Piecewise-quadratic and piecewise-linear Lyapunov functions for discontinuous systems†

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Abstract
This paper develops necessary and sufficient conditions for the existence of Lyapunov functions of tho Lur'e type (‘ piecewise-quadratic ’) for relay-control systems and gives explicit formulae- for ‘optimum ’functions of this form for estimating regions of asymptotic stability in special cases, comparing these with results of numerical optimization. Certain similarities and advantages of ‘piecewise-linear ’Lyapunov functions are discussed. Among their benefits is algebraic simplification which allows one to compute ranges of time varying and non-linear parameters which preserve asymptotic stability in a given region in the state space; an example of this type is given.