Minimum variation solutions for sliding vector fields on the intersection of two surfaces in \(\mathbb{R}^3\)
Te authors consider model problems of piecewise smooth systems in \(\mathbb{R}^3\) and propose minimum variation approaches to find a Filippov sliding vector field on the intersection \(\Sigma\) of two discontinuity surfaces. Their idea is to look at the minimum variation solution in the \(H^1-\)norm, among either all admissible sets of coefficients for a Filippov vector field, or among all Filippov vector fields. They compare the resulting solutions to other possible Filippov sliding vector fields (including the bilinear and moments solutions). The authors further show that -- in the absence of equilibria -- also these other techniques select a minimum variation solution, for an appropriately weighted \(H^1-\)norm, and relate this weight to the change of the time variable giving orbital equivalence among the different vector fields. Finally, they give details of how to build a minimum variation solution for a general piecewise smooth system in \(\mathbb{R}^3\).
scientific article; zbMATH DE number 6479591
scientific article