Differential Equations and The - Braun, Martin 7579 · Problem 2

Mathematical Control Theory: Deterministic Finite Dimensional Systems — Braun, Martin 7579

Mathematics — Calculus · sturm-liouville-not-selfadjoint

Problem

Here is the outline of a proof that

L

[see Equation (3)] is not selfadjoint if

b(x)\neq a'(x)

. Observe first that if

b(x)\neq a'(x)

and we can choose

u

and

v

so that

v(x)u'(x)-u(x)v'(x)=b(x)-a'(x),

then the integral in Equation (10) reduces to

\int_\alpha^\beta [b(x)-a'(x)]^2\,dx\neq 0.

The only problem is that

u

and

v

have to satisfy the boundary conditions (3'). To modify the proof, (a) Observe that

v(x)u'(x)-u(x)v'(x)=v^2(x)\left(\frac{u}{v}\right)'.

(b) Choose

u

v

, and constants

c_1,c_3,\ldots,c_{k+1}

(with

k

to be determined) so that

\left(\frac{u}{v}\right)'=c_1[b(x)-a'(x)]+c_3[b(x)-a'(x)]^3+c_5[b(x)-a'(x)]^5+\cdots,

u

and

v

satisfy the boundary conditions (3'), and the integral in Equation (10) is

\neq 0