Research On Oscillation Of Functional Differential Equations And Stabilising Properties For Switched Systems

The theory of functional Differential equations is one of the new research areas in recent years, there are many mathematicians major in this theory, and the fundamental theories are obtained high development. Among these theories, oscillation theories for functional differential equation and partial functional differential equations are the most important qualitative theories for these equations, which have profound physical backgrounds. As we know, the comparison and separation theories of zeros distribution for second order homogeneous linear differential equations established by G. Sturm lay a foundation of oscillation theory for differential equations. During one and a half century, oscillation theory of differential equations has developed quickly and played an important role in qualitative theories and theory of boundary value problems. There are many mathematicians are major in this subject and they obtained many useful results.On the other hand, as one of the important branches, theories of delayed differential equations are the key research areas for many scientists. The existence of delay arguments makes the analysis of stability of the system more difficult. A switched system is a special hybrid system which consists of several continuous time (or discrete time) subsystems and a rule that orchestrates the switching among them. Switched systems possess good system structures which may lead to fundamental theoretical interest and important practical value. The switching law plays an important role in the performance of the switched system. In fact, switching among unstable subsystems could probably make an asymptotically stable switched system; similarly, when all subsystems are stable, the switched system could be unstable, depending on a particular switching signal. Main contributions of this paper are as follows:1. Using a generalized Riccati transformation and the method of integral average, new oscillation criteria for second order partial differential equations with delays are obtained, these results can be considered as the improvements and generalizations of Kamenev type oscillation criterion and Philos type oscillation criterion for ordinary differential equations.2. For second order partial differential equations with delays, we use integral averaging method and Riccati transformation to give oscillation criteria of interval type. These criteria are different from most known ones in the sense that they are based on the information only on a sequence of sub-interval of[t0,∞), rather than on the whole half-line. Our results are sharper than some of previous results and can handle the cases which are not covered by known criteria.3. For second order neutral differential equations, by using differential inequality, we skillfully dealt with the neutral term, using the Riccati transformation and integral averaging method, we obtain some oscillation criteria, which can be considered as generalizations and improvements for known results of neutral delay differential equations.4. The stabilisability problem of a class of single-input switched linear systems is considered. The dimension of the system is reduced with the variable-structure control. The sufficient conditions of the uniform stabilisation of the systems and the existence of the admissible stabilising strategies of the systems are obtained through the study of the sliding mode of the reduced systems. And the detailed admissible stabilising strategy sets are proposed. The completely admissible stabilising strategies for second-order switched systems are given as an application. The dynamic behavior of the second-order closed-loop switched systems near the switching boundary is considered based on the planar geometry theory and some new conceptions. And the sufficient conditions of whether sliding mode occurs or not on the switching boundary are given. Some numerical simulations validate the main results.