Name
Abstract | ||
|---|---|---|
We introduce a class of scattering passive linear systems motivated by examples from mathematical physics. The state space of the system is X = H circle plus 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 = [0 -L L* G-1/2 K* K], where L is an element of L(E-0, H) and K is an element of L(E-0, U) are such that [L K], with domain E-0, is closed as an unbounded operator from E to H circle plus U. The operator G is an element of L(E-0, E-0') is such that Re < G(w0), w(0)> <= 0 for all w(0) is an element of 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 = [z w] 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) over dot = Ax + Bu satisfy d/d t parallel to x(t)parallel to(2) = parallel to u(t)parallel to(2) - parallel to y(t)parallel to(2) + 2Re < Gw, w >. Moreover, the dual system satisfies a similar power balance equation. Hence, this system is scattering conservative if and only if Re < Gw(0), w(0)> = 0 for all w(0) is an element of E-0. We give two examples involving the beam equation and one with Maxwell's equations. |
Year | DOI | Venue |
|---|---|---|
2012 | 10.1137/110846403 | SIAM JOURNAL ON CONTROL AND OPTIMIZATION |
Keywords | Field | DocType |
scattering passive system,scattering conservative system,system node,Cayley transform,beam equation,Maxwell's equations | Hilbert space,Mathematical optimization,Unbounded operator,Matrix (mathematics),Mathematical analysis,Cayley transform,Operator (computer programming),Continuous embedding,Semigroup,State space,Mathematics | Journal |
Volume | Issue | ISSN |
50 | 5 | 0363-0129 |
Citations | PageRank | References |
10 | 1.03 | 6 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Olof J. Staffans | 1 | 66 | 15.08 |
| George Weiss | 2 | 316 | 64.49 |