Löwner partial order construction for nonzero-sum games of nonlinear system and event-triggered-based actor-critic neural networks research

This paper pioneers the application of algebraic matrix partial order, Löwner partial order, to the multi-player nonzero-sum games of nonlinear systems (NMPNZS) with unmatched uncertainty, with extended investigations incorporating event-triggered mechanism (ETM) control method integrated with actor-critic neural networks (NNs). To handle unmatched uncertainties and construct appropriate auxiliary system are critically important issues. However, the construction of the auxiliary system by exploiting the seemingly unrelated Löwner partial order constitutes a particularly innovative and technically challenging methodology proposed in this study, distinguishing it from existing approaches. Moreover, through rigorous theoretical analysis, it is proved that under the framework of Löwner partial order-constructed auxiliary system with redefined value function, robust control issue can be transformed into optimal control issue. Then based on the designed ETM condition, in contrast to traditional time-triggered mechanism (TTM) control it reduces both computational burden and communication bandwidth. Furthermore, a kind of composite actor-critic NNs architecture is synergistically embedded within the control-theoretic framework to operationalize Nash equilibrium attainment, with optimization of control policy. By constructing suitable Lyapunov functions, it is demonstrated that both the weight approximation errors and the considered NMPNZS system are uniformly ultimately bounded (UUB). Meanwhile, the Zeno phenomenon is excluded by proving the inter-event time is bounded. Finally, a nonlinear numerical example and the canonical reinforcement learning testing environment ‘Pendulum-v1’ are employed to validate the operational feasibility and effectiveness of the developed control strategy.