# Thread: Optimal Control Problem: Constrained Optimzation with Fixed Endpoints

1. ## Optimal Control Problem: Constrained Optimzation with Fixed Endpoints

Hi. This is a problem I'm having hard time solving.
The following "autonomous" optimization problem can also be solved using Calculus of Variations, but I prefer the Optimal Control approach:
Determine $x=x(t)$ solving $max\int_{0}^{2}(x-\dot{x})dt$ subject to $\dot{x}\in [0,1]$ and $x(0)=0$ and $x(2)=1$, where $\dot{x}=\frac{dx}{dt}.$
Now in optimal control theory, the conventional approach to solve such a Bounded problem is to put $u=\dot{x}$ as the control variable and $x$ as the state variable and recast the problem as below:
$max\int_{0}^{2}(x-u)dt,$ subject to $0 \leq u \leq 1$ and $x(0)=0$ and $x(2)=1.$
Then form the Lagrangian: $L=(x-u)+ \lambda u+ \mu _{1}(1-u)+\mu _{2}(u-0)$

But, instead, I decided to have a look on the unbounded version of the same u. In other words, initially, I tried to solve the same problem in a way as if there were no limits on the amount of $u$. Hence, instead of the Lagriangian, I found the Hamiltonian as below:

$H=(x-u)+ \lambda u$ and then applying the optimality conditions, we'll have:
$(I)$ $\frac{\partial H}{\partial u}=-1+ \lambda=0\Rightarrow \lambda=1$

$(II)$ $\frac{-\partial H}{\partial x}={\lambda}'(t)\Rightarrow -1={\lambda}'(t)\Rightarrow \lambda=-t+constant$

Since I and II yield two different $\lambda$s, one independent of time and the other, a linear function of t, I concluded that the unbounded version of the initial problem has no solution.
Now, question is: can I also conclude that since there is no solution for an unbounded u, a fortiori, there is no bounded control, u, that can satisfy the given conditions?
A classmate of mine suggested that there is a solution for the original problem and that I should be looking for a discontinuous "Bang Bang Control" solution, but I'm not convinced that my argument is incorrect. (And, if there is a "Bang Bang" control, satisfying the given constraints, how am I supposed to find it?)
Any help will be appreciated.