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

Filippov systems;Moments solution;Co-dimension 2;Co-dimension 3;Minimum variation solutions;Nonsmooth differential systems;Discontinuous differential equations;Regularization

Difonzo, Fabio Vito ; Dieci, Luca Mathematics de la Llave, Rafael Haddad, Wassim M. Kang, Sung Ha Zhou, Hao Min ; Dieci, Luca

University:Georgia Institute of Technology

Department:Mathematics

关键词: Filippov systems;    Moments solution;    Co-dimension 2;    Co-dimension 3;    Minimum variation solutions;    Nonsmooth differential systems;    Discontinuous differential equations;    Regularization;   

Others  :  https://smartech.gatech.edu/bitstream/1853/54388/1/DIFONZO-DISSERTATION-2015.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

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.