Stabilization of boundary control systems

Slemrod, Marshall

doi:10.1016/0022-0396(76)90036-x

Cited by 66 publications

(24 citation statements)

References 13 publications

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“…Interest in this particular model is motivated by various problems such as stabilization, see for example [8], [35], [36], [39], solvability of Riccati equations [18], minimum-phase property [23] and compensator problems with partial observations [19]. It is well-known that A generates a C 0 -semigroup of contractions in H 1/2 × H , where H 1/2 = D(A 1/2 0 ) is equipped with the norm x 1/2 = A 1/2 0 x H , and thus the spectrum of A is located in the closed left half plane.…”

Section: A Linear Equation Describing Transverse Motions Of a Thin Bementioning

confidence: 99%

Analyticity and Riesz basis property of semigroups associated to damped vibrations

2008

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Second order equations of the formz(t) + A 0 z(t) + Dż(t) = 0 are considered. Such equations are often used as a model for transverse motions of thin beams in the presence of damping. We derive various properties of the operator matrix A = 0 I −A 0 −D associated with the second order problem above. We develop sufficient conditions for analyticity of the associated semigroup and for the existence of a Riesz basis consisting of eigenvectors and associated vectors of A in the phase space.

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Section: A Linear Equation Describing Transverse Motions Of a Thin Bementioning

confidence: 99%

Analyticity and Riesz basis property of semigroups associated to damped vibrations

2008

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“…This implies that, denoting E = − [5], Balakrishnan [6,7], Russell [30], and Slemrod [33,34]. In this case, G is positive and (by Proposition 2.1) the feedback u = −κy + v stabilizes in an input-output sense.…”

Section: H(s)(i + H(s))mentioning

confidence: 93%

Exponential stabilization of well-posed systems by colocated feedback

Curtain¹

2006

SIAM J. Control Optim.

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Abstract. We consider well-posed linear systems whose state trajectories satisfyẋ = Ax + Bu, where u is the input and A is an essentially skew-adjoint and dissipative operator on the Hilbert space X. This means that the domains of A * and A are equal and A * + A = −Q, where Q ≥ 0 is bounded on X. The control operator B is possibly unbounded, but admissible and the observation operator of the system is B * . Such a description fits many wave and beam equations with colocated sensors and actuators, and it has been shown for many particular cases that the feedback u = −κy + v, with κ > 0, stabilizes the system, strongly or even exponentially. Here, y is the output of the system and v is the new input. We show, by means of a counterexample, that if B is sufficiently unbounded, then such a feedback may be unsuitable: the closed-loop semigroup may even grow exponentially. (Our counterexample is a simple regular system with feedthrough operator zero.) However, we prove that if the original system is exactly controllable and observable and if κ is sufficiently small, then the closed-loop system is exponentially stable.

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“…Interest in this particular model is motivated by various problems such as stabilization, see for example [6,26,27,29], solvability of the Riccati equations [11], minimum-phase property [16] and compensator problems with partial observations [12].…”

Section: Trunkmentioning

confidence: 99%