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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.

MSC:
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
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