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I am trying to minimize:

[tex]

\int_0^\infty{\exp(-t)(t\,f'(t)-f(t))^2\,dt}

[/tex]

by choice of [tex]f[/tex], subject to [tex]f(0)=1[/tex] and [tex]f'(x)>0[/tex] for all [tex]x[/tex].

The (real) solution to the Euler-Lagrange differential equation is:

[tex]

f(t)={C_1}t

[/tex]

rather unsurprisingly. However, this violates [tex]f(0)=1[/tex].

If we constrain to solutions of the form:

[tex]

f(t)=1+t^p

[/tex]

then numerical optimisation puts [tex]p=1.2848\cdots[/tex] at which point the target is [tex]0.6327\cdots[/tex].

Is it possible to solve this problem without restricting to a particular solution form?

Thanks in advance,

Tom

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# Maximizing a functional when the Euler-Lagrange equation's solution violates ICs

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