The Filippov moments solution on the intersection of two and three manifolds

Difonzo, Fabio Vito

Title:

The Filippov moments solution on the intersection of two and three manifolds

dc.contributor.advisor	Dieci, Luca
dc.contributor.author	Difonzo, Fabio Vito
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.contributor.department	Mathematics
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.date.updated	2016-01-07T17:25:21Z
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.description.degree	Ph.D.
dc.format.mimetype	application/pdf
dc.identifier.uri	http://hdl.handle.net/1853/54388
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	Text
dc.type.genre	Dissertation
dspace.entity.type	Publication
local.contributor.advisor	Dieci, Luca
local.contributor.corporatename	College of Sciences
local.contributor.corporatename	School of Mathematics
relation.isAdvisorOfPublication	eaae8ecc-0596-42e8-9e81-2b1037bbe6de
relation.isOrgUnitOfPublication	85042be6-2d68-4e07-b384-e1f908fae48a
relation.isOrgUnitOfPublication	84e5d930-8c17-4e24-96cc-63f5ab63da69
thesis.degree.level	Doctoral