Polynomial decay rate estimate for bilinear parabolic systems under weak observability condition

Abstract

In this paper, we shall study the stability for distributed bilinear systems on a Hilbert state space that can be decomposed in two subspaces: unstable finite-dimensional and stable infinite-dimensional with respect to the evolution generator. Then, we shall show under a weaker observability assumption that stabilizing such a system with a feedback control of the form \(p_{r}(t)=-\Vert y(t)\Vert ^{-r}\langle y(t),By(t)\rangle \) for \(r<2\), reduces stabilizing only its projection on the finite-dimension subspace which make the whole system stable. To this end, we shall propose a new family of continuous feedback controls that guarantee the uniform stabilizability with an explicit optimal decay rate estimate of the stabilized state. Two illustrating examples and simulations are provided.