Description

The investigation traverses the domain of discontinuous dynamical systems, elucidating the concept of Finite-Time Input-to-State Stability (FTISS) in the context of Networked Euler-Lagrange Systems. Utilizing the framework of extended Filippov’s solutions and non-smooth Lyapunov functions, the analytical methodology is robustly developed. Employing non-smooth and set-valued analysis substantiates the existence of a finite-time input-to-state Lyapunov function, thereby establishing the FTISS of the systems under consideration.

Progressing further, the scope encompasses Multi-Agent Systems (MASs) incorporating Euler-Lagrange dynamics. Through rigorous analysis, a seminal theorem is articulated, elucidating the finite-time input-to-state stability landscape of these MASs. The presence of discontinuities in the dynamical systems necessitates a nuanced approach, transcending conventional smooth mechanics to address scenarios of velocity jumps and force discontinuities.

The practical implications of the findings are manifold. The revealed principles find applicability in the domain of robotic manipulators interacting with their environment, offering a pathway to controlled stability in the face of inherent discontinuities. These findings have significant bearing on control problems characterized by time-dependent or discontinuous feedbacks.

The discourse extends to sliding mode and variable structure control methodologies, demonstrating the potential for stabilization in systems dictated by discontinuous feedbacks and control inputs. The results delineate a pathway into optimal and adaptive control arenas, where discontinuous switching algorithms find utilization in sketching optimal trajectories, enhancing system robustness, and ensuring boundedness of estimated variables.

Illustratively, a novel algorithm is proposed for finite-time leader-following consensus of nonlinear second-order MASs. This algorithm’s relevance extends to decentralized sliding mode controllers, promising finite-time consensus achievement in nonlinear second-order MASs, a noteworthy advancement over existing paradigms.

In summation, the investigation not only contributes a vital theorem to the academic corpus but also initiates a discourse on the stabilization and regulation of discontinuous dynamical systems. The practical ramifications are extensive, with potential to significantly influence the fields of robotics and control systems. The elucidation of Finite-Time Input-to-State Stability of Discontinuous Dynamical Systems is a pivotal stride towards comprehending the complex dynamics inherent in such systems, thereby inching closer to mastering the regulation of discontinuities within a networked framework. This endeavor, thus, transcends mere theoretical exploration, laying a substantive foundation for future scholarly and practical explorations in the realm of discontinuous dynamical systems.