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The discrete calculus of variations
Abstract
In this dissertation we seek necessary and sufficient conditions on a function y so that a discrete functional $J\lbrack y\rbrack$ is maximized or minimized. We use right focal boundary conditions for y, i.e., given a discrete interval $\lbrack a, b+2\rbrack$ we fix y(a) and $\Delta y(b+1)$. We first discuss the scalar case for a functional of the form$$J\lbrack y\rbrack=\sum\sbsp{t=a+1}{b+2}f(t,y(t),\Delta y(t-1)).$$We obtain analogs of the Euler-Lagrange and Legendre necessary conditions and establish a sufficient condition. Motivated by the discrete analog of the Jacobi equation, we prove the equivalence of the positive definiteness of the second variation and the C-disfocality of Mu(t) = 0 for an appropriately defined operator M. This leads to a second sufficient condition. We then prove analogs for two comparison theorems and for the Weierstrass Integral Formula. After discussing the special case when $J\lbrack y\rbrack$ is a quadratic form, we give two examples of functionals and find their proper global minima. We next establish corresponding results for the vector problem. In the vector case, one of the key theorems requires the use of the Legendre-Clebsch transformation rather than the Riccati equation. We also consider general boundary conditions for the scalar case, specifically we define the set of admissible functions to be$$\eqalign{{\cal G}{=}\{y:\lbrack a,b{+}2\rbrack\ \to\Re\vert\alpha y(a){+}\beta\Delta y (a) &= A\ \rm where\ \alpha\not=\beta,\ \alpha\sp2{+}\beta\sp2>0\cr \rm and\ \gamma y(b{+}1)+\delta\Delta y(b+1) &= B, where\ \delta\not=0,\ \gamma\sp2{+}\delta\sp2>0\}\cr.}$$Finally we consider the (2, 2) right focal boundary value problem, in which the functional we wish to extremize has the form$$J\lbrack y\rbrack=\sum\sbsp{t=a+2}{b+4}f(t,y(t), \Delta\sp2y(t-1))$$and the boundary conditions are$${\cal F}{=}\{y:\lbrack a,b+4\rbrack\to\Re:y(a){=}A,\Delta y(a){=}B,\Delta\sp2y(b+2){=}C,\Delta\sp3y(b+1){=}D\}.$$
Subject Area
Mathematics
Recommended Citation
Harmsen, Betty Jean, "The discrete calculus of variations" (1995). ETD collection for University of Nebraska-Lincoln. AAI9604414.
https://digitalcommons.unl.edu/dissertations/AAI9604414