Regularization and frequency-domain stability of well-posed systems

Abstract

Abstract. We study linear control systems with unbounded control and observation operators using certain regularization techniques. This allows us to introduce a modification of the transfer function for the system also if the input and output operators are not admissible in the usual sense. The modified transfer function is utilized to show exponential stability of sufficiently smooth solutions for the internal system under appropriate admissibility conditions on the system operators and appropriately modified stabilizability and detectability assumptions. If the internal system satisfies additional regularity properties, then we even obtain its uniform exponential stability.

Well-posed systems—The LTI case and beyond

“…Other classes of systems that contain the well-posed ones (and are contained in resolvent linear systems) are the systems with n-admissible control and observation operators discussed in Latushkin et al [44], the systems that are strictly proper with an integrator, introduced in Weiss and Zhao [85], and the system nodes, presented below.…”

Operator Valued Fourier Multipliers and Stability of Strongly Continuous Semigroups

“…However, cf. [15], the results of this section hold even if these assumptions are relaxed to allow that B : U → X and C : X → Y are bounded operators for a triple of Banach spaces X → X → X such that certain conditions of "regularity" are satisfied to cover the Weiss-regular or Pritchard-Salamon classes.…”

“…Finally, we prove that, under stabilizability and detectability assumptions, the exponential stability ω 1 (A) < 0 of the nominal system is equivalent to the fact that an appropriate modification of the transfer function is an L p -Fourier multiplier. Related developments of this theme can be found in [15].…”

Observability of polynomially stable systems

“…Recall that a Riesz basis of a Hilbert space H is a sequence (φ n ) n∈N in H such that φ n = Se n for an invertible operator S ∈ B(H) and an orthonormal basis (e n ) n∈N of H. A Riesz-spectral operator A on H is an operator possessing a Riesz basis of eigenvectors. We also note that in the following result one can weaken the assumption that C is admissible to α + 1-admissibility; i.e, C is admissible for the restriction of T (·) to X α , see [15]. Proposition 3.9.…”