Fixed-Time Synchronization of Different Dimensional Filippov Systems

This article aims to study the fixed-time (FxT) synchronization of different dimensional Filippov systems. New FxT stability lemmas containing the classical inequality <inline-formula> <tex-math notation="LaTeX">$\dot {V}\leq {-c}_{1}V^{a}-c_{2}V^{b}$ </tex-math></inline-formula> proposed by Polyakov are established. Different from the previous FxT stability lemmas in the literature, the proposed one shows the new conclusion that the settling times can be larger or smaller as long as <inline-formula> <tex-math notation="LaTeX">$c_{1}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$c_{2}$ </tex-math></inline-formula> satisfy the certain relationships, which synchronously reveals that the relationships between the control gains can lead to different settling times. Besides, a generalized economical inequality condition <inline-formula> <tex-math notation="LaTeX">$\dot {V} \leq -c_{1}V-c_{2}V^{\upsilon +{\mathrm{ sign}}(V-r)}$ </tex-math></inline-formula> is proposed and a novel FxT stability lemma is also established. The complete theoretical proof is given to reveal that <inline-formula> <tex-math notation="LaTeX">$r=1$ </tex-math></inline-formula> leads to desired settling time, some previous related results are improved. Based on the new FxT stability lemmas and differential inclusion theory, algebraic inequality conditions are provided to guarantee the FxT synchronization, which reports the first result on the FxT synchronization of different dimensional Filippov systems. Finally, numerical examples are provided to verify the correctness of the main theoretical results.