Simultaneous Exact Controllability and Some Applications

Tucsnak, Marius

doi:10.1137/s0363012999352716

Cited by 40 publications

(24 citation statements)

References 19 publications

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“…For more details on Gramians we refer to Hansen and Weiss [12], Jacob and Partington [14], Russell and Weiss [28], and Grabowski [8]. For more details on exact controllability in an operator-theoretic setting we also refer to Avdonin and Ivanov [3], Jacob and Zwart [15], Rebarber and Weiss [27], Tucsnak and Weiss [29], and the references therein. In the PDE setting, the relevant literature is overwhelming, and we mention the books of Lions [20], Lagnese and Lions [17], Bensoussan et al [6], Komornik [16], and the paper of Bardos, Lebeau, and Rauch [5].…”

Section: Suppose That C Is An Infinite-time Admissible Observation mentioning

confidence: 99%

How to Get a Conservative Well-Posed Linear System Out of Thin Air. Part II. Controllability and Stability

Tucsnak¹,

Weiss²

2003

SIAM J. Control Optim.

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Abstract. Let A 0 be a possibly unbounded positive operator on the Hilbert space H, which is boundedly invertible. Let C 0 be a bounded operator from D(A 1/2 0 ) (with the norm z 2 1/2 = A 0 z, z ) to another Hilbert space U . In Part I of this work we have proved that the system of equationsdetermines a well-posed linear system Σ with input u and output y, input and output space U , andMoreover, Σ is conservative, which means that a certain energy balance equation is satisfied both by the trajectories of Σ and by those of its dual system. In this paper we show that Σ is exactly controllable if and only if it is exactly observable, if and only if it is exponentially stable. Moreover, if we denote by A the generator of the contraction semigroup associated with Σ (which acts on X), then Σ is exponentially stable if and only if one of the entries in the second column of (iωI − A) −1 is uniformly bounded as a function of ω ∈ R. We also show that, under a mild assumption, Σ is approximately controllable if and only if it is approximately observable, if and only if it is strongly stable, if and only if the dual system is strongly stable. We prove many related results and we give examples based on wave and beam equations. 1. Introduction and main results. This paper is a continuation of our paper [35] in which we have investigated a class of conservative linear systems with a special structure, which occur often in applications. These systems are described by a second order differential equation (in a Hilbert space) and an output equation, and they may have unbounded control and observation operators. The main aim of [35] was to prove the wellposedness, conservativity, and other regularity properties of such systems. Here we investigate conditions under which such systems are exponentially stable or strongly stable. It turns out that these stability properties are equivalent to certain controllability and observability properties as well as to certain estimates.We recall the construction from the paper [35] in order to be able to state the new results. Let H be a Hilbert space, and let A 0 : D(A 0 ) → H be a self-adjoint, positive, and boundedly invertible operator. We introduce the scale of Hilbert spaces H α , α ∈ R, as follows : for every α ≥ 0, H α = D(A

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Section: Suppose That C Is An Infinite-time Admissible Observation mentioning

confidence: 99%

How to Get a Conservative Well-Posed Linear System Out of Thin Air. Part II. Controllability and Stability

Tucsnak¹,

Weiss²

2003

SIAM J. Control Optim.

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“…The inequality proved in [3] implies a particular case of assertion (b) in Theorem 1.5, namely the fact that, for almost all ξ, the states in W s , s > 0, which vanish at x = ξ, can be reached in time T > max{4ξ, 4(1 − ξ)}. The reachability of all the states in W s in time T > max{4ξ, 4(1 − ξ)} was first proved in [17].…”

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confidence: 99%

Simultaneous controllability in sharp time for two elastic strings

Avdonin

Tucsnak²

2001

ESAIM: COCV

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Abstract.We study the simultaneously reachable subspace for two strings controlled from a common endpoint. We give necessary and sufficient conditions for simultaneous spectral and approximate controllability. Moreover we prove the lack of simultaneous exact controllability and we study the space of simultaneously reachable states as a function of the position of the joint. For each type of controllability result we give the sharp controllability time.Mathematics Subject Classification. 93B, 35L, 42.

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“…The following result has been proved in Tucsnak and Weiss [22] in the case of a finite dimensional input space U and in the general form below in [23,Theorem 10.3.6].…”

Section: Basic Concepts and Auxiliary Resultsmentioning

confidence: 94%

“…By Hautus lemma (see [9]), we conclude that (A P Y N , C Y N ) is exactly observable for all τ > 0. Finally, since A N and A P Y N have no common eigenvalues, we can apply Theorem 3.3 of [22] to deduce that (A P , C) is exactly observable in any time τ > 0.…”

Section: Exact Observability For the Plate Equationmentioning

confidence: 99%

Solving Inverse Source Problems Using Observability. Applications to the Euler–Bernoulli Plate Equation

Alves¹,

Silvestre²,

Takahashi³

et al. 2009

SIAM J. Control Optim.

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Abstract. The aim of this paper is to provide a general framework allowing to use exact observability of infinite dimensional systems to solve a class of inverse source problems. More precisely, we show that if a system is exactly observable, then we can identify a source term in this system by knowing the corresponding intensity and appropriate observations which often correspond to the measure of some boundary traces. This abstract theory is then applied to a system governed by the EulerBernoulli plate equation. Using a different methodology, we show that exact observability can be used to identify both the locations and the intensities of combinations of point sources in the plate equation.

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Simultaneous Exact Controllability and Some Applications

Cited by 40 publications

References 19 publications

How to Get a Conservative Well-Posed Linear System Out of Thin Air. Part II. Controllability and Stability

How to Get a Conservative Well-Posed Linear System Out of Thin Air. Part II. Controllability and Stability

Simultaneous controllability in sharp time for two elastic strings

Solving Inverse Source Problems Using Observability. Applications to the Euler–Bernoulli Plate Equation

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