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dc.contributor.advisorDieci, Luca
dc.contributor.authorDifonzo, Fabio Vito
dc.date.accessioned2016-01-07T17:25:21Z
dc.date.available2016-01-07T17:25:21Z
dc.date.created2015-12
dc.date.issued2015-11-18
dc.date.submittedDecember 2015
dc.identifier.urihttp://hdl.handle.net/1853/54388
dc.description.abstractIn 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.mimetypeapplication/pdf
dc.language.isoen_US
dc.publisherGeorgia Institute of Technology
dc.subjectFilippov systems
dc.subjectMoments solution
dc.subjectCo-dimension 2
dc.subjectCo-dimension 3
dc.subjectMinimum variation solutions
dc.subjectNonsmooth differential systems
dc.subjectDiscontinuous differential equations
dc.subjectRegularization
dc.titleThe Filippov moments solution on the intersection of two and three manifolds
dc.typeDissertation
dc.description.degreePh.D.
dc.contributor.departmentMathematics
thesis.degree.levelDoctoral
dc.contributor.committeeMemberde la Llave, Rafael
dc.contributor.committeeMemberHaddad, Wassim M.
dc.contributor.committeeMemberKang, Sung Ha
dc.contributor.committeeMemberZhou, Hao Min
dc.date.updated2016-01-07T17:25:21Z


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