Problems of Nonlinear Control

P.I.将研究非线性控制和微分对策领域的问题。微分对策的研究将主要集中在反馈形式的Stackelberg解和无限时间范围内的Nash均衡解。这里的主要目的是推广线性二次型情形下的理论，研究更一般的具有鲁棒性的非线性模型。第二个主要研究领域是集值演化的控制。将考虑这样的模型，其中集合的增长可以受到分布式控制的影响，或者可以通过实时构建屏障来限制。特别是，P.I.和一名合作者将研究在哪些情况下集合可以始终一致有界，以及哪些是最优的限制策略。差分对策研究的主要动机来自经济学。在将要考虑的典型模式中，主要参与者--比如政府或中央银行--提前宣布其政策，而从属参与者--比如私营公司--为了实现自身利润最大化，选择自己的战略作为最佳回应。如果要实施的政策参考了未来将观察到的特定参数，如通货膨胀率或失业率，为主要参与者确定最佳选择将导致具有挑战性的数学问题。这些都将在本研究范围内进行调查。作为进一步的方向，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传播的模型推动的。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.