Exponential Stability of Maxwell’s Equations

Abstract

The control of electromagnetic systems, which is governed by Maxwell's equations, is of interest for various reasons such as controlling plasma in a nuclear fusion reactor or magnetically trapping antiparticles for observation. Proving that a controlled system is exponentially stable is of particular interest, as exponential stability infers that the system will converge asymptotically to a steady state solution. A tool called the multiplier method is considered, which allows for exponential stability to be demonstrated by defining an auxiliary functional and proving it is bounded by the system energy and the energy's time derivative in a particular way. If an appropriate bound is shown, exponential stability is not only guaranteed, but the exponential function which bounds the L2 norm of the system variables will be fully determined in terms of the systems parameters. Currently, there is ongoing work into generalizing the multiplier method approach for a class of problems known as Port-Hamiltonian systems. This thesis aims to contribute to this work by formulating Maxwell's equations as a Port-Hamiltonian system, and using this formulation as a basis for determining how to choose the auxiliary functional needed in the multiplier method.