An explicit Floquet-type representation of Riccati aperiodic exponential semigroups

Abstract

The article presents a rather surprising Floquet-type representation of time-varying transition matrices associated with a class of nonlinear matrix differential Riccati equations. The main difference with conventional Floquet theory comes from the fact that the underlying flow of the solution matrix is aperiodic. The monodromy matrix associated with this Floquet representation coincides with the exponential (fundamental) matrix associated with the stabilizing fixed point of the Riccati equation. The second part of this article is dedicated to the application of this representation to the stability of matrix differential Riccati equations. We provide refined global and local contraction inequalities for the Riccati exponential semigroup that depend linearly on the spectral norm of the initial condition. These refinements improve upon existing results and are a direct consequence of the Floquet-type representation, yielding what seems to be the first results of this type for this class of models.

On the stability of matrix-valued Riccati diffusions

“…This estimate in a sense generalises the well-known bounds Φ υ ≤ φ t (Q) ≤ Φ υ for some Φ υ , Φ υ > 0 and t ≥ υ > 0; see e.g. [8,10]. Note that the uniform estimates independent of the initial condition stated throughout, involve some arbitrarily small, positive time parameter υ, which is related to the notion of a so-called observability/controllability interval; for further details on this topic we refer to [8].…”

“…The proof of the r.h.s. uniform estimate is in [8,10]; e.g. it is easy to verify φ t (Q) ≤ c ( P ∞ ∨ Q ).…”

“…The proof of the preceding Theorem is provided in Section 5.3. Recall that the proof of (1.9) is an easy consequence of the inequality (4.5), see also Theorem 1.3 in [12], and the uniform estimates in [8,10].…”

“…the time horizon. For further discussion on these stability properties we refer to [8,10] and the references therein. We point to [3,4,37] for precise definitions of controllable, stabilizable, and their duals, observable, detectable.…”

“…The stability of (A − P t S), and also of φ t (Q) → t→∞ P ∞ , is directly related [8,10] to the contractive properties of the deterministic semigroup E s,t (Q). The stochastic equation (1.6) implies that the stability properties of EnKF state estimators with ̟ = 0 depend on the stability properties of the stochastic exponential semigroup E ǫ s,t (Q).…”