Université de Genève
Stabilization of the hidden dynamics in discontinuous differential equations
This talk considers ordinary differential equations with discontinuous right-hand side, where the discontinuity of the vector field takes place on smooth surfaces of the phase space. The main emphasis is put on what happens in the intersection of two of these surfaces. To avoid the
ambiguity of the Filippov approach, bilinear convex combinations of the adjacent vector fields are considered.
The hidden dynamics is given by a two-dimensional dynamical system (in the case of codimension-2 sliding modes) and describes the smooth transition of solution directions, which occurs instantaneously in the jump discontinuity of vector fields. This talk studies solutions of the hidden dynamics and, in particular, the stability of stationary points which correspond to codimension-2 sliding modes.
Close to the intersection of discontinuity surfaces, the hidden dynamics is realized by standard space regularizations. If a stationary point of the hidden dynamics is unstable, this results in high oscillations of frequency proportional to the inverse of the regularization parameter. A simple modification will be presented that permits to avoid this instability and makes the numerical treatment much more efficient.
This is joint work with Nicola Guglielmi from the University of L’Aquila.