Do Duc Thuan

Stability radius of implicit dynamic equations with constant coefficients on time scales

This paper deals with the stability radii of implicit dynamic equations on time scales when the structured perturbations act on both the coefficient of derivative and the right-hand side. Formulas of the stability radii are derived as a unification and generalization of some previous results. A special case where the real stability radius and the complex stability radius are equal is studied. Examples are derived to illustrate results.

STABILITY AND ROBUST STABILITY OF LINEAR TIME-INVARIANT DELAY DIFFERENTIAL-ALGEBRAIC EQUATIONS ∗

Necessary and sufficient conditions for exponential stability of linear time-invariant delay differential-algebraic equations are presented. The robustness of this property is studied when the equation is subjected to structured perturbations and a computable formula for the structured stability radius is derived. The results are illustrated by several examples.

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.

Stability Analysis of Implicit Difference Equations Under Restricted Perturbations

The stability analysis for linear implicit

The structured controllability radius of linear delay systems

m

Radius of approximate controllability of linear retarded systems under structured perturbations

th order difference equations is discussed. We allow the leading coefficient to be singular, i.e., we include the situation that the system does not generate an explicit recursion. A spectral condition for the characterization of asymptotic stability is presented and computable formulas are derived for the real and complex stability radii in the case that the coefficient matrices are subjected to structured perturbations.

Controllability Radii of Linear Systems with Constrained Controls Under Structured Perturbations

In this article, we shall deal with the problem of calculation of the controllability radius of a delay dynamical systems of the form x′(t) = A 0 x(t) + A 1 x(t − h 1) + ··· + A k x(t − h k ) + Bu(t). By using multi-valued linear operators, we are able to derive computable formulas for the controllability radius of a controllable delay system in the case where the systems coefficient matrices are subjected to structured perturbations. Some examples are provided to illustrate the obtained results.

Exponential Stability and Robust Stability for Linear Time-Varying Singular Systems of Second Order Difference Equations

Abstract In this paper we shall deal with the problem of calculation of the radius of approximate controllability in the Banach state space K n × L 2 ( [ − h k , 0 ] , K n ) for linear retarded systems of the form x ( t ) = A 0 x ( t ) + A 1 x ( t − h 1 ) + ⋯ + A k x ( t − h k ) + B u ( t ) . By using multi-valued linear operators we are able to derive computable formulas for this radius when the system’s coefficient matrices are subjected to structured perturbations. Some examples are provided to illustrate the obtained results.

Controllability radii of linear neutral systems under structured perturbations

In this paper, the robust controllability for linear systems with constrained controls

On data dependence of stability domains, exponential stability and stability radii for implicit linear dynamic equations

\dot x =Ax + Bu, u\in \Omega,