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Positivity of Quadratic Functionals on Time Scales: Necessity
Author(s) -
Hilscher Roman
Publication year - 2001
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200106)226:1<85::aid-mana85>3.0.co;2-h
Subject(s) - mathematics , controllability , quadratic equation , normality , hamiltonian (control theory) , discrete time and continuous time , interval (graph theory) , mathematical optimization , combinatorics , statistics , geometry
In this work we establish that disconjugacy of a linear Hamiltonian system on time scales is a necessary condition for the positivity of the corresponding quadratic functional. We employ a certain minimal normality (controllability) assumption. Hence, the open problems stated by the author in [17], [18] are solved with the result that positivity of the quadratic functional is equivalent to disconjugacy of the Hamiltonian system on the interval under consideration. The general approach on time scales involves, as special cases, the well–known continuous case for = ℝ and recently developed discrete one for = ℤ so that they are unified. As applications, Sturmian type separation and comparison theorems on time scales are also provided.