Suppose I have an objective function: $min(-\int_0^T((C*x_2 + f(x_1))*V) dt$
with the constraints
$\dot x_1 = x_2 $
and
$ \dot x_2 = -{C\over m} *x_2 -{f(x_1)\over m}-A $
The standard base excitation equations of a mass spring damper system with an unknown spring function. Here A is the acceleration input to the system, and V is the integral of A, velocity. And both are known inputs to the system, of the form $ A sin(\omega t)$.
The Hamiltonian is then
$H = C*x_2*V +f(x_1)*V + \lambda_1*x_2+\lambda_2*({-Cover m}*x_2 -{f(x_1)\over m}-A ) $
Solving my Costate equations gives:
$∂H/∂x_1 = -\dot\lambda_1 = f`(x_1)*V - \frac{\lambda_2}{m}*f`(x_1) \rightarrow \dot\lambda_1 = -f`(x_1)*V + \frac{\lambda_2}{m}*f`(x_1) $ and
$∂H/∂x_2 = -\dot\lambda_2 = C*V+\lambda_1 -\frac{C}{m}\lambda_2 $ and for the optimality condition
$∂H/∂f(x_1) = 0 \rightarrow V-\frac{\lambda_2}{m} = 0$
Now here is where I run into trouble. When solving these three equations the $ f(x_1)$ terms cancel out. I have tried several formulations of the objective function only to have the same result, $ f(x_1)$ canceling. So my question is, what does this mean? Am I formulating the question incorrectly? violating an assumption? Any help would be greatly appreciated.