The non-symmetric discrete algebraic Riccati equation and canonical factorization of rational matrix functions on the unit circle.

Canonical factorization of a rational matrix function on the unit circle is described explicitly in terms of a stabilizing solution of a discrete algebraic Riccati equation using a special state space representation of the symbol. The corresponding Riccati difference equation is also discussed. © The Author(s)

Multipoint Schur algorithm and orthogonal rational functions: convergence properties, I

Classical Schur analysis is intimately connected to the theory of orthogonal polynomials on the circle [Simon, 2005]. We investigate here the connection between multipoint Schur analysis and orthogonal rational functions. Specifically, we study the convergence of the Wall rational functions via the development of a rational analogue to the Szeg\H o theory, in the case where the interpolation points may accumulate on the unit circle. This leads us to generalize results from [Khrushchev,2001], [Bultheel et al., 1999], and yields asymptotics of a novel type.Comment: a preliminary version, 39 pages; some changes in the Introduction, Section 5 (Szeg\H o type asymptotics) is extende

Automatique et traitement du signal

Quelques réflexions sont présentées sur le thème "Automatique et Traitenent du Signal". Bien qu'une discipline s'appuie sur la notion de système et l'autre sur celle de signal, il est montré que les points de vue se complètent et que les techniques mathématiques sont souvent communes

Fast cooling for a system of stochastic oscillators

We study feedback control of coupled nonlinear stochastic oscillators in a force eld. We rst consider the problem of asymptotically driving the system to a desired steady state corresponding to reduced thermal noise. Among the feedback controls achieving the desired asymptotic transfer, we nd that the most e cient one from an energy point of view is characterized by time-reversibility. We also extend the theory of Schr\uf6dinger bridges to this model, thereby steering the system in nite time and with minimum e ort to a target steady-state distribution. The system can then be maintained in this state through the optimal steady-state feedback control. The solution, in the nite-horizon case, involves a space-time harmonic function \u3c6, and 12 log \u3c6 plays the role of an arti cial, time-varying potential in which the desired evolution occurs. This framework appears extremely general and exible and can be viewed as a considerable generalization of existing active control strategies such as macromolecular cooling. In the case of a quadratic potential, the results assume a form particularly attractive from the algorithmic viewpoint as the optimal control can be computed via deterministic matricial di erential equations. An example involving inertial particles illustrates both transient and steady state optimal feedback contro