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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.

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