- #1

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We have a function [itex]x^*[/itex] that maximizes or minimizes the integral:

[tex]\int^{t_1}_{t_0} F(t,x(t),\dot{x}(t))dt[/tex]

Our end point conditions are:

[tex]x(t_0) = x_0, x(t_1) \geq x_1[/tex]

I am told that [itex]x^*[/itex] has to satisfy the Euler equation. That I can fully understand since [itex]x^*(t_1)[/itex] can be equal to [itex]x_1[/itex]. However, then it gives me the transversality condition:

[tex]\left(\frac{\partial F}{\partial \dot{x}}\right)_{t=t_1} \leq 0 \text{ ( = 0 if $x^*(t_1) > x_1$)}[/tex]

I can understand the statement in the parentheses. However, I do not understand why [itex]\left(\frac{\partial F}{\partial \dot{x}}\right)_{t=t_1}[/itex] must be less than or equal to zero if [itex]x^*(t_1) = x_1[/itex]. Why can it not be more than zero?