In this paper, we characterize the stabilization of some delay systems. The proof of the main result uses the method introduced in Ammari and Tucsnak (ESAIM COCV 6:361-386, 2001) where the exponential stability for the closed loop problem is reduced to an observability estimate for the corresponding uncontrolled system combined with a boundedness property of the transfer function of the associated open loop system.

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We introduce polynomial stabilizability and detectability of wellposed systems in the sense that a feedback produces a polynomially stable C0-semigroup. Using these concepts, the polynomial stability of the given C0-semigroup governing the state equation can be characterized via polynomial bounds on the transfer function. We further give sufficient conditions for polynomial stabilizability and detectability in terms of decompositions into a polynomial stable and an observable part. Our approach relies on a recent characterization of polynomially stable C0-semigroups on a Hilbert space by resolvent estimates.Mathematics Subject Classification (2010). Primary: 93D25. Secondary: 47A55, 47D06, 93C25, 93D15.

In this paper we will generalize the Kalman rank condition for the null controllability to n-coupled linear degenerate parabolic systems with constant coefficients, diagonalizable diffusion matrix, and m-controls. For that we prove a global Carleman estimate of the solution of a scalar 2n-order equation then we infer from it an observability inequality for the corresponding adjoint system, and thus the null controllability.

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