Research On The Stability Criteria And Applications Of Two Classes Of Retarded Dynamical Systems

This thesis studies the stability criteria and applications of two classes of retarded dynamical systems, which are linear differential-difference systems of retarded type with two delays and neutral type with multiple delays. Theses systems, featuring with transcendental characteristic equations and infinite-dimensional solution spaces, may lose stability due to the time delays inherited in systems. The stability criteria and preliminary applications of the above systems have been developed in this thesis. The main work is as follows:1. The stability of a class of two-delay systems of retarded type has been studied. For the delay-independent stability, necessary and sufficient algebraic criteria have been obtained in terms of characteristic functions. The key of the criteria is to verify that a polynomial has no nonzero real roots. For the delay-dependent stability, a critical delay formula has been found. The delegate point method for determining the stability of subdivisions has been summarized and the gradient method for determining the instability of subdivisions has been put forward.2. Based on the theory of structured singular value, this thesis provides necessary and sufficient algebraic criteria for the delay-independent stability of a class of multiple-delay systems of neutral type. This criteria contains two parts. The first is to check the delay-independent stability of a class of multiple-delay linear difference systems. The key of this part is to verify the structured singular value of a constant matrix is not equal to one. The second is to assure a multiple-parameter matrix characteristic equation has no nonzero pure imaginary roots. The key of this part is to verify that the structured singular value of a one-unknown matrix function is not equal to one. This work can be accomplished by a frequency sweeping test method. With the spectral theory of matrices, three sufficient algebraic criteria have been developed for the above multiple-delay linear difference systems. The key of the criteria is to verify that the sum of the spectrum or norm of system matrices is less than 1. With the characteristic function method, necessary and sufficient algebraic criteria have been developed for a class of three-delay linear difference systems. The key of the criteria is to verify that the valueof a trigonometric polynomial is less than one. This work can be fulfilled by curve plotting.3. With the above stability criteria , this thesis studies the stability of three systems which are a delayed stirred tank control system, a delayed ecosystem, and a lossless transmission line. This thesis also studies the stability of a delayed control system with one driving leg mainly base on the classic theory on delay stability switches. These researches have extended pioneers' work.