网络蠕虫治理的状态反馈控制动力学研究

Around the world, computer worms have caused immense economic losses and have become the most dangerous threat to network security. Computer worm uses a computer network to spread itself, similar to the spread of pathogen among human populations. Therefore, computer worm propagation can be modeled as a continuous dynamical system. Anti-virus software, patching the operating system, and releasing controllable worms are currently being used as prevention/control measures for computer worms. Among these measures, except anti-virus software, patching the system and releasing controllable worms do not function continuously. Once the number of infected nod reaches a threshold, computer worm can be detected and these two measures will be implemented instantly. This process can be well defined as a state feedback impulsive dynamical system initiated by applying an impulsive turbulence on a continuous dynamical system. To optimize the control strategy, the basic theory of state feedback impulsive dynamical system has to be elucidated, such as equilibrium, periodic solution, limit cycle, and its stability etc. The results and conclusion obtained with this project will not only greatly improve the present understanding of impulsive dynamical system, but also have practical applications by providing solutions for multiple problems in biomathematics.

近年来，蠕虫病毒对世界范围内的计算机和互联网造成了巨大的影响，逐渐成为网络安全最大的潜在危机，威胁着人们的个人隐私和财产安全。蠕虫病毒在网络上的扩散与传染病在人群间的传播类似，可以用连续的微分动力系统描述，并据此研究其传播规律。目前防治蠕虫病毒的主要措施有杀毒软件、为系统打补丁和发布可控蠕虫等三种。杀毒软件的作用亦可考虑成一个连续变化的过程；但为系统打补丁、发布可控蠕虫这两种防治措施不是连续的行为，而是当网络中感染蠕虫病毒的节点达到一定数量被发现后，施加的瞬时（脉冲）防治策略，即在一个连续的微分动力系统中，以变量状态作为反馈条件施加脉冲扰动，形成一个状态反馈脉冲动力系统。为了得到蠕虫病毒防控的最优方案，必须先理清状态反馈脉冲动力系统的基础理论，如平衡态、周期解、极限环及其稳定性等。由此获得的结论将大大推动脉冲动力系统的研究，提出蠕虫防治的优化方案，也为解决其他生物问题提供必要的理论支撑。

随着人类对网络依赖程度的快速增长和互联网技术的日新月异，作为保护个人隐私和财产安全的必然要求，防治网络蠕虫病毒已成为一项长期而艰巨的工作。建立描述蠕虫病毒传播和防治的模型，完善状态反馈脉冲动力系统的基础理论，研究蠕虫治理的有效模式，是一项兼具理论意义和实践意义的项目。. 研究发现蠕虫的网络传播亦存在饱和现象，特别是传播能力较强的蠕虫病毒，能够在很短时间内达到饱和传染率。本项目建立了饱和传染状态下的蠕虫病毒传播的动力仿真系统。依据不同的蠕虫防治手段，建立了饱和传染率下蠕虫病毒传播及防治的状态脉冲反馈控制模型，得出模型具有阶一周期解的条件，研究了阶一周期解的稳定性。这一研究结果进一步完善了状态反馈脉冲控制的数学理论，同时也对网络蠕虫防治提供了理论支撑。此外项目后期对双边状态反馈脉冲控制问题进行了预探索，讨论了双边控制问题的背景、研究意义及可行性，初步确定了双边控制所需研究的科学问题和技术路线，为进一步开展后续研究工作奠定了一定的基础。