A physically motivated class of scattering passive linear systems

We introduce a class of scattering passive linear systems motivated by examples from mathematical physics. The state space of the system is X = H φ E, where H and E are Hilbert spaces. We also have a Hilbert space E ₀ which is dense in E, with continuous embedding, and E' ₀ is the dual of E ₀ with respect to the pivot space E. The input space is the same as the output space, and it is denoted by U. The semigroup generator has the structure A = [ ⁰L*G-1/2 ^-LK*K] where L ε L(E ₀,H) and K ε L(E ₀,U) are such that [L/K], with domain E ₀, is closed as an unbounded operator from E to H φ U. The operator G ε L(E ₀, E' ₀) is such that Re(Gω ₀,ω ₀) ≤ 0 for all ω ₀ ε E ₀. The observation operator is C = [0 -K], the control operator is B = -C _*, and the output equation is y = Cx + u = -Kw + u, where u is the input function, x = [zω] is the state trajectory, and y is the corresponding output function. We show that this system is scattering passive (hence, well-posed) and that classical solutions of the system equation x = Ax + Bu satisfy d /dt ||x(t)|| ² = ||u(t)|| ² - ||y(t)|| ² + 2Re (Gω,ω). Moreover, the dual system satisfies a similar power balance equation. Hence, this system is scattering conservative if and only if Re(Gω ₀, ω ₀) = 0 for all ω ₀ ε E ₀. We give two examples involving the beam equation and one with Maxwell's equations.