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.