Problems of Nonlinear Control

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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.