Publication details

Definiteness of quadratic functionals for Hamiltonian and symplectic systems: A survey



Year of publication 2009
Type Article in Periodical
Magazine / Source International Journal of Difference Equations
MU Faculty or unit

Faculty of Science

Field General mathematics
Keywords Linear Hamiltonian system; Discrete symplectic system; Time scale; Time scale symplectic system; Quadratic functional; Conjoined basis; Focal point; Nonnegativity; Positivity
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Description In this paper we provide a survey of characterizations of the nonnegativity and positivity of quadratic functionals arising in the theory of linear Hamiltonian and symplectic systems. We study these functionals on traditional continuous time domain (under and without controllability), on discrete domain, and on time scale domain which unifies and extends both previous types. For each case we distinguish functionals with zero, separated, and jointly varying endpoints. The presented conditions are formulated in terms of the properties of a special conjoined basis of the considered linear system. It is now easy to compare all the results - between continuous, discrete, and time scale cases, between the zero, separated, and jointly varying endpoits, and between the nonnegativity and positivity.
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