This paper presents a new technique for constructing Lyapunov functions for estimating the domain of asymptotic stability of nonlinear control systems. A machine program assigns figures of merit (based on the quality of the stability-boundary estimates) to the members of a set of Lyapunov functions and then searches for the one which maximizes this measure. The method is applied to relay-control systems, which are not tractable by other systematic techniques such as the method of Zubov. The paper is divided into two parts. The first part contains a discussion of the motions of relay-control systems and the technique of investigating their stability domains with Lyapunov functions. The second part contains a discussion of existing methods of generating Lyapunov functions and a description of the new method, along with a number of second and third-order examples.

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