Characterizations of complete stabilizability

TL;DR

This paper provides new characterizations of complete stabilizability for control systems using weak observability inequalities, extending to unbounded control operators and time-periodic systems, with applications to non-null controllable yet stabilizable systems.

Contribution

It introduces novel characterizations of stabilizability via weak observability inequalities, including for unbounded and time-periodic control systems, with spectral conditions and practical examples.

Findings

Characterizations via weak observability inequalities for stabilizability

Extension to unbounded control operators and time-periodic systems

Examples of systems that are stabilizable but not null controllable

Abstract

We present several characterizations, via some weak observability inequalities, on the complete stabilizability for a control system $[A,B]$, i.e., $y'(t)=Ay(t)+Bu(t)$, $t\geq 0$, where $A$ generates a $C_0$-semigroup on a Hilbert space $X$ and $B$ is a linear and bounded operator from another Hilbert space $U$ to $X$. We then extend the aforementioned characterizations in two directions: first, the control operator $B$ is unbounded; second, the control system is time-periodic. We also give some sufficient conditions, from the perspective of the spectral projections, to ensure the weak observability inequalities. As applications, we provide several examples, which are not null controllable, but can be verified, via the weak observability inequalities, to be completely stabilizable.