Filippov systems with nondifferentiable and unbounded delays : periodicity, stabilization, and energy consumption estimation

In this article, a general Filippov system with nondifferentiable and unbounded delays is studied, which is different from the previous delayed Filippov systems. First, based on the differential inclusion theory and Kakutani's fixed point theorem, the periodicity of solutions is proved and sufficient criteria are derived. Second, fixed-time (FxT) stabilization is studied by designing a new control law with a unified steepness exponent, which is more simpler than the previous ones with two steepness exponents. Since the delays are nondifferentiable and unbounded, rather than constructing the Lyapunov-Krasovskii functions, by constructing a suitable Lyapunov function, the FxT stabilization is analyzed and estimation of the settling-time (ST) is given through analyzing the state variables inside and outside the unit spherical area. The energy consumption is also estimated when the FxT stabilization is achieved under the designed controller. Third, given that smaller ST and lower energy consumption are usually preferred, the optimization problem is further considered. The optimal control parameters are selected based on the normalization approach and maximum principle. Finally, the validity of the theoretical results is demonstrated by a numerical example, where the delay function is composed of the Takagi function and the absolute value function.