The behaviour of optimal Lyapunov functions

The use of Lyapunov's direct method in obtaining regions of asymptotic stability of non-linear autonomous systems is well-known. This thesis is an investigation into the optimization of some function of these systems over different classes of Lyapunov functions. In Chapter 2 bounds on the transient response of two systems are optimized over a subset of quadratic Lyapunov functions and numerical work is carried out to compare several bounds. Zubov's equation is the subject of Chapter 3. The non-uniformity of the series-construction procedure is studied analytically and a new approach is made to the solution of the equation by finite difference methods. Chapters 4, 5 and 6 have a common theme of optimizing the RAS over a class of Lyapunov functions. Chapter 4 is restricted to optimal quadratics which are investigated analytically and numerically, two algorithms being developed. An optimal quadratic algorithm and a RAS algorithm are proposed in Chapter 5 for high order systems. Extensions are made in Chapter 6 to optimal Lyapunov functions of general degree and relay control systems and systems of Lur'e form are considered.