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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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