New problems in nonlinear control

这项研究将在控制和微分对策领域寻求一些新的方向。主要研究三个问题：(I)非合作微分对策的反馈形式的纳什均衡解。这些将通过查看值函数的哈密顿-雅可比方程组，并使用同伦方法将微分对策的解与相应的最优控制问题的解联系起来来研究。(Ii)以主动约束的方式控制机械系统。特别提出要研究系统的全局可控性、最优控制以及渐近镇定到平衡点的问题。(Iii)用微分包含的形式描述野火蔓延的新数学模型。这里假设可以通过沿平面上的可校正曲线实时地建造障碍物来阻止火灾传播。在这方面，我们将研究最小化烧毁面积的最优策略的存在性和性质。微分对策的研究将专门集中在价格-库存博弈上，以提供更好的理解应该是什么？理性策略？处于竞争的经济形势中，不同的生产者和消费者群体试图为了自己的利益而操纵市场。机械系统的研究主要受到流体中类似游泳运动的应用的推动。通过定期改变身体形状来实现运动的游泳者的机动性和最佳泳姿将被研究。最后，对火灾传播模型的分析将为在森林大火沿大前线推进的情况下如何实时优化管理灭火资源提供指导。

This research will pursue some new directions in the area of control and differential games. Three main topics will be investigated: (i) Nash equilibrium solutions in feedback form, for non-cooperative differential games. These will be studied by looking at the system of Hamilton-Jacobi equations for the value functions, and using a homotopy approach to connect solutions of the differential game to the solution of a corresponding optimal control problem. (ii) The control of mechanical systems by means of active constraints. In particular, it is proposed to study the issues of global controllability, optimal control, and the asymptotic stabilization to an equilibrium position. (iii) A new mathematical model describing the spreading of a wild fire in terms of a differential inclusion. It is here assumed that fire propagation can be stopped by constructing barriers, in real time, along rectifiable curves in the plane. In this connection the existence and the properties of optimal strategies, which minimize the total area of the burned region, will be investigated.The research on differential games will specifically focus on price-inventory games, providing a better understanding of what should the ?rational strategies? be in a competitive economic situation, where different groups of producers and consumers try to manipulate the market to their own advantage. The study of mechanical systems is primarily motivated by applications to swim-like motion in fluids. Maneuverability and optimal strokes for a swimmer that achieves locomotion by periodically changing its body shape will be investigated. Finally, the analysis of fire propagation models will provide indications on how to optimally manage fire-fighting resources in real time, in the presence of a forest fire advancing along a large front.