State in Hilbert Space

Resolution space, recently introduced for the study of causality in an operator theoretic setting, is employed to formulate an abstract state concept which generalizes the state space theory commonly used in the study of finite-dimensional dynamical systems. After formulating the state concept as an...

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Bibliographic Details
Published in:SIAM review 1973-04, Vol.15 (2), p.283-308
Main Author: Saeks, R.
Format: Article
Language:English
Subjects:
Causality
Differential operators
Factorization
Hilbert space
Hilbert spaces
Input output
Linear transformations
Mathematical set theory
Optimal control
Systems theory
Trajectories
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Summary:Resolution space, recently introduced for the study of causality in an operator theoretic setting, is employed to formulate an abstract state concept which generalizes the state space theory commonly used in the study of finite-dimensional dynamical systems. After formulating the state concept as an operator factorization and verifying the existence of a family of transition operators on the resultant state space, the integral representation for operators on a resolution space is employed to develop "convolution-like" formulas for the state of an operator resulting from a specified excitation and an operator factorization theorem. The theory is illustrated via a study of stability and the state optimal control problem in the resolution space context.
ISSN:0036-1445
1095-7200
DOI:10.1137/1015030