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Computation of Lyapunov functions for smooth nonlinear systems using convex optimization. (English) Zbl 0971.93069

This work presents a computational approach for searching a Lyapunov function for the equilibrium point x=0 for a class of nonautonomous nonlinear systems x ˙=f(x,θ), where x n is the state vector, θ α is a possibly time-varying parameter vector, f(0,θ)=0, and for all θ, f(x,θ) is smooth. The Lyapunov function is considered in the form V(x)=x T P(x)x, where P(x)= i=1 N P i ρ i (x), ρ i (x) are smooth basis-functions and P i are parameter matrices. The parameter matrices are sought from the condition, V ˙-γV(x), where γ>0. The last condition is the condition of exponential stability of the equilibrium point.

The main contribution of this paper is the utilization of a flexible and general smooth parameterization of the Lyapunov function candidates that does not introduce significant conservativeness and the problem is reduced to a convex optimization problem involving linear inequality constraints at each point in the state space.

93D30Scalar and vector Lyapunov functions
93C10Nonlinear control systems
93B40Computational methods in systems theory
15A39Linear inequalities of matrices
90C25Convex programming