Abstract
Let A be a linear bounded operator in H and let E be a finite-dimensional space, equipped with a scalar product. We will consider a system defined by the following conditions:
where ψ : E → H, φ : H → E are linear mappings of E into H and of H into E respectively. The input u(t) and the corresponding output v(t) (t0 ≤ t ≤ t) are vector-functions with values belonging to the space E, and the corresponding inner state f (t) is an H-valued vector-function. We assume that the following“energy“ balance law holds:
where σ = σ* is a linear selfadjoint operator in E. The scalar products (σu, u) and (σv,v) can be interpreted as the energy flows through the input and through the output correspondingly.
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© 1995 Springer Science+Business Media Dordrecht
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Livšic, M.S., Kravitsky, N., Markus, A.S., Vinnikov, V. (1995). Open Systems and Open Fields. In: Theory of Commuting Nonselfadjoint Operators. Mathematics and Its Applications, vol 332. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8561-3_3
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DOI: https://doi.org/10.1007/978-94-015-8561-3_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4585-0
Online ISBN: 978-94-015-8561-3
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