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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 $x\in\bbfR^n$ is the state vector, $\theta\in \bbfR^\alpha$ 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, $\dot V\le -\gamma V(x)$, 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:
93D30Scalar and vector Lyapunov functions
93C10Nonlinear control systems
93B40Computational methods in systems theory
15A39Linear inequalities of matrices
90C25Convex programming
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References:
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