## 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 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 = [ ^{0}L*G-1/2 ^{-L}K*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.

Original language | English |
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Pages (from-to) | 3083-3112 |

Number of pages | 30 |

Journal | SIAM Journal on Control and Optimization |

Volume | 50 |

Issue number | 5 |

DOIs | |

State | Published - 2012 |

## Keywords

- Beam equation
- Cayley transform
- Maxwell's equations
- Scattering conservative system
- Scattering passive system
- System node