Abstract A new sliding mode control (SMC) design approach using measure theory and Lyapunov functional candidate is presented for nonlinear control problems. A Lyapunov function is supposed for designing a sliding surface (SS). In fact the problem that is considered is as follows. A state trajectory from a given initial point reaches into a given point on a sliding surface in the minimum time, and then tends to the origin (equilibrium point) along the sliding surface. A measure theory approach with embedding process is used to solve such a problem in two phases. In the first phase, after designing an appropriate SS by a suggested Lyapunove function, and using measure theoty, an embedding is constructed to solve a time optimal control problem such that the system trajectory reaches a SS in minimum time, then in the second phase, using SS, a control is designed such that the system trajectory tends to the origin along the SS. A numerical example is presented to illustrate the effectiveness of the proposed method.

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