Minimal and optimal linear discrete time-invariant dissipative scattering systems. (English) Zbl 0898.47005
Summary: For an operator-valued function \(\theta\) in the Schur class a new geometric proof, using state space considerations only, of the construction of a minimal and optimal realization is given. A minimal and optimal realization also appears as a restricted shift realization where the state space is the completion of the range of the associated Hankel operator in the de Branges-Rovnyak norm associated with \(\theta\). It is also shown that minimal and optimal, and minimal and star-optimal realizations of a rational matrix function in the Schur class are intimately connected to the extremal positive solutions of the associated Kalman-Yakubovich-Popov operator inequality.
MSC:
| 47A40 | Scattering theory of linear operators |
| 47A57 | Linear operator methods in interpolation, moment and extension problems |
| 47A48 | Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc. |
| 47A56 | Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) |
| 93A99 | General systems theory |
| 47A63 | Linear operator inequalities |
Keywords:
Schur class; state space; minimal and optimal realization; Hankel operator; de Branges-Rovnyak norm; star-optimal realizations of a rational matrix function; Kalman-Yakubovich-Popov operator inequalityReferences:
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