On a class of strongly stabilizable systems of neutral type

International audienceWe consider the strong stabilizability problem for delayed system of neutral type. For simplicity the case of one delay in state is studied. We separate a class of such systems and give a constructive solution of the problem in this case, without the derivative of the localized delayed state. Our results are based on an abstract theorem on the strong stabilizability of contractive systems in Hilbert space. An illustrating example is also given

Strong stabilizability for a class of linear time delay systems of neutral type

International audienceWe consider the strong stabilizability, i.e. asymptotic non-exponential stabilizability, problem for a class of neutral type systems. A constructive solution of the feedback law is given, without the derivative of the localized delayed state. Our results are based on an abstract theorem on the strong stabilizability of contractive systems in Hilbert space. The paper is an extended version of the article published in "Applied Mathematics Letters", vol. 18, 4 (2005), pp. 463--469, cf. hal-00819335-v1

Dynamics and control of a class of underactuated mechanical systems

This paper presents a theoretical framework for the dynamics and control of underactuated mechanical systems, defined as systems with fewer inputs than degrees of freedom. Control system formulation of underactuated mechanical systems is addressed and a class of underactuated systems characterized by nonintegrable dynamics relations is identified. Controllability and stabilizability results are derived for this class of underactuated systems. Examples are included to illustrate the results; these examples are of underactuated mechanical systems that are not linearly controllable or smoothly stabilizable

On strong stability and stabilizability of systems of neutral type

International audienceFor linear stationary systems, the infinite dimensional framework allows one to distinguish different notions of stability: weak, strong or exponential. The purpose of this chapler is to investigate the problem of strong stability, i.e. asymptotic non-exponential stability for linear systems of neutral type in order to use this characterization in the study of the stabilizability problem for this type of systems. An important tool in this investigation is the Riesz basis property of generalized eigenspaces of the neutral system, because that the generalized eigenvectors do not form, in general, a Riesz basis. This allows one to describe more precisely asymptotic non-exponential stability of neutral systems. For a particular case, conditions of strong stabilizability of neutral type systems are given with a feedback law without derivative of the delayed state

On Pole Assignment and Stabilizability of Neutral Type Systems

In this note we present a systematic approach to the stabilizability problem of linear infinite-dimensional dynamical systems whose infinitesimal generator has an infinite number of instable eigenvalues. We are interested in strong non-exponential stabilizability by a linear feed-back control. The study is based on our recent results on the Riesz basis property and a careful selection of the control laws which preserve this property. The investigation may be applied to wave equations and neutral type delay equations

On strong regular stabilizability for linear neutral type systems

International audienceThe problem of strong stabilizability of linear systems of neutral type is investigated. We are interested in the case when the system has an infinite sequence of eigenvalues with vanishing real parts. This is the case when the main part of the neutral equation is not assumed to be stable in the classical sense. We discuss the notion of regular strong stabilizability and present an approach to stabilize the system by regular linear controls. The method covers the case of multivariable control and is essentially based on the idea of infinite-dimensional pole assignment proposed in [G.M. Sklyar, A.V. Rezounenko, A theorem on the strong asymptotic stability and determination of stabilizing controls, C. R. Acad. Sci. Paris Ser. I Math. 333 (8) (2001) 807-812]. Our approach is based on the recent results on the Riesz basis of invariant finitedimensional subspaces and strong stability for neutral type systems presented in [R. Rabah, G.M. Sklyar, A.V. Rezounenko, Stability analysis of neutral type systems in Hilbert space, J. Differential Equations 214 (2) (2005) 391-428]

The revision and extension of the R-MS ring for time delay systems

This paper is aimed at reviewing the ring of retarded quasipolynomial meromorphic functions (R-MS) that was recently introduced as a convenient control design tool for linear, time-invariant time delay systems (TDS). It has been found by the authors that the original definition does not constitute a ring and has some essential deficiencies, and hence it could not be used for an algebraic control design without a thorough reformulation which i.e. extends the usability to neutral TDS and to those with distributed delays. This contribution summarizes the original definition of RMS, simply highlights its deficiencies via examples, and suggests a possible new extended definition. Hence, the new ring of quasipolynomial meromorphic functions (R-QM) is established to avoid confusion. The paper also investigates and introduces selected algebraic properties supported by some illustrative examples and concisely outlines its use in controller design.European Regional Development Fund under the project CEBIA-Tech Instrumentation [CZ.1.05/2.1.00/19

The stability and stabilization of infinite dimensional Caputo-time fractional differential linear systems

We investigate the stability and stabilization concepts for infinite dimensional time fractional differential linear systems in Hilbert spaces with Caputo derivatives. Firstly, based on a family of operators generated by strongly continuous semigroups and on a probability density function, we provide sufficient and necessary conditions for the exponential stability of the considered class of systems. Then, by assuming that the system dynamics is symmetric and uniformly elliptic and by using the properties of the Mittag-Leffler function, we provide sufficient conditions that ensure strong stability. Finally, we characterize an explicit feedback control that guarantees the strong stabilization of a controlled Caputo time fractional linear system through a decomposition approach. Some examples are presented that illustrate the effectiveness of our results.publishe

Mathematical control of complex systems

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