Publication details

Nabla time scale symplectic systems and related quadratic functionals

Authors

ŠIMON HILSCHER Roman ZEIDAN Vera Michel

Year of publication 2010
Type Article in Periodical
Magazine / Source Differential Equations and Dynamical Systems
MU Faculty or unit

Faculty of Science

Citation
Field General mathematics
Keywords Time scale; Time scale symplectic system; Nabla derivative; Nabla dynamic equation; Quadratic functional; Controllability; Normality; Conjoined basis; Riccati equation; Reid roundabout theorem; Linear Hamiltonian system; Discrete symplectic system
Description In this paper we present the theory of nabla time scale symplectic systems. In particular, we establish conditions characterizing the positivity and nonnegativity of the quadratic functionals associated with such systems. These results are parallel (or dual) to the ones obtained recently by the authors for the delta time scale symplectic systems without normality assumption. A surprising outcome of this theory is the fact that some of the known results for the delta time scale case and the presented new results for the nabla time scale case do not coincide in the special cases of both continuous linear Hamiltonian systems and discrete symplectic systems. To the contrary, the nabla time scale results are also new in the latter two special cases. As applications of the obtained positivity and nonnegativity results we derive the Reid roundabout theorems for nabla time scale symplectic systems.
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