Implicit Euler Time-Discretization of a Class of Lagrangian Systems with Set-Valued Robust Controller

A class of Lagrangian continuous dynamical systems with set-valued controller and subjected to a perturbation force has recently been thoroughly studied by the authors [Well-posedness, robustness and stability analysis of a set-valued controller for Lagrangian systems, SIAM J. Control Optim. 51(2) (2013) 1592--1614]. In this paper, we study the time discretization of these set-valued systems with an implicit Euler scheme. Under some mild conditions, the well-posedness (existence and uniqueness of solutions) of the discrete-time scheme, as well as the convergence of the sequences of discrete positions and velocities in finite steps are assured. Furthermore, the approximate piecewise linear function generated by these discrete sequences is shown to converge to the solution of the continuous time differential inclusion with order 1/2. Some numerical simulations on a two-degree of freedom example illustrate the theoretical developments.