V. Z. Tsalyuk
abstract:
Quadratic variational problems are considered in the space of functions on the
segment $[a,b]$. They are transformed to extremal problems in the space
$\bold{L}_2$ by the $W$-substitution $x = \bold{W}z + X\alpha$, where $\bold{W}$
is Green's operator of some boundary value problem for differential equation of
the $n$-th order, and $(X\alpha)(t) = \alpha^1 x_1(t) + \dots + \alpha^n x_n(t)$,
$x_i(t)$ being suitable fundamental system of solutions of the corresponding
homogeneous equation. This substitution allows one to satisfy $n$ constraints.
If the number of linear constraints exceeds the order $n$, the transformed
extremal problem in $\bold{L}_2$ contains constraints not satisfied by the
substitution. For such a case, two ways are considered to satisfy all
constraints and, hence, to deal with a problem without constraints at all. Those
are the modified $W$-substitution, and the so called double $W$-substitution.
We show that they both give a quadratic extremal problems in subspaces of
$\bold{L}_2$, which are easy to study and solve. The paper, mainly, is devoted
to the comparison of techniques based on these substitutions and relation
between them.
Mathematics Subject Classification: 49N10, 49K27, 34B27, 34K10
Key words and phrases: Quadratic variational problem, extremal problem, Hilbert space, functional differential equation, boundary value problem, Green operator