Controllability and Stabilizability Theory for Linear Partial Differential Equations: Recent Progress and Open Questions

Russell, David L.

doi:10.1137/1020095

Cited by 946 publications

(609 citation statements)

References 79 publications

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“…Hence, we may interpret T 0 as the time necessary for the signal to propagate from x = 0 to x = l, and T * as the time necessary for the signal to propagate from x = 0 to x = l and back. If we consider a string with the tension that is a function of coordinate x only, we find the well-known formulas (see [28], [2], [5, Ch.V])…”

Section: Lemma 4 the Solutions Of Slp (22) Normalized By Imposing Cmentioning

confidence: 99%

Controllability of a Nonhomogeneous String and Ring under Time Dependent Tension

Avdonin

Belinskiy

Pandolfi

2010

Math. Model. Nat. Phenom.

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Abstract. We study controllability for a nonhomogeneous string and ring under an axial stretching tension that varies with time. We consider the boundary control for a string and distributed control for a ring. For a string, we are looking for a control f (t) ∈ L 2 (0, T ) that drives the state solution to rest. We show that for a ring, two forces are required to achieve controllability. The controllability problem is reduced to a moment problem for the control. We describe the set of initial data which may be driven to rest by the control. The proof is based on an auxiliary basis property result.

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Section: Lemma 4 the Solutions Of Slp (22) Normalized By Imposing Cmentioning

confidence: 99%

Controllability of a Nonhomogeneous String and Ring under Time Dependent Tension

Avdonin

Belinskiy

Pandolfi

2010

Math. Model. Nat. Phenom.

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“…It is known (see for instance [15] or [12, Th. 5.1]) that for T = 2b/c there exists a unique boundary control f of minimal L 2 (0, T )-norm which drives the solution u of system (3.16) to rest at time T, that is,…”

Section: Uniformly Controlling All the Initial Data In Lmentioning

confidence: 99%

Some remarks on homogenization and exact boundary controllability for the one-dimensional wave equation

Pedregal

Periago

2006

Quart. Appl. Math.

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Abstract. This paper contains three results concerning the homogenization and exact controllability for the one-dimensional wave equation. First, we give sufficient conditions on the initial data to ensure the convergence of the conormal derivatives associated with the wave equation with a rapidly oscillating coefficient and zero Dirichlet boundary conditions. Secondly, we apply this result to prove the existence of a class of initial data whose associated boundary controls are uniformly bounded and obtain some information (in particular, its limit behavior) on this class of data. Finally, we prove that all initial data in L 2 × H −1 may be uniformly controlled but at the price of adding an internal feedback control in our system. The main advantage of this last procedure is that we have explicit formulae for both states and controls.

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“…Dolecki and Russell [6]). Controllability and observability are important properties of a distributed parameter system, which have been extensively studied in the literature, see for example [2], [14] and [19].…”

Section: Introductionmentioning

confidence: 99%

Observability of polynomially stable systems

Jacob¹,

Schnaubelt²

2007

Systems & Control Letters

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Abstract. For finite-dimensional systems the Hautus test is a well-known and easy checkable condition for observability. Russell and Weiss (SIAM J. Control Optim. 32:1-23, 1994) suggested an infinite-dimensional version of the Hautus test, which is necessary for exact infinite-time observability and sufficient for approximate infinite-time observability of exponentially stable systems. In this paper the notion of observability is studied for polynomially stable systems. Several known results for exponentially stable systems are extended to the setting of polynomially stable systems. By means of an example the obtained results are illustrated.

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Controllability and Stabilizability Theory for Linear Partial Differential Equations: Recent Progress and Open Questions

Cited by 946 publications

References 79 publications

Controllability of a Nonhomogeneous String and Ring under Time Dependent Tension

Controllability of a Nonhomogeneous String and Ring under Time Dependent Tension

Some remarks on homogenization and exact boundary controllability for the one-dimensional wave equation

Observability of polynomially stable systems

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