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Existence and controllability for nondensely defined partial neutral functional differential inclusions. (English) Zbl 1363.34264

In the paper infinite-dimensional, semilinear, partial neutral functional differential nonstationary inclusion control system with diffusion and finite delays in the state variable defined in Banach spaces is considered. The semilinear differential inclusion contains both pure linear and pure nonlinear parts. First of all, using methods and concepts taken directly from functional analysis and theory of delayed partial differential equations the existence of mild solution is proved and its properties are discussed. Next, the concept of exact relative controllability on a given finite time interval for delayed control systems is recalled. The main results of the paper are sufficient conditions for relative exact controllability proved by using operator method and techniques based on fixed-point theorems, such that multivalued contractions theorem and Frigon’s fixed-point methods. Finally, an example of exactly controllable dynamical semilinear system which illustrates theoretical considerations is presented. It should be pointed out, that similar approximate controllability considerations for semilinear infinite-dimensional functional differential dynamical systems with delay can be found in the recent papers [L. Wang, Sci. China, Ser. F 52, No. 7, 1095–1102 (2009; Zbl 1182.93026); J. Optim. Theory Appl. 143, No. 1, 185–206 (2009; Zbl 1176.93018)].

MSC:

34K35 Control problems for functional-differential equations
93B05 Controllability
34K30 Functional-differential equations in abstract spaces
34K09 Functional-differential inclusions
34K40 Neutral functional-differential equations
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