基于矩阵博弈的潜艇防御模型

Submarine Defense Model Based on Matrix Game

摘要: 潜艇作为现代海战中不可或缺的作战平台，承担着重要的战略和战术任务。其隐蔽性和高机动性使其成为海上作战中的强大力量。然而，随着鱼雷技术的不断进步，防御鱼雷攻击成为潜艇在水下作战中的一项重大挑战。本文研究了潜艇在水下作战中防御鱼雷攻击的最优策略问题，将潜艇与鱼雷的对抗转化为追逃博弈模型，假设敌方鱼雷攻击潜艇时，潜艇通过改变航向来规避攻击。通过应用博弈论中的零和博弈理论和矩阵博弈模型，本文分析了潜艇和鱼雷在有限策略集合下的最优对策。研究表明，基于潜艇和鱼雷的运动学特性及其有限策略，构建的收益矩阵能够为双方提供最优策略。最后通过具体算例与数值仿真验证了模型的合理性与可行性。

Abstract: Submarines constitute an indispensable asset in modern naval warfare, executing pivotal strategic and tactical missions. Their inherent stealth and superior maneuverability empower them to operate covertly in contested maritime environments. However, continuous advancements in torpedo technology have rendered the development of effective countermeasures against torpedo attacks a critical challenge in underwater combat. This paper formulates the submarine torpedo counter-measure problem as an optimal control problem within the framework of pursuit-evasion game theory. Specifically, the engagement is modeled as a zero-sum differential game where the submarine employs evasive maneuvers, principally through heading adjustments to mitigate the threat posed by an incoming torpedo. By employing a matrix game model defined over a finite discrete strategy set, we derive the Nash equilibrium solutions. The constructed payoff matrix, based on the kinematic constraints and maneuverability limitations of the submarine and torpedo, facilitates the determination of optimal strategies for both adversaries. Numerical simulations and case studies further validate the analytical robustness and practical feasibility of the proposed model in realistic underwater combat scenarios.