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In this paper, we propose a new convex approach to stability analysis ofnonlinear systems with polynomial vector fields. First, we consider anarbitrary convex polytope that contains the equilibrium in its interior. Then,we decompose the polytope into several convex sub-polytopes with a commonvertex at the equilibrium. Then, by using Handelman's theorem, we derive a newset of affine feasibility conditions -solvable by linear programming- on eachsub-polytope. Any solution to this feasibility problem yields a piecewisepolynomial Lyapunov function on the entire polytope. This is the first resultwhich utilizes Handelman's theorem and decomposition to construct piecewisepolynomial Lyapunov functions on arbitrary polytopes. In a computationalcomplexity analysis, we show that for large number of states and large degreesof the Lyapunov function, the complexity of the proposed feasibility problem isless than the complexity of certain semi-definite programs associated withalternative methods based on Sum-of-Squares or Polya's theorem. Using differenttypes of convex polytopes, we assess the accuracy of the algorithm inestimating the region of attraction of the equilibrium point of thereverse-time Van Der Pol oscillator.

Constructing Piecewise Polynomial Lyapunov Functions for Local Stability of Nonlinear Systems Using Handelman's Theorem