Stochastically Adaptive Control and Synchronization: From Globally One-Sided Lipschitzian to Only Locally Lipschitzian Systems

Abstract

The mathematical framework of stochastically adaptive feedback control, which is generally applicable to significant problems in nonlinear dynamics such as stabilization and synchronization, has been previously established but only for systems whose vector fields satisfy the global Lipschitzian condition. Nonlinear dynamical systems arising from physical, chemical, or biological situations are typically described by vector fields that are only locally Lipschitzian. To generalize the mathematical theory of stochastically adaptive control to realistic systems is quite a challenging and formidable task. We meet this challenge by proving rigorously that stabilization and synchronization can be achieved with probability one for only locally Lipschitzian systems. The result holds not only for one-dimensional but also for any finite-dimensional white noises. Representative examples and an application to synchronization-based parameter identification are presented to illustrate the broad applicability of the developed mathematical criteria. Our successful relaxation of the mathematical condition from globally to locally Lipschitzian provides a rigorous guarantee of the stability of stochastically adaptive control in physical systems with significant implications to the design and realization of engineering control systems.