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This paper presents a proof that existence of a polynomial Lyapunov functionis necessary and sufficient for exponential stability of sufficiently smoothnonlinear ordinary differential equations on bounded sets. The main resultstates that if there exists an n-times continuously differentiable Lyapunovfunction which proves exponential stability on a bounded subset of R^n, thenthere exists a polynomial Lyapunov function which proves exponential stabilityon the same region. Such a continuous Lyapunov function will exist if, forexample, the right-hand side of the differential equation is polynomial or atleast n-times continuously differentiable. The proof is based on ageneralization of the Weierstrass approximation theorem to differentiablefunctions in several variables. Specifically, we show how to use polynomials toapproximate a differentiable function in the Sobolev norm W^{1,\infty} to anydesired accuracy. We combine this approximation result with the second-orderTaylor series expansion to find that polynomial Lyapunov functions canapproximate continuous Lyapunov functions arbitrarily well on bounded sets. Ourinvestigation is motivated by the use of polynomial optimization algorithms toconstruct polynomial Lyapunov functions.

Exponentially Stable Nonlinear Systems have Polynomial Lyapunov Functions on Bounded Regions