- #1
Avatrin
- 245
- 6
Hey, I have a theorem I cannot prove.
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?
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?