Dynamical systems with random influences mixing logic and physics: a framework enabling control engineers to design for resilient autonomy

这个项目的目标是开发必要的数学工具来描述和分析混合了逻辑和物理，同时包括随机影响的动态系统。这些类型的系统有时被称为随机混合系统；其中随机指的是随机影响的存在，而混合指的是基于逻辑的决策和决定系统演化的物理规律的混合。这些数学工具将非常有用，因为它们将提供一个框架，使控制工程师能够更好地设计在面对不确定性时具有弹性的自主动力系统。这一建议的结果有可能影响广泛应用领域的研究人员，在这些领域，随机性与最坏情况的影响相互作用，需要严格的分析工具来证明工程系统中的期望行为。一个可能的应用领域是控制基因表达和其他特殊的生物现象。另一个应用领域是多代理资源分配算法，其中代理在其分配策略中调用随机性来保证对网络中的恶意代理的健壮性。事实上，随机混合系统可以用来模拟复杂的生物系统、金融系统、资源分配系统和交通管理系统，仅举几例。通过这项提案产生的所有成果都将在主要会议场所和期刊上发表，为将工作过渡到应用提供途径。除了对科学文献的贡献外，拟议的工作还将增加现有的随机和混合动力系统的研究生课程。研究生将通过直接财政支持进行培训，但更广泛的学生群体将通过教材和围绕取得的突破而开发的课程来接受教育。这些学生包括所有工程领域的研究生。为了广泛地接触到学生，PI设想了一本关于从这项工作中产生的随机混合系统的研究生教科书，以补充最近关于非随机混合系统的书。这项工作将促进与国际研究人员的合作，并将加强与几所国际大学的学生交换项目。这些研究进展将应用于对广大人口重要的领域。此外，PI将利用这个机会考虑如何让一年级的工程学学生接触到广泛的动力系统原理。这项工作将延续之前在一次新生研讨会上教授工程学和非工程学学生如何将最优化原理应用于现实世界问题的经验。该项目的具体技术目标是建立一个数学框架和解决方案的概念，该框架和解决方案的概念既易于掌握又足够普遍，可以广泛应用。变分分析领域的工具对这一发展至关重要。目标是继续开发广泛的分析工具，这些工具可用于证明工程随机混合系统的适当行为。这些技术将特别集中于对稳定性属性的Lyapunov分析，如概率和递归的渐近稳定性；也将考虑其他对非随机混合系统卓有成效的想法。其次，我们的目的是建立所考虑模型解的强序列紧性结果，并由这些结果建立不变原理和逆李雅普诺夫定理。这些结果将与最近关于非随机混合系统的非常有用的结果相类似。最后的任务是开始明确展示开发的框架和分析工具如何用于设计先进的、有弹性的、自主的控制系统。

The goal of this project is to develop the mathematical tools necessary to describe and analyze dynamical systems that mix logic and physics while including random influences. These types of systems are sometimes called stochastic hybrid systems; where stochastic refers to the presence of random influences and hybrid refers to the mix of logic-based decision making and physical laws that determine the system evolution. These mathematical tools would be extremely useful, as they would provide a framework enabling control engineers to better design autonomous dynamical systems that are resilient in the face of uncertainty. The results of this proposal have the potential to impact researchers in a wide variety of application domains, where randomness interacts with worst-case effects and rigorous analysis tools are needed to certify desirable behavior in an engineered system. One possible application domain is the control of gene expression and other particular biological phenomena. Another application area is multi-agent resource allocation algorithms where agents invoke randomness in their allocation strategies to guarantee robustness to malicious agents in the network. Indeed, stochastic hybrid systems can be used to model complicated biological systems, financial systems, resource allocation systems, and traffic management systems, to name just a few. All of the results developed through this proposal will be published in the leading conference venues and journals, to provide avenues for the transition of the work to applications. Beyond contributions to the scientific literature, the proposed work will add to the existing graduate curriculum on stochastic and hybrid dynamical systems. Graduate students will be trained through direct financial support, but a broader group of students will be educated through teaching materials and a course that will be developed around the produced breakthroughs. These students include graduate students in all areas of engineering. To reach students broadly, the PI envisions a graduate textbook on stochastic hybrid systems emerging from this work, to complement the recent book on non-stochastic hybrid systems. The work will spawn collaboration with international researchers, and will enhance student exchange programs with several international universities. The research developments will be applied to areas that are important to the broad population. Moreover, the PI will use this opportunity to consider how to expose freshman engineering students to a broad range of dynamical systems principles. This effort will follow up the previous experience teaching a freshman seminar, to engineering and non-engineering students, on the application of optimization principles to real-world problems. The aim here aligns with the societal goal of keeping younger students engaged in topics related to science, technology, engineering, and mathematics.The specific technical objectives of the project start with establishing a mathematical framework and solution concept that is tractable yet general enough to be widely applicable. Tools from the field of variational analysis are crucial to this development. The objectives continue with the task of developing a wide range of analysis tools that can be used to certify appropriate behavior of an engineered stochastic hybrid system. The techniques will especially focus on a Lyapunov analysis for stability properties like asymptotic stability in probability and recurrence; other ideas that have been fruitful for non-stochastic hybrid systems also will be considered. Next, the aim is to establish strong sequential compactness results for the set of solutions to the models considered and, from these results, establish an invariance principle and converse Lyapunov theorems. These results would parallel recent, very useful results for non-stochastic hybrid systems. The final task is to begin to show explicitly how the developed framework and analysis tools can be used to engineer advanced, resilient, autonomous control systems.