Finite-time stability of jump diffusion system and its application in drill-bit tracking control

This paper investigates the finite-time stability of a jump diffusion system with Brownian motion and random jump components. First, we establish a novel Lyapunov-type finite-time stability theorem, which provides a direct way for selecting an appropriate Lyapunov function. It is important to note that our finite-time stability result is significantly distinct from the case involving only Brownian motion. The presence of the jump term disrupts the continuity of the system’s solution paths, thereby introducing additional complexities in the analysis of finite-time stability. Subsequently, we employ this theorem to design a finite-time controller to ensure the finite-time stochastic stability of the tracking error in a drill-bit system. The proposed control strategy guarantees that the tracking error converges to the origin within finite time and remains there thereafter with probability one. Finally, simulation results are presented to validate the effectiveness of the proposed control law in achieving precise drill-bit tracking control.