We examine linear programming (LP) based relaxations for synthesizing polynomial Lyapunov functions to prove the stability of polynomial ODEs. Our approach starts from a desired parametric polynomial form of the polynomial Lyapunov function. Subsequently, we encode the positive-definiteness of the function, and the negation of its derivative, over the domain of interest. We first compare two classes of relaxations for encoding polynomial positivity: relaxations by sum-of-squares (SOS) programs, against relaxations based on Handelman representations and Bernstein polynomials, that produce linear programs. Next, we present a series of increasingly powerful LP relaxations based on expressing the given polynomial in its Bernstein form, as a linear combination of Bernstein polynomials. 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 techniques with approaches based on SOS on a suite of automatically synthesized benchmarks. Posi-
Abstract. In this paper we propose a new method for reachability analysis of the class of discrete-time polynomial dynamical systems. Our work is based on the approach combining the use of template polyhedra and optimization [1,2]. These problems are non-convex and are therefore generally difficult to solve exactly. Using the Bernstein form of polynomials, we define a set of equivalent problems which can be relaxed to linear programs. Unlike using affine lower-bound functions in [2], in this work we use piecewise affine lower-bound functions, which allows us to obtain more accurate approximations. In addition, we show that these bounds can be improved by increasing artificially the degree of the polynomials. This new method allows us to compute more accurately guaranteed over-approximations of the reachable sets of discrete-time polynomial dynamical systems. We also show different ways to choose suitable polyhedral templates. Finally, we show the merits of our approach on several examples.
Abstract. This paper deals with the computation of polytopic invariant sets for polynomial dynamical systems. An invariant set of a dynamical system is a subset of the state space such that if the state of the system belongs to the set at a given instant, it will remain in the set forever in the future. Polytopic invariants for polynomial systems can be verified by solving a set of optimization problems involving multivariate polynomials on bounded polytopes. Using the blossoming principle together with properties of multi-affine functions on rectangles and Lagrangian duality, we show that certified lower bounds of the optimal values of such optimization problems can be computed effectively using linear programs. This allows us to propose a method based on linear programming for verifying polytopic invariant sets of polynomial dynamical systems. Additionally, using sensitivity analysis of linear programs, one can iteratively compute a polytopic invariant set. Finally, we show using a set of examples borrowed from biological applications, that our approach is effective in practice.
Abstract-This paper deals with the computation of polyhedral positive invariant sets for polynomial dynamical systems. A positive invariant set is a subset of the state-space such that if the initial state of the system belongs to this set, then the state of the system remains inside the set for all future time instances. In this work, we present a procedure that constructs an invariant set, iteratively, starting from an initial polyhedron that forms a "guess" at the invariant. At each iterative step, our procedure attempts to prove that the given polyhedron is a positive invariant by setting up a non-linear optimization problem for each facet of the current polyhedron. This is relaxed to a linear program through the use of the blossoming principle for polynomials. If the current iterate fails to be invariant, we attempt to use local sensitivity analysis using the primal-dual solutions of the linear program to push its faces outwards/inwards in a bid to make it invariant. Doing so, however, keeps the face normals of the iterates fixed for all steps. In this paper, we generalize the process to vary the normal vectors as well as the offsets for the individual faces. Doing so, makes the procedure completely general, but at the same time increases its complexity. Nevertheless, we demonstrate that the new approach allows our procedure to recover from a poor choice of templates initially to yield better invariants. I. INTRODUCTIONThis paper presents an iterative algorithm for discovering polyhedral positive invariant sets for polynomial dynamical systems. A set P is a positive invariant iff for all x ∈ P , the trajectory initialized to x(0) = x satisfies x(t) ∈ P for all t ≥ 0. Positive invariants are useful in the analysis of dynamical systems for proving properties. The problem of finding the smallest positive invariant that contains a given set (and/or excludes a set of unsafe states) is a useful primitive in reachability analysis. Likewise, the problem of finding the largest positive invariant containing an attractor is useful in bounding the region of attraction.However, finding positive invariants is often a hard problem for nonlinear systems. Existing approaches rely on relaxations such as sum-of-squares programming to find a polynomial p such that p ≥ 0 is a positive invariant [PJP07]. In this work, we find a polyhedron that is positive invariant for a polynomial system by solving a series of linear programming (LP) problems obtained by relaxing nonlinear optimization problems. The relaxation uses the blossom principle of writing a polynomial p as a multiaffine polynomial over a larger set of variables. Our overall approach can be seen as a gradient descent to discover the facets of a positive invariant polyhedron Ax ≤ b. Starting from an initial guess, each step checks using the LP relaxation, whether the current iterate A j x ≤ b j satisfies the positive invariance conditions.
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