Finite determination of accessibility and singular points of nonlinear systems: An algebraic approach

摘要

Exploiting tools from algebraic geometry, the problem of determination of accessibility/strong accessibility is investigated for polynomial systems and also for analytic systems that are immersible into polynomial systems. The results are constructive, and algorithms are given to find the maximum depth of Lie brackets necessary for deciding accessibility/strong accessibility of the system at any point, called here accessibility/strong accessibility index of the system, and known as the degree of non-holonomy in the literature. Alternatively, upper bounds on the accessibility/strong accessibility index are obtained, which can be computed easier. In each approach, the entire set of accessibility/strong accessibility singular points are obtained, as a limiting algebraic set of a strictly increasing chain of ideals, that stabilizes in finite time. Several examples demonstrate the applicability of the results using computer algebra tools. (C) 2019 Elsevier B.V. All rights reserved.