TY - JOUR

T1 - Maxwell's equations as a scattering passive linear system

AU - Weiss, George

AU - Staffans, Olof J.

PY - 2013

Y1 - 2013

N2 - We consider Maxwell's equations on a bounded domain Ω ⊂ 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 (Formula Presentsd.) , where B and D are the magnetic and electric flux densities, and the state space of the system is X = E+E, where E = L2(Ω;3). We assume that Γ0 and Γ1 are disjoint, relatively open subsets of Γ such that (Formula Presented.). We consider Γ0 to be a superconductor, which means that on Γ0 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 L2 on Γ1. 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 (Formula Presented.), where L = rot (with a suitable domain), Γ is the tangential component trace operator restricted to Γ1, R is a strictly positive pointwise multiplication operator on U (that can be chosen arbitrarily), and (Formula Presented.) 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.

AB - We consider Maxwell's equations on a bounded domain Ω ⊂ 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 (Formula Presentsd.) , where B and D are the magnetic and electric flux densities, and the state space of the system is X = E+E, where E = L2(Ω;3). We assume that Γ0 and Γ1 are disjoint, relatively open subsets of Γ such that (Formula Presented.). We consider Γ0 to be a superconductor, which means that on Γ0 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 L2 on Γ1. 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 (Formula Presented.), where L = rot (with a suitable domain), Γ is the tangential component trace operator restricted to Γ1, R is a strictly positive pointwise multiplication operator on U (that can be chosen arbitrarily), and (Formula Presented.) 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.

KW - Boundary control

KW - Boundary observation

KW - Maxwell's equations

KW - Scattering conservative system

KW - Scattering passive system

UR - http://www.scopus.com/inward/record.url?scp=84890481006&partnerID=8YFLogxK

U2 - 10.1137/120869444

DO - 10.1137/120869444

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:84890481006

SN - 0363-0129

VL - 51

SP - 3722

EP - 3756

JO - SIAM Journal on Control and Optimization

JF - SIAM Journal on Control and Optimization

IS - 5

ER -