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- W1969449464 abstract "Let the operator 5 describe a linear causal system. Then, roughly, the system is dissipative if it satisfies the scattering condition I − SS∗ ⩾ 0 [relation (4.2)] or the immittance condition S + S∗ ⩾ 0 [relation (4.1)]. In either case, the resulting kernel is a real, tempered, and causal matrix distribution. More specifically, s belongs to D′L2 ∩ D′+ ⊗ L in one case and w ϵ D′L∞ ∩ D′+ ⊗ L in the other. The passivity of the system is reflected in the respective positivity statements that 1δ − s ∗ ≈sT (the kernel of 1 − SS∗) or w + w̃T (the kernel of S + S∗) are positive in the sense of Bochner. The Laplace transform of the kernel exists and dissipativity shows that either the scattering matrix S(p) is bounded-real or that the immittance matrix W(p) is positive-real. Conversely, every bounded-real or positive-real matrix is the scatterings or immittance matrix of a dissipative system. In the frequency domain, these conditions translate into the assertion that either 1 − S(ω) S(ω)T is a.e. nonnegative definite (equivalently, ∥S(ω)∥ ⩽ 1 a.e.), or that Wω + W̄ωT defines a positive matrix valued measure. In the lossless case, we must have S(ω) be a.e. unitary. We saw that S(p), W(p) ϵ H+ and have S′ boundary values, component by component. The real and imaginary parts of these boundary values satisfy precise dispersion relations." @default.
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- W1969449464 date "1967-08-01" @default.
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- W1969449464 title "Linear dissipative systems, nonnegative definite distributional kernels, and the boundary values of bounded-real and positive-real matrices" @default.
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- W1969449464 doi "https://doi.org/10.1016/0022-247x(67)90118-7" @default.
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