Problems of Nonlinear Control

P.I.将研究非线性控制和微分对策领域的问题。微分对策研究的主要焦点将是反馈形式的Stackelberg解和无限时间范围内的纳什均衡解。这里的主要目标是扩展现有的理论在线性二次的情况下，研究更一般的非线性模型具有鲁棒性。第二个主要研究领域是集值进化的控制。当一个集合的增长受到分布式控制的影响，或者被实时构建的障碍所限制时，将考虑模型。特别是，P.I.和合作者将研究在哪些情况下集合可以在所有时间内呈现一致有界，以及哪些是最优约束策略。研究微分博弈的主要动机来自经济学。在我们将要讨论的一个典型模型中，主要参与者（比如政府或中央银行）提前宣布其政策，而次要参与者（比如私营公司）选择其策略作为最佳对策，以实现自身利润最大化。如果要实施的政策引用了未来将观察到的特定参数，例如通货膨胀率或失业率，那么确定主要参与者的最佳选择将导致具有挑战性的数学问题。这些将在本研究中进行调查。作为进一步的研究方向，动态阻塞问题的研究是由描述森林火灾或化学污染的空间蔓延的模型驱动的。在遏制努力中，资源的最佳分配提出了新的数学问题，本项目也将解决这些问题。

The P.I. will study problems in the area of nonlinear control and differential games. A major focus of the research on differential games will be on Stackelberg solutions in feedback form, and on Nash equilibrium solutions in infinite time horizon. Here the main goal is to extend the theory available in linear-quadratic case, studying more general nonlinear models with robustness properties. A second main area of research is the control of set-valued evolutions. Models will be considered where the growth of a set can be influenced by a distributed control, or restrained by constructing barriers in real time. In particular, the P.I. and a collaborator will study in which cases the set can be rendered uniformly bounded for all times, and which are the optimal confinement strategies. The primary motivation for the research on differential games comes from economics. In a typical model that will be considered, the leading player--say, a government, or a central bank--announces its policy in advance, while a subordinate player--say, a private company--chooses its strategy as a best reply, in order to maximize its own profit. If the policy to be implemented makes reference to specific parameters that will be observed in the future, such as inflation or unemployment rates, determining the optimal choice for the leading player leads to challenging mathematical problems. These will be investigated within the present research. As a further direction, the study of dynamic blocking problems is motivated by models describing the spatial spreading of a forest fire, or of a chemical contamination. The optimal allocation of resources, in the containment effort, poses novel mathematical questions which will also be addressed by the present project.