Nguyen Khoa Son

Stability radii of higher order positive difference systems

In this paper we study stability radii of positive polynomial matrices under affine perturbations of the coefficient matrices. It is shown that the real and complex stability radii coincide. Moreover, explicit formulas are derived for these stability radii and illustrated by some examples.

A unified approach to constrained approximate controllability for the heat equations and the retarded equations

Abstract The aim of this paper is to study controllability of the linear infinite-dimensional system x = Ax + Bu , where A is the infinitesimal generator of a C 0 -semigroup of linear bounded operators in the Banach space X ; the control u is restricted to lie in a subset Ω of the Banach space U ; Ω need not assumed to be convex or to contain 0 in its interior. Some necessary and sufficient conditions for approximate controllability to the whole space X are given. The proof of the main result is based on the spectral decomposition method developed by Fattorini and the generalized open mapping theorem due to Robinson. The obtained results are used to consider some controllability problems, with constrained controls, for the class of linear systems described by partial differential equations of parabolic type with bounded domain, and for the class of retarded functional differential equations in the state space M P . Particularly, in the case of the heat equation with positive scalar controls and in the case of the retarded equation with finite discrete delays, the general result leads to easily verifiable tests for approximate controllability, expressed in terms of the system matrices.

Stability Radii of Positive Linear Functional Differential Equations under Multi-Perturbations

We study stability radii of linear retarded systems described by general linear functional differential equations. A lower and an upper bound for the complex stability radius with respect to multi-perturbations are given. Furthermore, in some special cases concerning the structure matrices, the complex stability radius can precisely be computed via the associated transfer function. Then, the class of positive linear retarded systems is studied in detail. It is shown that for this class, complex, real and positive stability radius under multi-perturbations or multi-affine perturbations coincide and can be computed by simple formulae expressed in terms of the system matrices.

Stability Radii of Linear Systems Under Multi-perturbations

Abstract In this paper, we study stability radii of linear systems under multi-perturbation of the coefficient matrices. Formulas for complex stability radius are derived. We then consider linear positive systems and prove that for this class of systems, the complex stability radius is equal to the real stability radius which can be computed via a simple formula. We illustrated the obtained results by two examples.

Stability radius of linear delay systems

We study robust stability of linear time delay systems under structured parameter uncertainty. A formula for complex stability radius is derived. We then consider linear positive delay systems and prove that for this class of systems the complex stability radius is equal to the real stability radius which can be computed via a simple formula. An illustrative example is given.

Stability radii of positive higher order difference system in infinite dimensional spaces

Abstract In this paper we study stability radii of positive higher order difference system under affine perturbations of the coefficient operators. It is shown that the real and complex stability radii coincide. Moreover, explicit formulas are derived for these stability radii and illustrated by some examples. The results obtained are extensions of the recent work in [D. Hinrichsen, P.H.A. Ngoc, N.K. Son, Stability radii of positive higher order difference system, Systems Control Lett. 49 (2003) 377–388] to Banach spaces.

Stability radii of delay difference systems under affine parameter perturbations in infinite dimensional spaces

Abstract In this paper, we study the stability radii of positive difference systems with delays under arbitrary affine parameter perturbations in infinite dimensional spaces. The obtained results are extensions of the recent results in [P.H.A. Ngoc, N.K. Son, Stability radii of positive linear difference equations under affine parameter perturbation, Appl. Math. Comput. 134 (2003) 577–594; A. Fischer, Stability radii of infinite-dimensional positive systems, Math. Control Signals Syst. 10 (1997) 223–236].

Stability radii of positive linear time-delay systems under fractional perturbations

In this paper we study stability radii of positive linear delay systems under fractional perturbations. It is shown that the three stability radii: complex, real and positive stability radii coincide and can be computed by a simple formula. Finally, a simple example is given to illustrate the obtained results.

The structured distance to uncontrollability under multi-perturbations: An approach using multi-valued linear operators

Abstract In this paper we develop a unifying approach for computing the distance to uncontrollability of linear control systems. By using multi-valued linear operators in representing and estimating the system’s equations and matrices we are able to derive computable formulas of the distance from a controllable linear system to the nearest uncontrollable system under the assumption that the system’s matrices are subjected to structured multi-perturbations and measured by arbitrary operator norms. In the case of spectral norms, the obtained results unify and extend some previous works as well as a recent interesting result in [M. Karrow, D. Kressner, On the structured distance to uncontrollability, Systems Control Lett. 58 (2009) 128–132]. Some illustrating examples are given.

On the real stability radius of positive linear discrete-time systems

Robust stability of linear discrete-time systems invariant with respect to a convex cone in R n is considered. An implicit formula for the real stability radius is established and proved to coincide with the complex stability radius for wide classes of vector norms.