and \textit{S. Marshal Anthoni}, Comput. Math. Appl. 41, No. 10--11, 1223--1235 (2001; Zbl 0990.93007)

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### References:

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