Observability of polynomially stable systems

Abstract

Abstract. For finite-dimensional systems the Hautus test is a well-known and easy checkable condition for observability. Russell and Weiss (SIAM J. Control Optim. 32:1-23, 1994) suggested an infinite-dimensional version of the Hautus test, which is necessary for exact infinite-time observability and sufficient for approximate infinite-time observability of exponentially stable systems. In this paper the notion of observability is studied for polynomially stable systems. Several known results for exponentially stable systems are extended to the setting of polynomially stable systems. By means of an example the obtained results are illustrated.

Characterization by observability inequalities of controllability and stabilization properties

“…By Theorem 3, for every α ∈ (0, 1), there is T > 0 such that µ T α < +∞. Defining T α = inf{T > 0 | µ T α < +∞}, by (14) we have ω * = inf ln α T α α ∈ (0, 1) .…”

“…We have provided in Theorem 3 a characterization of exponential stabilizability. The C 0 -semigroup (S(t)) t 0 is said to be polynomially stable when there exist constants γ, δ > 0 such that S(t)(A − βid) −γ L(X) M t −δ for every t 1, for some M > 0 and some β ∈ ρ(A) (see [14] where polynomial stability is compared with observability). Finding a dual characterization of polynomial stabilizability in terms of an observability inequality is an open issue, which may be related to the previous question on Hautus tests.…”

On the Hautus Test for Exponentially Stable $C_0$-Groups

“…Here C − denotes the open left half plane. The Hautus test (HT) is sufficient for approximate observability of exponentially stable systems [14] and for polynomially stable systems [6]. Further, (HT) is sufficient for exact observability of strongly stable Riesz-spectral systems with finite-dimensional output spaces [7], for exponentially stable systems with A bounded on H [14], and for exponentially stable systems if the constant m in (HT) equals 1 [3]; a short proof of this last result can be found in section 4.…”

Observability for port-Hamiltonian systems

“…Here C − denotes the open left half plane. The Hautus test (HT) is sufficient for approximate observability of exponentially stable systems [14] and for polynomially stable systems [5]. Further, the Hautus test (HT) is sufficient for exact observability of strongly stable Riesz-spectral systems with finite-dimensional output spaces [6], for exponentially stable systems with A is bounded on H [14], and for exponentially stable systems if the constant m in (HT) equals one [2].…”