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 solving subject to and and, where

Now in optimal control theory, the conventional approach to solve such aBoundedproblem is to put as the control variable and as the state variable and recast the problem as below:

subject to and and

Then form the Lagrangian:

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 . Hence, instead of the Lagriangian, I found the Hamiltonian as below:

and then applying the optimality conditions, we'll have:

Since I and II yield two different 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.