Weakly coprime factorization and continuous-time systems

Abstract

We generalize the classical theory on algebraic Riccati equations and optimization to infinite-dimensional well-posed linear systems, thus completing the work of George Weiss, Olof Staffans and others. We show that the optimal control is given by the stabilizing solution of an integral Riccati equation. If the input operator is not maximally unbounded, then this integral Riccati equation is equivalent to the algebraic Riccati equation.Using the integral Riccati equation, we show that for (nonsingular) minimization problems the optimal state-feedback loop is always well-posed. In particular, the optimal state-feedback operator is admissible also for the original semigroup, not only for the closed-loop semigroup (as has been known in some cases); moreover, both settings are well-posed with respect to an external input. This leads to the positive solution of several central, previously open questions on exponential, output and dynamic (aka. "internal") stabilization and on coprime factorization of transfer functions.Our theory covers all quadratic (possibly indefinite) cost functions, but the optimal state feedback need not be well-posed (admissible) unless the cost function is uniformly positive or the system is sufficiently regular.

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

“…In the operatorvalued case "gcd-coprime" coincides with "(Bézout) r.c." [19], so the term "weakly coprime" can be reserved, instead, to our definition (of "w.r.c.") also in that case.…”

“…The continuous-time counterparts of the results in this article (and more) are provided in [19], which is built on the results in this article, with the class of well-posed linear systems [30,35,36,44] as the class of realizations. 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).…”

“…If N and M are scalar-valued and we only require (1) to hold for scalar-valued functions V , then we get the classical definition of a "weakly right coprime factorization" [9,12,32]. The same holds in the matrix-valued case too if we only require (1) to hold for square-matrix-valued functions V [19,Theorem 2.17(c) ].…”

“…6 the relations between different forms of coprimeness will be treated (the claims on (1) partially in [19]) and the above will be shown equivalent to the classical definition in the matrix-valued case. The property (2) is very important from the control-theoretic point of view.…”

“…1 In [19] it will be shown that an equivalent definition is obtained with, e.g., any H p space in place of H 2 in (2).…”