Stabilization of unbounded linear control systems in Hilbert space with monotone feedback

Abstract

We consider linear control systems of the form y′(t)=Ay(t)+Bu(t) where A generates a strongly continuous semigroup of contractions (e^tA)_t≥0 on an infinite-dimensional Hilbert space Y. We suppose that the control is unbounded in the sense that the linear control operator B is bounded from the (Hilbert) control space U to some larger space W such that Y⊂W, but not from U into Y. Taking into account eventual control saturation, we study the problem of stabilization by (possibly nonlinear) monotone feedback of the form u(t)=−f(B^*y(t)). We extend to the unbounded monotone feedback context weak and strong stability results which have been established for linear feedback systems. These results are based on weak observability involving the unstable subspace. This fact is illustrated by a heat equation with singular control. In the particular case where the system can be reduced to a second-order evolution equation of the form z″(t)+𝒜z(t)=ℬu(t), we establish decay estimates with (eventually) saturating feedback and under a strong observability property. We show that when the initial data are sufficiently regular, we obtain exponential and polynomial decay. We establish also that when the control operator ℬ is bounded, the regularity assumption can be dispensed with. Applications to the wave equation with distributed control and pointwise control are considered. We establish for these systems non-uniform exponential decay with general saturating feedback. For a string, we present an explicit saturating feedback leading to exponential decay of the energy. Moreover, the degree of regularity of the initial data are related to the Diophantine properties of the actuator position.