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Linear Relaxations of Polynomial Positivity for Polynomial Lyapunov Function Synthesis

In this paper, we examine linear programming (LP) based relaxations for synthesizing polynomial Lyapunov functions to prove the stability of polynomial ODEs. A common approach to Lyapunov function synthesis starts from a desired parametric polynomial form of the polynomial Lyapunov function. Subsequently, we encode the positive-definiteness of the function, and the negative-definiteness of its derivative over the domain of interest. Therefore, the key primitives for this encoding include: (a) proving that a given polynomial is positive definite over a domain of interest, and (b) encoding the positive definiteness of a given parametric polynomial, as a constraint on the unknown parameters. We first examine two classes of relaxations for proving polynomial positivity: relaxations by sum-of-squares (SOS) programs, against relaxations that produce linear programs. We compare both types of relaxations by examining the class of polynomials that can be shown to be positive in each case. Next, we present a progression of increasingly more powerful LP relaxations based on expressing the given polynomial in its Bernstein form, as a linear combination of Bernstein polynomials. The well-known bounds on Bernstein polynomials over the unit box help us formulate increasingly precise LP relaxations that help us establish the positive definiteness of a polynomial over a bounded domain. Subsequently, we show how these LP relaxations can be used to search for Lyapunov functions for polynomial ODEs by formulating LP instances. We compare our approaches to synthesizing Lyapunov functions with approaches based on SOS programming relaxations.