Criteria for the existence of impulse responses and kernel representations for linear maps. (English) Zbl 0724.47008
Summary: Results are given that establish, for the first time, necessary and sufficient conditions for the existence of impulse responses and kernel representations for linear not-necessarily-time-invariant systems described by input-output operator equations.
These results concern systems whose inputs and outputs are real-valued functions on the real line \({\mathbb{R}}\), the half-line \([0,\infty)\) or \({\mathbb{R}}^ m\). They deal with causal as well as non-causal maps and considerably extend related previous results which concern causal maps defined on functions on the half-line.
These results concern systems whose inputs and outputs are real-valued functions on the real line \({\mathbb{R}}\), the half-line \([0,\infty)\) or \({\mathbb{R}}^ m\). They deal with causal as well as non-causal maps and considerably extend related previous results which concern causal maps defined on functions on the half-line.
MSC:
47A50 | Equations and inequalities involving linear operators, with vector unknowns |
47B38 | Linear operators on function spaces (general) |
93C05 | Linear systems in control theory |
Keywords:
necessary and sufficient conditions for the existence of impulse responses and kernel representations for linear not-necessarily-time- invariant systems described by input-output operator equations; time-invariantReferences:
[1] | Sandberg, IEEE Trans. Circuits and Systems CAS-35 pp 201– (1988) |
[2] | Real and Complex Analysis, 3rd Edn, McGraw-Hill, New York, 1981. |
[3] | and I. W. Sandberg, ’g- and h-representations for nonlinear maps’, J. Math. Anal. Appl. in the press. · Zbl 0732.93036 |
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