SUMMARYNew algorithms based upon the asymptotic numerical method (ANM) are proposed to solve unilateral contact problems. ANM leads to a representation of a solution path in terms of series or Padé approximants. To get a smooth solution path, a hyperbolic relation between contact forces and clearance is introduced. Three key points are discussed: the influence of the regularization of the contact law, the discretization of the contact force by Lagrange multipliers and prediction-correction algorithms. Simple benchmarks are considered to evaluate the relevance of the proposed algorithms.

This paper describes a variable structure control for fractional-order systems with delay in both the input and state variables. The proposed method includes a fractional-order state predictor to eliminate the input delay. The resulting state-delay system is controlled through a sliding mode approach where the controller uses a sliding surface defined by fractional order integral. Then, the proposed control law ensures that the state trajectories reach the sliding surface in finite time. Based on recent results of Lyapunov stability theory for fractional-order systems, the stability of the closed loop is studied. Finally, an illustrative example is given to show the interest of the proposed approach.

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