AI 中文总结
本文针对带线性代价、正系统动力学和输入约束的连续时间一般和非合作博弈，提出刻画有限与无限时间范围反馈纳什均衡的方法，给出迭代求解策略，并通过大规模污染博弈验证该方法。
AI 中文摘要
本文研究一类连续时间一般和非合作博弈，该博弈具有线性代价、正系统线性动力学以及逐元素线性输入约束。在有限时间范围情形下，我们提出一个验证定理，用于刻画反馈纳什均衡，该均衡由耦合向量值常微分方程组的绝对连续解表示，通过时变反馈律实现。与基于里卡蒂方程、其均衡随状态维度呈二次增长的线性二次微分博弈不同，本文提出的公式随状态维度呈线性增长。不过，由此产生的分段常数反馈在其约束边界之间饱和，而非平滑变化，且在刻画微分方程的解时会出现额外的数学挑战，这些解通常因反馈增益的切换特性而不连续。本工作研究切换仅发生在孤立时间 instants 的情形。在无限时间范围情形下，在可镇定假设下，均衡由耦合向量值代数方程刻画。针对该博弈，我们提出迭代方法以计算有限和无限时间范围的均衡。通过一个大规模污染博弈示例说明该方法。
英文摘要
This paper studies a continuous-time general-sum non-cooperative game with linear costs, positive linear system dynamics, and elementwise linear input constraints. In the finite-horizon case, we present a verification theorem characterizing feedback Nash equilibria, in terms of absolutely continuous solutions of a coupled system of vector-valued ordinary differential equations, realized by time-varying feedback laws. Unlike linear-quadratic differential games, whose Riccati-based equilibria scale quadratically with the state dimension, this formulation scales linearly. However, the resulting piecewise-constant feedback saturates between its constraint bounds rather than varying smoothly, and additional mathematical challenges arise when characterizing the solutions of the differential equations, which are generally discontinuous due to the switching nature of the feedback gains. In this work, we study the case where switching occurs only at isolated time instants. In the infinite-horizon case, under stabilizability assumptions, the equilibrium is characterized by coupled vector-valued algebraic equations. For this game, we propose iterative methods to compute both finite and infinite-horizon equilibria. The approach is illustrated through a large-scale pollution game.
Comments6 pages and 3 figures. Accepted for presentation at the 65th IEEE Conference in decision and control 2026 (CDC)
