Quadratic optimal control

of stable well-posed linear systems

Author:
Olof J. Staffans

Journal:
Trans. Amer. Math. Soc. **349** (1997), 3679-3715

MSC (1991):
Primary 49J27, 93A05, 47B35

DOI:
https://doi.org/10.1090/S0002-9947-97-01863-1

MathSciNet review:
1407712

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Abstract | References | Similar Articles | Additional Information

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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Additional Information

**Olof J. Staffans**

Affiliation:
Department of Mathematics, Åbo Akademi University, FIN-20500 Åbo, Finland

Email:
Olof.Staffans@abo.fi

DOI:
https://doi.org/10.1090/S0002-9947-97-01863-1

Keywords:
Spectral factorization,
inner-outer factorization,
Wiener-Hopf factorization,
algebraic Riccati equation,
state feedback

Received by editor(s):
January 30, 1995

Received by editor(s) in revised form:
March 20, 1996

Article copyright:
© Copyright 1997
American Mathematical Society