Admissible observation operators for linear semigroups

Weiss, George

doi:10.1007/bf02788172

Cited by 308 publications

(238 citation statements)

References 13 publications

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“…By a result due to Salamon [35] (and apparently discovered independently by Weiss [43,44]), every well-posed linear system Ψ has a well-defined (unbounded) control operator B and a well-defined (unbounded) observation operator C, and formulas (3), (4), (8), (9), (11), (12), (13), and (15) hold in a weak sense (see Remark 30 below). In order to present this result we need some additional definitions.…”

Section: The Generators Of a Well-posed Linear Systemmentioning

confidence: 95%

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Quadratic optimal control of stable well-posed linear systems

Staffans

1997

Trans. Amer. Math. Soc.

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Abstract. We consider the infinite horizon quadratic cost minimization problem for a stable time-invariant well-posed linear system in the sense of Salamon and Weiss, and show that it can be reduced to a spectral factorization problem in the control space. More precisely, we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution. If both the system and the spectral factor are regular, then the feedback operator can be expressed in terms of the Riccati operator, and the Riccati operator is a positive self-adjoint solution of an algebraic Riccati equation. This Riccati equation is similar to the usual algebraic Riccati equation, but one of its coefficients varies depending on the subspace in which the equation is posed. Similar results are true for unstable systems, as we have proved elsewhere.

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Section: The Generators Of a Well-posed Linear Systemmentioning

confidence: 95%

“…This theory has been developed in [33], [34], [35], [8], [11], and [43], [44], [45], [46] (and many other papers), and we refer the reader to these sources for additional reading. (Salamon calls these systems "well-posed semigroup control systems" and Weiss calls them "abstract linear systems".)…”

Section: Well-posed Linear Systems and Time-invariant Operatorsmentioning

confidence: 99%

Quadratic optimal control of stable well-posed linear systems

Staffans

1997

Trans. Amer. Math. Soc.

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“…Given a complex valued sequence (c n ), we define Cx = n c n x n . This operator is admissible for A if and only if (c n ) is bounded, thanks to the Carleson measure criterion applied to A − I, see Proposition 7.1 in [22] and the references therein. We thus assume that (c n ) is bounded.…”

Section: Applications To Diagonal Systemsmentioning

confidence: 99%

“…In order to guarantee this, we assume that C is an admissible observation operator for T (·). The notion of admissible observation operators was introduced by Weiss [22] as follows. Definition 1.1.…”

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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“…We say that A, B, C, D are the generating operators of the well-posed linear system. In fact, any well-posed linear system has uniquely defined generating operators A, B, C where A is the infinitesimal generator of the strongly continuous semigroup A and B and C are in general unbounded operators (see Weiss [15], [16]). In general a feedthrough operator 'D' may not exist.…”

Section: Well-posed Linear Systemsmentioning

confidence: 99%

New Riccati equations for well-posed linear systems

Opmeer

Curtain

2004

Systems & Control Letters

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We consider the classic problem of minimizing a quadratic cost functional for well-posed linear systems over the class of inputs that are square integrable and that produce a square integrable output. As is well-known, the minimum cost can be expressed in terms of a bounded nonnegative selfadjoint operator X that in the finite-dimensional case satisfies a Riccati equation. Unfortunately, the infinite-dimensional generalization of this Riccati equation is not always well-defined. We show that X always satisfies alternative Riccati equations, which are more suitable for algebraic and numerical computations.

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Admissible observation operators for linear semigroups

Cited by 308 publications

References 13 publications

Quadratic optimal control of stable well-posed linear systems

Quadratic optimal control of stable well-posed linear systems

Observability of polynomially stable systems

New Riccati equations for well-posed linear systems

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