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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 \(\dot x= f(x,\theta)\), where \(x\in\mathbb{R}^n\) is the state vector, \(\theta\in \mathbb{R}^\alpha\) is a possibly time-varying parameter vector, \(f(0,\theta)= 0\), and for all \(\theta\), \(f(x,\theta)\) is smooth. The Lyapunov function is considered in the form \(V(x)= x^T P(x)x\), where \(P(x)= \sum^N_{i=1} P_i\rho_i(x)\), \(\rho_i(x)\) are smooth basis-functions and \(P_i\) are parameter matrices. The parameter matrices are sought from the condition, \(\dot V\leq -\gamma V(x)\), 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.

93D30 Lyapunov and storage functions
93C10 Nonlinear systems in control theory
93B40 Computational methods in systems theory (MSC2010)
15A39 Linear inequalities of matrices
90C25 Convex programming
Full Text: DOI
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