Approximate Reachability Computation for Polynomial Systems

Abstract

Abstract. In this paper we propose an algorithm for approximating the reachable sets of systems defined by polynomial differential equations. Such systems can be used to model a variety of physical phenomena. We first derive an integration scheme that approximates the state reachable in one time step by applying some polynomial map to the current state. In order to use this scheme to compute all the states reachable by the system starting from some initial set, we then consider the problem of computing the image of a set by a multivariate polynomial. We propose a method to do so using the Bézier control net of the polynomial map and the blossoming technique to compute this control net. We also prove that our overall method is of order 2. In addition, we have successfully applied our reachability algorithm to two models of a biological system.

Hybridization methods for the analysis of nonlinear systems

“…Another promising direction is to use mixed rectangular-simplicial meshes in order to achieve a good trade-off between accuracy and computational cost. In addition, the convergence can be improved by using higher degree approximants, such as piecewise quadratic, and the reachability method for polynomial systems [17] can then be used. Finally, an important theoretical question to address is whether other new properties can be verified using the hybridization approach.…”

“…The methods of the first category try to approximate reachable sets as accurately as possible by tracking their evolution under the continuous flows using some set represention (such as polyhedra, ellipsoids, level sets). This results in a variety of approximation schemes (such as [6,13,15,17,19,27,28,36,39,44,57]), and implemented by a number of tools such as Coho [28], CheckMate [15], d/dt [7], VeriShift [13], HYSDEL [58], MPT [45], HJB toolbox [44]. Since accurate reachable set approximations are computationally expensive, the methods of the second category seek approximations that are sufficiently good to prove the property of interest (such as barrier certificates [46], polynomial invariants [56]).…”

“…This results in a piecewise bilinear system with uncertain inputs, which is then handled by a reachability analysis method using the Maximum principle. Alternatively, one can apply our recent method for polynomial systems [17] which exploits the geometric properties of polynomial maps without resorting to variable projection.…”

Image Computation for Polynomial Dynamical Systems Using the Bernstein Expansion

“…Many existing reachability computation methods can be seen as an extension of numerical integration. That is, one has to solve the above equation (1) This problem was previously considered in the work [24], which was inspired by modeling techniques from Computer Aided Geometric Design (CADG) and tried to exploit special geometric properties of polynomials. The drawback of the Bézier simplex based method proposed in this work is that it requires expensive mesh computation, which restricts its application to systems of dimensions not higher than 3, 4.…”

“…The drawback of the Bézier simplex based method proposed in this work is that it requires expensive mesh computation, which restricts its application to systems of dimensions not higher than 3, 4. In this paper, we pursue the direction which was initiated in [24] and make use of a special class of polyhedra. These polyhedra can be thought of as local meshes of fixed form.…”