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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 $\stackrel{˙}{x}=f\left(x,\theta \right)$, where $x\in {ℝ}^{n}$ is the state vector, $\theta \in {ℝ}^{\alpha }$ is a possibly time-varying parameter vector, $f\left(0,\theta \right)=0$, and for all $\theta$, $f\left(x,\theta \right)$ is smooth. The Lyapunov function is considered in the form $V\left(x\right)={x}^{T}P\left(x\right)x$, where $P\left(x\right)={\sum }_{i=1}^{N}{P}_{i}{\rho }_{i}\left(x\right)$, ${\rho }_{i}\left(x\right)$ are smooth basis-functions and ${P}_{i}$ are parameter matrices. The parameter matrices are sought from the condition, $\stackrel{˙}{V}\le -\gamma V\left(x\right)$, where $\gamma >0$. 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:
 93D30 Scalar and vector Lyapunov functions 93C10 Nonlinear control systems 93B40 Computational methods in systems theory 15A39 Linear inequalities of matrices 90C25 Convex programming