Carleman Estimates for Systems of Partial Differential Equations with Application to the Control of Coupled Systems

这个项目的目的是发展偏微分方程系统的Carleman估计。 Carleman估计是加权能量估计，是控制理论和逆问题中的重要工具。 特别是，Carleman估计是必不可少的不适定柯西问题的研究。 这类问题在标量偏微分方程中很容易理解。 然而，偏微分方程组由于具有多重特征而更难处理。 本课题主要研究数学物理学中的经典偏微分方程系统的估计，包括麦克斯韦方程和可变非光滑系数的弹性系统。为了避免物理系统的不必要行为，如大的振动或裂纹的形成，需要准确预测系统的演变。 这涉及非破坏性测量，例如在系统的边界上。 该项目关注的是更好地理解为了准确预测物理系统的行为需要多少信息。 在工程和材料科学的许多领域的进步已经产生了涉及偏微分方程系统的现实数学模型，这是本数学研究的主题。

The aim of this project is to develop Carleman estimates for systems of partial differential equations. Carleman estimates are weighted energy estimates that are crucial tools in control theory and inverse problems. In particular, Carleman estimates are indispensable in the study of ill-posed Cauchy problems. Such problems are well understood for scalar partial differential equations. However, systems of partial differential equations are much harder to treat because of multiple characteristics. This project develops estimates for classical systems of partial differential equations arising in mathematical physics, including Maxwell's equations and the system of elasticity with variable non-smooth coefficients.In order to avoid unwanted behavior of a physical system, such as large vibrations or formation of cracks, an accurate prediction of the evolution of the system is desirable. This involves nondestructive measurements, for example on the boundary of the system. This project is concerned with a better understanding of how much information is needed in order to predict the behavior of a physical system accurately. Advances in many areas of engineering and material science have produced realistic mathematical models that involve systems of partial differential equations, which are the subject of this mathematical investigation.