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Finite codimensional controllability and optimal control problems with endpoint state constraints. (English. French summary) Zbl 1441.93033

Summary: In this paper, motivated by the study of optimal control problems for infinite dimensional systems with endpoint state constraints, we introduce the notion of finite codimensional (exact/approximate) controllability. Some equivalent criteria on the finite codimensional controllability are presented. In particular, the finite codimensional exact controllability is reduced to deriving a Gårding type inequality for the adjoint system, which is new for many evolution equations. This inequality can be verified for some concrete problems (and hence applied to the corresponding optimal control problems), say the wave equations with both time and space dependent potentials. Moreover, under some mild assumptions, we show that the finite codimensional exact controllability of this sort of wave equations is equivalent to the classical geometric control condition.

MSC:

93B05 Controllability
93C20 Control/observation systems governed by partial differential equations
49J20 Existence theories for optimal control problems involving partial differential equations
93B07 Observability
49K20 Optimality conditions for problems involving partial differential equations
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