Linear Retarded Systems Research Articles

New double integral inequality with application to stability analysis for linear retarded systems

This paper presents the development of a new double integral inequality (II) with the motivation of yielding quadratic approximation. It is well known that approximating integral quadratic terms with quadratic terms involves a certain degree of conservatism. In this paper, a sufficient gap has been identified in the approximation of two recent IIs reported in the literature, thereby leading to the new double II. The developed inequality has been applied to access the stability of a linear retarded system to estimate a maximum delay upper-bound. Furthermore, a mathematical relationship of the new double II with existing inequalities is discussed to show that the developed inequality is more general, effective and bears less computational burden. Four numerical examples are given to validate the authors' claim with regard to the effective estimate of delay bound results for a linear retarded system.

On the Stability of a Linear Retarded Differential-Difference Equation

In the present paper, we give a necessary and sufficient condition for the zero solution of a linear retarded system dx(t)/dt = Ax(t) + Bx(t − τ) to be asymptotically stable. Here A is a real-valued n × n matrix and B = bI, where b is a scalar parameter and I is the n × n unit matrix.The stability analysis is reduced to deriving a necessary and sufficient condition for all the roots of a characteristic equation z − α − βe−z = 0 to have negative real parts. Here α is a complex number defined by α = τλ with an eigenvalue λ of A, and β = τb. Our stability criterion is a natural extension of that for the widely-known case where α is a real number.

Nonoscillations in retarded systems

This note is concerned with the existence of nonoscillatory solutions of a linear retarded system. Several criteria for nonoscillations are obtained, some of them regarding specific classes of continuous and differentiable delay functions.

Oscillatory retarded functional systems

This note is concerned with the oscillatory behavior of a linear retarded system. Several criteria are obtained for having the system oscillatory. Conditions regarding the existence of nonoscillatory solutions are also given.

Feedback stabilization of linear systems with distributed delays in state and control variables

Solving a generalized characteristic matrix equation (GCME) of a linear retarded system is the crux for stabilizing the system using a reduction technique. This note gives the solution of GCME under the general condition that the eigenvalues occur in the spectrum of the system with algebraic and geometric multiplicities being greater than or equal to one. The main idea is to transform the GCME into a group of linear algebraic equations. A sufficient condition for the existence of the solution to the linear algebraic equations is established. A key lemma for stabilizing the systems under the above general condition is also proved.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the closure in W1q of the attainable subspace of linear time lag systems

This paper presents the necessary and sufficient condition for the closedness in the topology of W 1 q ( T − 1, T; R n ) of the set of all complete final states x [ T − 1, T] of a linear retarded system, which can be attained by L p controls. The criterion is fully computable. The analyticity of the matrix coefficients of the system equation is assumed. A number of related results are presented, including a criterion for the closedness of the set of all trajectories of a nondelayed linear system with measurable coefficients.