标题： 在线性二次博弈中连接有限与无限时域纳什均衡 显示英文标题

标题： Bridging Finite and Infinite-Horizon Nash Equilibria in Linear Quadratic Games

摘要： 有限时间线性二次（LQ）博弈存在唯一的纳什均衡，而无限时间情形可能有多个。 我们通过将有限时间均衡解释为非线性动力系统来澄清这两种情况之间的关系。 在此框架下，我们证明其平衡点正好是无限时间均衡，并且通过适当选择终端成本可以恢复任何这样的均衡。 我们进一步表明，当动力系统出现周期轨道时，它们对应于周期性纳什均衡，并提供了收敛到这些循环的数值证据。 最后，仿真结果揭示了三种渐近状态：收敛到静态均衡、收敛到周期性均衡以及有界不收敛轨迹。 这些发现为使用无限时间方法调整有限时间LQ博弈提供了新的见解和工具。 显示英文摘要

摘要： Finite-horizon linear quadratic (LQ) games admit a unique Nash equilibrium, while infinite-horizon settings may have multiple. We clarify the relationship between these two cases by interpreting the finite-horizon equilibrium as a nonlinear dynamical system. Within this framework, we prove that its fixed points are exactly the infinite-horizon equilibria and that any such equilibrium can be recovered by an appropriate choice of terminal costs. We further show that periodic orbits of the dynamical system, when they arise, correspond to periodic Nash equilibria, and we provide numerical evidence of convergence to such cycles. Finally, simulations reveal three asymptotic regimes: convergence to stationary equilibria, convergence to periodic equilibria, and bounded non-convergent trajectories. These findings offer new insights and tools for tuning finite-horizon LQ games using infinite-horizon.