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This paper presents a comprehensive method to analyze the sliding motion on a nonsmooth surface. The dynamic behavior of the trajectory on the surface is characterized by the solution of Filippovs differential inclusion. Based on the new criterions on the relation between system trajectories and a nonsmooth surface in the state space, we discuss the sliding motion along the surface, which is processed with the help of the cross product to calculate the direction of the nonsmooth region from one side to another. This result allows us to construct sliding mode controller for more general cases, of which the switching surface can be designed Lipschitz continuous for the nonsmooth sliding motion. Finally, we provide a numerical example to illustrate water drops motion on the nonsmooth surface. It shows that the water drops motion on the surface can be precisely estimated based on the new criteria in good accordance with the physical phenomena.

Gossiping is a well-studied distributed algorithm whose purpose is to enable the members of a group of autonomous agents to asymptotically determine in a decentralized manner, the average of their initial scalar-valued gossip variables. T-periodic gossiping is a gossiping protocol which stipulates that each agent must gossip with each of its neighbors exactly once every Ttime unit. Under suitable connectivity assumptions of a graph characterizing all allowable gossip pairs, a T-periodic gossip sequence will converge at a rate determined by the magnitude of the second largest eigenvalue of the stochastic matrix determined by the sequence of gossips which occurs over a period. It has been shown in the prior work that if the underlying graph of allowable gossips is a tree, this eigenvalue is the same for all possible T-periodic gossip sequences. The aim of this paper is to develop several properties for stochastic matrices induced by the sequence of gossips occurring over a Tperiod and reprove the result using these properties in a different and simpler argument.