International Journal of Difference Equations

  • Year: 2006
  • Volume: 1
  • Issue: 1

Implicit Riccati Equations and Quadratic Functionals for Discrete Symplectic Systems*

  • Author:
  • Roman Hilscher1,†, Viera Růžičková2
  • Total Page Count: 20
  • DOI:
  • Page Number: 135 to 154

1Department of Mathematical Analysis, Faculty of Science, Masaryk University, Janáčkovo nám. 2a, CZ-60200 Brno, Czech Republic. E-mail: hilscher@math.muni.cz

2Department of Mathematical Analysis, Faculty of Science, Masaryk University, Janáčkovo náam. 2a, CZ-60200 Brno, Czech Republic. E-mail: xruzicko@math.muni.cz

†Corresponding author. Research supported by the Ministry of Education, Youth, and Sports of the Czech Republic under grant 1K04001 and by the Grant Agency of the Academy of Sciences of the Czech Republic under grant KJB1019407.

* Research supported by the Czech Grant Agency under grant 201/04/0580.

Abstract

In this paper we study discrete (implicit) Riccati matrix equations associated with discrete symplectic systems and related quadratic functionals with variable endpoints. We derive these Riccati equations for nonnegative functionals with separable and jointly varying endpoints. The result for jointly varying endpoints is in terms of the nonaugmented Riccati operator. The method also allows to simplify implicit Riccati equations known for the positivity of . Finally, we establish a comparison result (Riccati inequality) for solutions of Riccati equations associated with two discrete symplectic systems.

Keywords

Discrete symplectic system, quadratic functional, non-negativity, positivity, Riccati inequality, Riccati equation, conjoined basis, Sturmian theorem