New problems in nonlinear control

本研究将在控制和微分对策领域探索一些新的方向。将研究三个主要主题：(i)反馈形式的纳什均衡解，用于非合作微分博弈。这些将通过观察值函数的Hamilton-Jacobi方程系统来研究，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。（ii）利用主动约束对机械系统进行控制。重点研究了系统的全局可控性、最优控制和平衡点的渐近镇定问题。（三）一个新的数学模型，根据差异包涵来描述野火的蔓延。这里假设火焰的传播可以通过在平面上沿可校正曲线实时设置障碍物来阻止。在这方面的存在和最优策略的性质，使烧伤区域的总面积最小，将进行研究。对差异博弈的研究将特别关注价格-库存博弈，以便更好地理解应该做什么？理性的策略?在一个竞争激烈的经济环境中，不同的生产者和消费者群体试图操纵市场，使其对自己有利。机械系统的研究主要是由流体中类似游泳运动的应用所驱动的。通过周期性改变身体形状来实现运动的游泳者的机动性和最佳划水将被研究。最后，对火灾传播模型的分析将为如何在森林火灾沿大前线推进的情况下优化实时管理消防资源提供指示。

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.