Equivalent conditions for stabilizability of infinite-dimensional systems with admissible control operators

Birgit Jacob, Hans Zwart

Research output: Contribution to journalArticleAcademicpeer-review

16 Citations (Scopus)
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Abstract

In this paper we study the optimizability of infinite-dimensional systems with admissible control operators. We show that under a weak condition such a system is optimizable if and only if the system can be split into an exponentially stable subsystem and an unstable subsystem that is exactly controllable in finite time. The state space of the unstable subsystem equals the span of all unstable (generalized) eigenvectors of the original system. This subsystem can be infinite-dimensional. Furthermore, the unstable poles satisfy a summability condition. The state space of the exponentially stable subsystem is given by all vectors for which the action of the original C0 -semigroup is stable.
Original languageEnglish
Pages (from-to)1419-1455
Number of pages37
JournalSIAM journal on control and optimization
Volume37
Issue number5
DOIs
Publication statusPublished - 1999

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Infinite-dimensional Systems
Stabilizability
Eigenvalues and eigenfunctions
Poles
Subsystem
Unstable
Operator
State Space
C0-semigroup
Summability
Eigenvector
Pole
If and only if

Keywords

  • Infinite-dimensional systems
  • Controllability
  • Optimizability
  • Stabilizability

Cite this

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Equivalent conditions for stabilizability of infinite-dimensional systems with admissible control operators. / Jacob, Birgit; Zwart, Hans.

In: SIAM journal on control and optimization, Vol. 37, No. 5, 1999, p. 1419-1455.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Equivalent conditions for stabilizability of infinite-dimensional systems with admissible control operators

AU - Jacob, Birgit

AU - Zwart, Hans

PY - 1999

Y1 - 1999

N2 - In this paper we study the optimizability of infinite-dimensional systems with admissible control operators. We show that under a weak condition such a system is optimizable if and only if the system can be split into an exponentially stable subsystem and an unstable subsystem that is exactly controllable in finite time. The state space of the unstable subsystem equals the span of all unstable (generalized) eigenvectors of the original system. This subsystem can be infinite-dimensional. Furthermore, the unstable poles satisfy a summability condition. The state space of the exponentially stable subsystem is given by all vectors for which the action of the original C0 -semigroup is stable.

AB - In this paper we study the optimizability of infinite-dimensional systems with admissible control operators. We show that under a weak condition such a system is optimizable if and only if the system can be split into an exponentially stable subsystem and an unstable subsystem that is exactly controllable in finite time. The state space of the unstable subsystem equals the span of all unstable (generalized) eigenvectors of the original system. This subsystem can be infinite-dimensional. Furthermore, the unstable poles satisfy a summability condition. The state space of the exponentially stable subsystem is given by all vectors for which the action of the original C0 -semigroup is stable.

KW - Infinite-dimensional systems

KW - Controllability

KW - Optimizability

KW - Stabilizability

U2 - 10.1137/S036301299833344X

DO - 10.1137/S036301299833344X

M3 - Article

VL - 37

SP - 1419

EP - 1455

JO - SIAM journal on control and optimization

JF - SIAM journal on control and optimization

SN - 0363-0129

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ER -