树状网络上一维非线性双曲守恒律系统的能控性

Nonlinear hyperbolic systems of conservation laws have strong practical background, they are widely applied on dynamics, biology, medicine, industry, etc.. The associated research on control problems for this kind of systems is a forward direction in modern mathematics, and is attracting more and more attentions. However, due to the difficulty in theoretical analysis on hyperbolic systems of conservation laws in the framework of entropy solutions, as well as the complexity and variety in reality, it is a big challenge to study control problems for such systems in the context of entropy solutions. Moreover, it is more difficult and more complicated to consider these problems on networks. This proposal is mainly concerned with two models arising from reality, and we are mainly interested in controllability for related hyperbolic systems of conservation laws on networks. These results would provide theoretical foundation and guidance for solving related reality problems.

非线性双曲守恒律系统具有很强的实际应用背景，被广泛应用于动力学、生物学、医学以及工业生产等各个领域。研究由这类系统支配的流体动力学模型的控制问题，是近代数学中一个备受关注的前沿研究方向。但由于对双曲守恒律系统的熵解在理论分析上有很大的难度，且实际问题往往具有很大的复杂性和多样性，研究这类系统的控制问题在数学上是一个重大的挑战，而在网络上研究这一类问题则具有更大的难度。本课题主要研究非线性双曲守恒律系统在两个具有很强实际背景的网络模型中的应用，研究其相关的能控性问题，为解决相应的实际问题提供理论依据及指导。

众所周知，非线性双曲守恒律系统是偏微分方程的一个重要分支，研究相应系统在网络上的控制问题是近代数学的一个热点。本项目研究了车辆交通网络的能控性和管道网络中一维等熵可压气体Euler方程的能控性。主要成果分为以下两个方面：一、对车辆交通网络模型在熵解的框架下建立了能控性，即构造一个满足初始状态的熵解，使得在某一时刻网络中所有道路上的车辆流密度达到目标状态；二、对管道网络中一维等熵可压气体Euler方程，研究了其在连接点处的相互作用。其次，我们提出了一类二阶拟线性双曲系统，通过构造性方法，得到了其在一般非线性边界条件下的能观性。对这两个具有很强应用背景的控制问题的研究，为解决相应的实际问题提供了必要的理论依据。