Characterizations of stabilizable sets for some parabolic equations in [formula omitted

Abstract

We consider the parabolic type equation in Rn:(0.1)(∂t+H)y(t,x)=0,(t,x)∈(0,∞)×Rn;y(0,x)∈L2(Rn), where H can be one of the following operators: (i) a shifted fractional Laplacian; (ii) a shifted Hermite operator; (iii) the Schrödinger operator with some general potentials. We call a subset E⊂Rn as a stabilizable set for (0.1), if there is a linear bounded operator K on L2(Rn) so that the semigroup {e−t(H−χEK)}t≥0 is exponentially stable. (Here, χE denotes the characteristic function of E, which is treated as a linear operator on L2(Rn).)This paper presents different geometric characterizations of the stabilizable sets for (0.1) with different H. In particular, when H is a shifted fractional Laplacian, E⊂Rn is a stabilizable set for (0.1) if and only if E⊂Rn is a thick set, while when H is a shifted Hermite operator, E⊂Rn is a stabilizable set for (0.1) if and only if E⊂Rn is a set of positive measure. Our results, together with the results on the observable sets for (0.1) obtained in [1,19,25,33], reveal such phenomena: for some H, the class of stabilizable sets contains strictly the class of observable sets, while for some other H, the classes of stabilizable sets and observable sets coincide. Besides, this paper gives some sufficient conditions on the stabilizable sets for (0.1) where H is the Schrödinger operator with some general potentials.