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Nabla time scale symplectic systems and related quadratic functionals

  • Roman HilscherEmail author
  • Vera Zeidan
Original Research

Abstract

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.

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 

Mathematics Subject Classification (2000)

34C10 39A12 49K15 15A63 

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Copyright information

© Foundation for Scientific Research and Technological Innovation 2010

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Faculty of ScienceMasaryk UniversityBrnoCzech Republic
  2. 2.Department of MathematicsMichigan State UniversityEast LansingUSA

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