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 $\displaystyle x=x(t)$ solving $\displaystyle max\int_{0}^{2}(x-\dot{x})dt$ subject to $\displaystyle \dot{x}\in [0,1] $ and $\displaystyle x(0)=0$ and $\displaystyle x(2)=1$**, where $\displaystyle \dot{x}=\frac{dx}{dt}.$

Now in optimal control theory, the conventional approach to solve such a *Bounded *problem is to put $\displaystyle u=\dot{x}$ as the control variable and $\displaystyle x$ as the state variable and recast the problem as below:

$\displaystyle max\int_{0}^{2}(x-u)dt,$ subject to $\displaystyle 0 \leq u \leq 1$ and $\displaystyle x(0)=0$ and $\displaystyle x(2)=1.$

Then form the Lagrangian: $\displaystyle L=(x-u)+ \lambda u+ \mu _{1}(1-u)+\mu _{2}(u-0)$

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 $\displaystyle u$. Hence, instead of the Lagriangian, I found the Hamiltonian as below:

$\displaystyle H=(x-u)+ \lambda u$ and then applying the optimality conditions, we'll have:

$\displaystyle (I)$ $\displaystyle \frac{\partial H}{\partial u}=-1+ \lambda=0\Rightarrow \lambda=1$

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

Since I and II yield two different $\displaystyle \lambda$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. :)