Explain krasovskii method for determination of Liapunov function.
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Analysis of the stability of dynamical systems plays a very important role in control system analysis and design. For linear systems, it is easy to verify the stability of equilibria. For nonlinear dynamical systems, proving stability of equilibria of nonlinear systems is more complicated than linear systems. One can use the Lyapunov function at the equilibriato determine the stability.
For an autonomous polynomial system of differential equations, how to compute the Lyapunov function at equilibria is a basic problem. In [1, 2], the author transformed the problem of computing the Lyapunov function into a quantifier elimination problem. The disadvantage of the method is that the computation complexity of quantifier elimination is doubly exponential in the number of total variables. In order to avoid this problem, She et al. [3] propose a symbolic method; they first construct a special semialgebraic system using the local properties of a Lyapunov function as well as its derivative and solving these inequations using cylindrical algebraic decomposition (CAD) introduced by Collins in [4]. The algorithm in [5] uses semidefinite programming to search for Lyapunov function. There are also other algorithms
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