Observation and Control for Operator Semigroups

Tucsnak, Marius; Weiss, George H.

doi:10.1007/978-3-7643-8994-9

Cited by 1,023 publications

(1,155 citation statements)

References 123 publications

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“…where A −1 is the extension of A to H, namely for all h ∈ H and ϕ ∈ D(A), A −1 h is the unique element in (D(A)) ′ (the duality is in the sense of H), such that (see for instance [21])…”

Section: The Wave Equation With Internal and Boundary Dampingsmentioning

confidence: 99%

Stability Results for Second-Order Evolution Equations with Switching Time-Delay

Nicaise

Pignotti

2014

J Dyn Diff Equat

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We consider second-order evolution equations in an abstract setting with intermittently delayed/not-delayed damping. We give sufficient conditions for asymptotic and exponential stability, improving and generalising our previous results from [19]. In particular, under suitable conditions, we can consider unbounded damping operators. Some concrete examples are finally presented.2000 Mathematics Subject Classification: 35L05, 93D15

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Section: The Wave Equation With Internal and Boundary Dampingsmentioning

confidence: 99%

Stability Results for Second-Order Evolution Equations with Switching Time-Delay

Nicaise

Pignotti

2014

J Dyn Diff Equat

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“…Denoting by S(t) the C 0 semi-group generated by A, it is well known (see, e.g., [13]) that the system (22) is exactly null controllable in time T if and only if one has the observability inequality…”

Section: Proof Of the Spectral Controllability Resultsmentioning

confidence: 99%

Randomised observation, control and stabilization of waves

2016

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The problems of observing, controlling and stabilizing wave processes arise in many different contexts ranging from structural mechanics to seismic waves. In a suitable functional setting, they are closely interconnected and sometimes completely equivalent. In a series of previous articles we have addressed the problem of the optimal design of sensors for purely conservative wave models. We analyzed a relaxed version of the optimal observation problem, considering the expectation of solutions under a randomisation procedure, rather than that where all possible solutions are considered in a purely deterministic setting. From an analytical point of view, this randomisation procedure had the advantage of leading to a spectral diagonalisation of the observations. In this way, using fine asymptotic spectral properties of the Laplacian, we disclosed the links between the geometric properties of the domain where waves propagate and the existence of optimal locations for the sensors or, by the contrary, the emergence of relaxation phenomena. Here we show that spectral randomised observability is equivalent to the property of spectral controllability by means of a discrete set of lumped controls acting everywhere on the domain, and distributed according to the shape of the eigenfunctions. Our results on optimal observation then find natural equivalents on the problem of optimal spectral control. We also give an interpretation of these results in terms of a feedback stabilization property, ensuring the exponential decay of the energy of solutions as time tends to infinity.

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“…Assume that ϕ satisfies (17) and (18). Then there exists ψ ∈ S that also satisfies (17) and (18) and that…”

Section: Distribution Spaces and The ϕ-Transformmentioning

confidence: 99%

The weighted Weiss conjecture and reproducing kernel theses for generalized Hankel operators

Jacob

Rydhe²,

Wynn

2013

J. Evol. Equ.

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The weighted Weiss conjecture states that the system theoretic property of weighted admissibility can be characterised by a resolvent growth condition. For positive weights, it is known that the conjecture is true if the system is governed by a normal operator; however, the conjecture fails if the system operator is the unilateral shift on the Hardy space H 2 (D) (discrete time) or the right-shift semigroup on L 2 (R + ) (continuous time). To contrast and complement these counterexamples, in this paper positive results are presented characterising weighted admissibility of linear systems governed by shift operators and shift semigroups. These results are shown to be equivalent to the question of whether certain generalized Hankel operators satisfy a reproducing kernel thesis.

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Observation and Control for Operator Semigroups

Cited by 1,023 publications

References 123 publications

Stability Results for Second-Order Evolution Equations with Switching Time-Delay

Stability Results for Second-Order Evolution Equations with Switching Time-Delay

Randomised observation, control and stabilization of waves

The weighted Weiss conjecture and reproducing kernel theses for generalized Hankel operators

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