Convergence of regularised solutions of piecewise smooth differential equations

Daniel Paul Tietz (Martin-Luther-Universität Halle-Wittenberg), Martin Arnold

We study piecewise smooth differential equations in which the discontinuity of the vector field occurs on two smooth surfaces of the phase space and may result in codimension-\(2\) sliding.
First we will regularise the associated differential inclusion with a small regularisation parameter \(\epsilon\). Based on the ideas presented in [1] and [2], especially some asymptotic expansion techniques, we will then discuss the linear convergence of the regularised solutions in \(\epsilon\) for the most common cases.
Finally we will analyse some problems given from electrical engineering and validate the theoretical result.
[1] N. Guglielmi und E. Hairer. Classification of hidden dynamics in discontinuous dynamical systems. In: SIAM Journal on Applied Dynamical Systems, 14(3): 1454-1477, 2015.
[2] N. Guglielmi und E. Hairer. Solutions leaving a codimension-2 sliding. In: Nonlinear Dynamics, 88(2): 1427-1439, 2017.