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

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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.