dc.contributor.advisor Dieci, Luca dc.contributor.author Difonzo, Fabio Vito dc.date.accessioned 2016-01-07T17:25:21Z dc.date.available 2016-01-07T17:25:21Z dc.date.created 2015-12 dc.date.issued 2015-11-18 dc.date.submitted December 2015 dc.identifier.uri http://hdl.handle.net/1853/54388 dc.description.abstract In this thesis, we study the Filippov moments solution for differential equations with discontinuous right-hand side. In particular, our aim is to define a suitable Filippov sliding vector field on a co-dimension $2$ manifold $\Sigma$, intersection of two co-dimension $1$ manifolds with linearly independent normals, and then to study the dynamics provided by this selection. More specifically, we devote Chapter 1 to motivate our interest in this subject, presenting several problems from control theory, non-smooth dynamics, vehicle motion and neural networks. We then introduce the co-dimension $1$ case and basic notations, from which we set up, in the most general context, our specific problem. In Chapter 2 we propose and compare several approaches in selecting a Filippov sliding vector field for the particular case of $\Sigma$ nodally attractive: amongst these proposals, in Chapter 3 we focus on what we called \emph{moments solution}, that is the main and novel mathematical object presented and studied in this thesis. There, we extend the validity of the moments solution to $\Sigma$ attractive under general sliding conditions, proving interesting results about the smoothness of the Filippov sliding vector field on $\Sigma$, tangential exit at first-order exit points, uniqueness at potential exit points among all other admissible solutions. In Chapter 4 we propose a completely new and different perspective from which one can look at the problem: we study minimum variation solutions for Filippov sliding vector fields in $\R^{3}$, taking advantage of the relatively easy form of the Euler-Lagrange equation provided by the analysis, and of the orbital equivalence that we have in the eventuality $\Sigma$ does not have any equilibrium points on it; we further remove this assumption and extend our results. In Chapter 5 examples and numerical implementations are given, with which we corroborate our theoretical results and show that selecting a Filippov sliding vector field on $\Sigma$ without the required properties of smoothness and exit at first-order exit points ends up dynamics that make no sense, developing undesirable singularities. Finally, Chapter 6 presents an extension of the moments method to co-dimension $3$ and higher: this is the first result which provides a unique admissible solution for this problem. dc.format.mimetype application/pdf dc.language.iso en_US dc.publisher Georgia Institute of Technology dc.subject Filippov systems dc.subject Moments solution dc.subject Co-dimension 2 dc.subject Co-dimension 3 dc.subject Minimum variation solutions dc.subject Nonsmooth differential systems dc.subject Discontinuous differential equations dc.subject Regularization dc.title The Filippov moments solution on the intersection of two and three manifolds dc.type Dissertation dc.description.degree Ph.D. dc.contributor.department Mathematics thesis.degree.level Doctoral dc.contributor.committeeMember de la Llave, Rafael dc.contributor.committeeMember Haddad, Wassim M. dc.contributor.committeeMember Kang, Sung Ha dc.contributor.committeeMember Zhou, Hao Min dc.date.updated 2016-01-07T17:25:21Z
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