A Physically Motivated Class of Scattering Passive Linear Systems
نویسندگان
چکیده
We introduce a class of scattering passive linear systems motivated by examples from mathematical physics. The state space of the system is X = H ⊕ E, where H and E are Hilbert spaces. We also have a Hilbert space E0 which is dense in E, with continuous embedding, and E ′ 0 is the dual of E0 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 = [ 0 −L L∗ G− 1 2 K∗K ] , where L ∈ L(E0, H) and K ∈ L(E0, U) are such that [ L K ] , with domain E0, is closed as an unbounded operator from E to H ⊕ U . The operator G ∈ L(E0, E′ 0) is such that Re 〈Gw0 , w0〉 ≤ 0 for all w0 ∈ E0. 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 = [ z w ] is the state trajectory and y is the corresponding output function. We show that this system is scattering passive (hence, wellposed), and that classical solutions of the system equation ẋ = Ax+Bu satisfy d dt‖x(t)‖ 2 = ‖u(t)‖2 − ‖y(t)‖2 + 2Re 〈Gw,w〉. Moreover, the dual system satisfies a similar power balance equation. Hence, this system is scattering conservative iff Re 〈Gw0 , w0〉 = 0 for all w0 ∈ E0. We give two examples involving the beam equation, and one with Maxwell’s equations.
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عنوان ژورنال:
- SIAM J. Control and Optimization
دوره 50 شماره
صفحات -
تاریخ انتشار 2012