Existence of quadratic type liapunov functions for a class of nonlinear systems

Abstract The paper deals with the construction of Liapunov functions for nonlinear systems related to the Lur'e problem. The method presented consists in developing a quadratic type Liapunov function— called a Spanning Liapunov Function (SLF)—for an associated linear system and modifying it suitably for the nonlinear case. Using this approach, conditions are derived for the existence of an SLF ensuring stability for the generalized Lur'e problem with m nonlinearities. For the casem = 1, these conditions are identical to those obtained by Popov [3] and Kalman [l]. Conditions for the existence of an SLF for the case of a single nonlinear gain l ( σ ) with0 l ( σ ) l are derived. The results are used to construct Liapunov functions for certain specific systems.