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Over and Under Approximations of Reachable Sets Within Hamilton-Jacobi Framework

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时间:2017-09-01  来源:数学机械化重点实验室

题目:           Over and Under Approximations of Reachable Sets Within Hamilton-Jacobi Framework

报告人:     佘志坤(北京航空航天大学)

时间地点:   2017.09.06  15:30pm  N210

摘要:           For dynamical systems, reachable sets can be described by solutions of Hamilton-Jacobi equations. In this paper, we discuss a methodology to compute approximations, defined by zero sub-level sets of polynomials, of time-bounded reachable sets (i.e., flowpipes) with arbitrary bounded errors for polynomial dynamical systems via solving derived Hamilton-Jacobi equations with inequality constraints. We start with evolution functions for describing the flowpipes of systems, and find their explicit Taylor expansions with respect to time. Then, we prove the existence of polynomial approximations to evolution functions with arbitrary bounded errors by investigating solutions of corresponding partial differential equations with derived inequality constraints, which shows the applicability of this methodology to obtain both over and under approximations of reachable sets with arbitrary precisions in Hausdorff metric. Afterwards, we propose two methods to compute polynomial template based evolution functions with constraints via using sum-of-squares decomposition and quantifier elimination, respectively. We test these two methods on some examples with comparisons to the advection operator based method. The computation and comparison results show that the QE based method to certain extent has better performance than the SOS based method and the advection operator based method.

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