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Controllability and extensions

Authors
Journal
Journal of the Franklin Institute
0016-0032
Publisher
Elsevier
Publication Date
Volume
309
Issue
5
Identifiers
DOI: 10.1016/0016-0032(80)90024-1
Disciplines
  • Mathematics

Abstract

Abstract Control problems in Hilbert spaces are treated in a measure-theoretical framework; instead of dealing with a set of admissible trajectory-control pairs, a set of measures defined by the boundary conditions and the differential equations of the problem are considered. The concept of weak controllability is introduced; a system has this property if every pair of initial and final points, ( t a, x a) and ( t b, x b) can be weakly joined; this is possible if a set of linear equalities involving measures has a solution. In turn, this is shown to be equivalent to the possibility of extending a linear functional in a positive manner. Necessary and sufficient conditions for controllability are derived, and applied to the study of a finite-dimensional system with the control appearing linearly.

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