New results for time reversed symplectic dynamic systems and quadratic functionals

In this paper, we examine time scale symplectic (or Hamiltonian) systems and the associated quadratic functionals which contain a forward shift in the time vari- able. Such systems and functionals have a close connection to Jacobi systems for calculus of variations and optimal control problems on time scales. Our results, among which we consider the Reid roundabout theorem, generalize the corresponding classical theory for time reversed discrete symplectic systems, as well as they complete the recently developed theory of time scale symplectic systems.

[1]  V. Zeidan,et al.  Nabla time scale symplectic systems and related quadratic functionals , 2010 .

[2]  Vera Zeidan,et al.  Applications of time scale symplectic systems without normality , 2008 .

[3]  Roman Simon Hilscher,et al.  Picone type identities and definiteness of quadratic functionals on time scales , 2009, Appl. Math. Comput..

[4]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[5]  Editors , 1986, Brain Research Bulletin.

[6]  R. Hilscher,et al.  Perturbation of time scale quadratic functionals with variableendpoints , 2007 .

[7]  A. Peterson,et al.  Dynamic Equations on Time Scales: An Introduction with Applications , 2001 .

[8]  V. Zeidan,et al.  Symplectic Structure of Jacobi Systems on Time Scales , 2010 .

[9]  A. Peterson,et al.  Dynamic Equations on Time Scales , 2001 .

[10]  Douglas R. Anderson,et al.  Higher-order self-adjoint boundary-value problems on time scales , 2006 .

[11]  Vera Zeidan,et al.  Weak maximum principle and accessory problem for control problems on time scales , 2009 .

[12]  V. Zeidan,et al.  Reid Roundabout Theorems for Time Scale Symplectic Systems , 2010 .

[13]  A. Peterson,et al.  Advances in Dynamic Equations on Time Scales , 2012 .

[14]  Martin Bohner,et al.  Disconjugacy and Transformations for Symplectic Systems , 1997 .

[15]  P. Hartman.  Ordinary Differential Equations , 1965 .

[16]  V. Zeidan,et al.  Time scale symplectic systems without normality , 2006 .

[17]  V. Zeidan,et al.  Riccati equations for abnormal time scale quadratic functionals , 2008 .

[18]  C. Ahlbrandt,et al.  Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations , 1996 .