Improved stability for linear SPDEs using mixed boundary/internal controls

Abstract

This paper is motivated by the asymptotic stabilization of abstract SPDEs of linear type. As a first step, it proposes an abstract contribution to the exact controllability (in a general L p -sense, p > 1) of a class of linear SDEs with general time-invariant rank control coefficient in the diffusion term. From this point of view, our paper generalizes some of the results in [13] where full and null rank were considered. Necessary conditions and sufficient ones are discussed and their hierarchy and connections with the approximate controllability are illustrated. Second, our paper illustrates, on relevant frameworks of linear SPDEs, a way to drive exactly to 0 their unstable part of dimension n ≥ 1 by using M internal, respectively N boundary controls such that max {M, N } < n. Extensive examples are presented as is the minimal gain for judicious control pairs.