Rapid stabilizability of infinite-dimensional control systems with time delays
Pith reviewed 2026-05-16 09:32 UTC · model grok-4.3
The pith
A constant time delay does not affect rapid stabilizability of infinite-dimensional control systems.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Under the assumptions that the state operator generates an immediately compact semigroup and that the delay coefficient is constant, the presence of a time-delay term does not affect the rapid stabilizability of the control system, that is, this property depends only on the state and control operators. Static feedback is sufficient to achieve rapid stabilization.
What carries the argument
The immediately compact semigroup generated by the state operator, which is used to prove that rapid stabilizability is independent of the constant delay term.
If this is right
- Static feedback suffices to achieve rapid stabilization whether or not the constant delay is present.
- The rapid stabilizability property is completely determined by the state and control operators alone.
- The same feedback operator stabilizes both the delayed system and its delay-free counterpart.
Where Pith is reading between the lines
- Controller synthesis for rapid stabilization can proceed by first ignoring the delay and then verifying the compactness condition.
- The independence result may simplify numerical approximations that drop small delays under compact semigroup assumptions.
- Extensions to mild delays or distributed delays would require separate analysis but could follow analogous compactness arguments.
Load-bearing premise
The state operator generates an immediately compact semigroup and the delay coefficient is constant.
What would settle it
A concrete system whose state operator generates an immediately compact semigroup that is rapidly stabilizable without delay but fails to be rapidly stabilizable once the constant delay term is added.
read the original abstract
In this paper, we investigate the rapid stabilizability of linear infinite-dimensional control systems with constant delays. Under the assumptions that the state operator generates an immediately compact semigroup and that the delay coefficient is constant, we establish two main results: (i) the presence of a time-delay term does not affect the rapid stabilizability of the control system, that is, this property depends only on the state and control operators; (ii) static feedback is sufficient to achieve rapid stabilization of the system. Applications are also presented.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper investigates rapid stabilizability of linear infinite-dimensional control systems with constant time delays. Under the assumptions that the state operator A generates an immediately compact semigroup and the delay coefficient is constant, it establishes that the time-delay term does not affect rapid stabilizability (which depends only on A and B) and that static feedback suffices for rapid stabilization, with applications presented.
Significance. If the results hold under the stated assumptions, the work would be significant for infinite-dimensional control theory: it reduces the rapid-stabilizability question for delayed systems to the non-delayed case via standard spectral perturbation arguments for immediately compact semigroups, thereby simplifying feedback design. The explicit separation of the delay effect from the essential spectrum is a clean contribution when the derivations are complete.
major comments (1)
- [Main result (likely §3 or Theorem 3.2)] The central perturbation argument (that the constant-delay term on the product space leaves the essential spectrum of the generator unchanged) must be spelled out explicitly, including the precise reference to the compact-semigroup perturbation theorem invoked and verification that the growth bound can still be made arbitrarily negative.
minor comments (2)
- [Abstract] The abstract states the two main results clearly but does not indicate the section where the applications are developed; a one-sentence pointer would improve readability.
- [Preliminaries] Notation for the product-space generator and the delay operator should be introduced once in a dedicated preliminary section to avoid repeated re-definition later.
Simulated Author's Rebuttal
We thank the referee for the careful reading of our manuscript and for the positive assessment of its significance. We appreciate the constructive suggestion regarding the presentation of the central perturbation argument and address it below.
read point-by-point responses
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Referee: [Main result (likely §3 or Theorem 3.2)] The central perturbation argument (that the constant-delay term on the product space leaves the essential spectrum of the generator unchanged) must be spelled out explicitly, including the precise reference to the compact-semigroup perturbation theorem invoked and verification that the growth bound can still be made arbitrarily negative.
Authors: We agree that greater explicitness in the perturbation argument will improve readability. In the revised version we will expand the proof of the main result (Theorem 3.2) by (i) citing the precise statement of the compact perturbation theorem for immediately compact semigroups that is applied (a standard result from the theory of C0-semigroups, e.g., the version appearing as Theorem 3.1 in Pazy’s book or the corresponding result in Engel–Nagel), (ii) verifying step-by-step that the constant-delay operator, viewed as a bounded perturbation on the product space, leaves the essential spectrum of the generator unchanged, and (iii) confirming that the growth bound of the resulting semigroup can be driven arbitrarily negative by the same feedback operator that works for the delay-free system, since only the essential spectrum is preserved and the spectral bound is controlled by the original pair (A,B). These additions will be purely expository and will not alter the statements or proofs of the theorems. revision: yes
Circularity Check
No significant circularity; derivation is self-contained via standard perturbation theory
full rationale
The paper's core claim—that rapid stabilizability depends only on the state operator A and control operator B, independent of a constant delay—follows from the assumption that A generates an immediately compact semigroup. Under this condition the essential spectrum of the generator on the product space remains unchanged by the bounded delay perturbation, so any feedback achieving arbitrary negative growth bound for the non-delay system continues to do so for the delayed system. This is a direct application of classical spectral and perturbation results for compact semigroups and does not reduce by construction to a redefinition of stabilizability, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The derivation chain is externally grounded in semigroup theory and remains independent of the target result.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption The state operator generates an immediately compact semigroup
- domain assumption The delay coefficient is constant
discussion (0)
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