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。三个主要议题将被调查：（一）反馈形式的纳什均衡解，非合作微分对策。这些将通过研究价值函数的Hamilton-Jacobi方程系统，并使用同伦方法将微分博弈的解与相应的最优控制问题的解联系起来。 (ii)通过主动约束对机械系统的控制。 特别是，它建议研究的问题的全局可控性，最优控制，和渐近稳定的平衡位置。 (iii)用微分包含描述野火蔓延的一个新的数学模型。这里假设，可以通过在真实的时间内，沿着平面中的可求长曲线构造障碍物来阻止火灾传播。 在这方面的存在性和最优策略，最大限度地减少烧伤区域的总面积的属性，将调查。微分对策的研究将特别集中在价格库存游戏，提供了一个更好的理解应该是什么？理性的战略？在竞争激烈的经济形势下，不同的生产者和消费者群体试图操纵市场，以达到自己的利益。机械系统的研究主要是由流体中的游泳运动的应用所激发的。 将研究游泳者通过定期改变体形来实现运动的机动性和最佳划水方式。最后，火灾传播模型的分析将提供如何最佳地管理消防资源在真实的时间，在存在的森林火灾推进沿着一个大的前线。

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.