Symplectic difference systems: Variable stepsize discretization and discrete quadratic functionals
type: Item
instance of: scholarly article
title: Symplectic difference systems: Variable stepsize discretization and discrete quadratic functionals
zbMATH Open document ID: 1021.39008
DOI: 10.1016/S0024-3795(02)00590-6
author: Roman Šimon Hilscher
author: Vera Zeidan
published in: Linear Algebra and its Applications
publication date: 2003-05-27T00:00:00Z
review text: The authors consider the positivity and the nonnegativity of some discrete quadratic functionals with variable endpoints for variable stepsize symplectic difference systems. Seventeen theorems are given for the above-mentioned properties of such functionals in terms of conjugate intervals, conjoined bases, and implicit and explicit Riccati equations with various forms of boundary conditions. Moreover, they show that a variable stepsize discretization of a continuous time nonlinear control problem of the form \[ \begin{cases} \text{minimize} \quad & G(x,u):=K\bigl(x(a), x(b)\bigr)+ \int^b_a g\bigl(t,x(t), u(t)\bigr) dt,\\ & x'(t)=f\bigl(t,x(t), u(t)\bigr),\;\psi\bigl( t,u(t) \bigr)=0;\;t\in[a,b],\\ & \varphi\bigl(x(a), x(b)\bigr)= 0\end{cases} \] leads to a discrete linear quadratic problem and a Hamiltonian difference system, which are special cases of their symplectic counterparts. A number of interesting remarks, lemmas and corollaries are also indicated herein.
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 93C55
Mathematics Subject Classification ID: 49J15
Mathematics Subject Classification ID: 65K10
zbMATH DE Number: 1917805
zbMATH Keywords: positive functionals
zbMATH Keywords: discrete quadratic functional
zbMATH Keywords: variable stepsize
zbMATH Keywords: linear Hamiltonian difference system
zbMATH Keywords: conjugate interval
zbMATH Keywords: conjoined basis
zbMATH Keywords: Riccati difference
zbMATH Keywords: equation
zbMATH Keywords: variable state endpoints
zbMATH Keywords: symplectic difference systems
MaRDI profile type: Publication
cites work: Hamiltonian systems on time scales
cites work: Q4338990
cites work: Q4716258
cites work: Linear Hamiltonian difference systems: Disconjugacy and Jacobi-type conditions
cites work: Q4802272
cites work: Disconjugacy and transformations for symplectic systems
cites work: Q4940908
cites work: Q2723195
cites work: Disconjugacy, transformations and quadratic functionals for symplectic dynamic systems on time scales
cites work: Disconjugacy for linear Hamiltonian difference systems
cites work: Q3980599
cites work: Q3197394
cites work: Q4303968
cites work: Analysis on measure chains - a unified approach to continuous and discrete calculus
cites work: Disconjugacy of symplectic systems and positive definiteness of block tridiagonal matrices
cites work: Reid roundabout theorem for symplectic dynamic systems on time scales
cites work: Second Order Sufficiency Criteria for a Discrete Optimal Control Problem
cites work: Discrete Optimal Control: Second Order Optimality Conditions
cites work: Coupled intervals in the discrete calculus of variations: necessity and sufficiency
cites work: A remark on discrete quadratic functionals with separable endpoints
cites work: Q4664452
cites work: Q4837108
cites work: Discrete Oscillation
cites work: Q4401802
cites work: An approach to nonlinear programming
cites work: Sufficient conditions for variational problems with variable endpoints: Coupled points
cites work: Coupled Points in the Calculus of Variations and Applications to Periodic Problems
cites work: Jacobi condition for elliptic forms in Hilbert spaces
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description: scientific article; zbMATH DE number 1917805