State-Feedback Stabilization of Well-Posed Linear Systems

Abstract

A finite-dimensional linear time-invariant system is output-stabilizable if and only if it satisfies the finite cost condition, i.e., if for each initial state there exists at least one L 2 input that produces an L 2 output. It is exponentially stabilizable if and only if for each initial state there exists at least one L 2 input that produces an L 2 state trajectory. We extend these results to well-posed linear systems with infinite-dimensional input, state and output spaces. Our main contribution is the fact that the stabilizing state feedback is well posed, i.e., the map from an exogenous input (or disturbance) to the feedback, state and output signals is continuous in L 2 loc in both open-loop and closed-loop settings. The state feedback can be chosen in such a way that it also stabilizes the I/O map and induces a (quasi) right coprime factorization of the original transfer function. The solution of the LQR problem has these properties. (2000). Primary 93D15, 49N10; Secondary 93C25. Mathematics Subject Classification

Weakly coprime factorization and state-feedback stabilization of discrete-time systems

“…Therefore, the "more natural" coprimeness concept, "gcd-coprime", has been called "weakly coprime" [12,32]. In the matrix-valued case that coincides with our definition, namely with the "quasi-coprime" of [14] and [15], by Theorem 6.14. In the operatorvalued case "gcd-coprime" coincides with "(Bézout) r.c."…”

“…is a strictly stronger tool than that provided by the dual inner- Proof of Theorem 6. 15 4. Therefore, F d is outer too.…”

“…The main exception is that in continuous time the Riccati condition in Theorem 1.2 (is slightly different and) may become very complicated if B is highly unbounded [15,46] (yet the rest of Theorem 1.2 holds). Moreover, in that article "proper" means "bounded on some right half-plane", and "some right half-plane" takes the role of "some neighborhood of the origin".…”

“…5 The transfer function of any output-stable system lies in H 2 strong . Conversely, any P ∈ H 2 strong (U, Y) has the output-stable realization (15) with state-space X := H 2 1/r,− (U) for any r > 1.…”

H ∞-functional calculus and models of Nagy–Foiaş type for sectorial operators

“…Then C is called admissible if it satisfies the estimate ∞ 0 Ce −t A x 2 dt ≤ K x 2 , x ∈ D(A), (7.2) for some positive constant K . Admissible (and exact) observation operators are important in linear Control Theory, in particular in the linear quadratic optimization problem, see [17,18,20] and references therein. In [12], see also [13, p. 204], admissible operators have been studied in terms of the admissibility of the operator √ A (in this case Y = H ), for bounded analytic semigroups (e −t A ) t>0 or equivalently when A is sectorial of type ω < π/2 (see [15]).…”