Maxwell's Equations as a Scattering Passive Linear System

George Weiss, Olof J. Staffans
2013 SIAM Journal of Control and Optimization  
We consider Maxwell's equations on a bounded domain Ω ⊂ R 3 with Lipschitz boundary Γ, with boundary control and boundary observation. Relying on an abstract framework developed by us in an earlier paper, we define a scattering passive linear system that corresponds to Maxwell's equations and investigate its properties. The state of the system is B D , where B and D are the magnetic and electric flux densities, and the state space of the system is X = E⊕E, where E = L 2 (Ω; R 3 ). We assume
more » ... Γ 0 and Γ 1 are disjoint, relatively open subsets of Γ such that Γ 0 ∪Γ 1 = Γ. We consider Γ 0 to be a superconductor, which means that on Γ 0 the tangential component of the electric field is forced to be zero. The input and output space U consists of tangential vector fields of class L 2 on Γ 1 . The input and output at any moment are suitable linear combinations of the tangential components of the electric and magnetic fields. The semigroup generator has the structure A = 0 −L L * G−γ * Rγ P , where L = rot (with a suitable domain), γ is the tangential component trace operator restricted to Γ 1 , R is a strictly positive pointwise multiplication operator on U (that can be chosen arbitrarily), and P −1 = μ 0 0 ε is another strictly positive pointwise multiplication operator (acting on X). The operator −G is pointwise multiplication with the conductivity g ≥ 0 of the material in Ω. The system is scattering conservative iff g = 0.
doi:10.1137/120869444 fatcat:vyzyr7uxn5b7ji6ogjoe3gnwbu