Caputo Barrier Functions and Their Applications to the Safety, Safety-and-Stability, and Input-to-State Safety of a Class of Fractional-Order Systems

Safety control based on barrier functions has gradually become one of the emerging and more important directions in the field of safety. Scholars are attempting to apply barrier functions to integer-order dynamical systems, such as general nonlinear systems, hybrid systems, linear systems, etc. Moreover, the introduction of barrier functions has even expanded the research approaches on safe reinforcement learning. However, there is very little research on the safety control problem of fractional-order dynamical systems. Based on our previous work, this article further explores, in depth, the problem of the transfer and adaptability of barrier functions for integer-order systems in fractional-order systems, and it also proposes the Caputo reciprocal barrier function and Caputo zeroing barrier function. And we established two theorems, which proved that we can also achieve uniform asymptotic stability or exponential stability with guaranteed safety. In the end, we created a new description for the definition of input-to-state safety under Caputo’s fractional-order systems, and we used this description and the above two Caputo barrier functions to construct two criteria of the Caputo input-to-state safety. Thus, we, finally, established the embryonic form of the theoretical framework of safety control based on barrier functions for fractional-order systems.