I have the following optimal control problem:
$$ \operatorname{Max} \int_{0}^{1} u(t) d t $$ s.t. $\dot{x}(t)=x(t)-u(t)^{2}, x(0)=1, x(1) \geq 0$
I am trying to find an expression for $\lambda(t). $The Hamiltonia is $u(t)+\lambda(t)\left[x-u(t)^{2}\right]$. The FOC are:
$$H_{u}=0 \quad \rightarrow \quad 1-2 \lambda u(t)=0$$
$$\dot{\lambda}(t)=-H_{x} \quad \rightarrow \quad \dot{\lambda}(t)=-\lambda(t)$$
By definition and the FOCs, $\lambda(t)=\lambda(0)+\int_{0}^{t}\dot{\lambda(t)}dt=\lambda(0)-\int_{0}^{t}\frac{1}{2u(t)}{}dt. $
Here, however, I become stuck—because no explicit expression for $u(t)$ has been given. Thus I cannot compute the right hand side object. How may I proceed?
Thank you
Generally when using this method to solve an optimal control problem one shifts the problem of finding the optimal input as a function of time ($u(t)$) to finding the best initial conditions for the co-state ($\lambda(0)$).
The equation you got for $H_u = 0$ should normally be used to solve for the "optimal" $u(t)$. Each obtained $u(t)$ will be optimal with respect to the cost function integral, but potentially with respect to different inequality constraints. In this case $\lambda(t)$ can be solved directly, since $\dot{\lambda}(t)$ is only a linear function of $\lambda(t)$ and it can be shown that the following expression solves it
$$ \lambda(t) = e^{-t}\,\lambda(0). \tag{1} $$
Substituting $(1)$ in $H_u = 0$ and solve it for $u(t)$ yields
$$ u(t) = \frac{e^{t}}{2\,\lambda(0)}. \tag{2} $$
Now I will leave it up to you to find which $\lambda(0)$ would actually yield the optimal solution for your problem. Hint: which sign should $u(t)$ and therefore $\lambda(0)$ have and what should $x(1)$ (when solving for $x(t)$ one can use the convolution integral) be under optimal control?
It can be noted that if $\dot{\lambda}(t)$ would be more complicated, such as also a function of $x(t)$ and $u(t)$ or nonlinear, one might not be able to find a solution for $\lambda(t)$ and one might need to resort to numerical methods, such as the shooting method.