Optimal study/advertising plan for a knowledge/goodwill evolution process with rank memory

We consider the problem of an ambitious student with a variable and bounded time to take his or her examination. We assume that the first passage over a specified knowledge threshold affects the subsequent knowledge evolution process permanently. This first passage time is used to transform the original nondifferentiable optimal control problem into a nonlinear programming problem, whose decision variables are the first passage time and the final time. A marketing interpretation of the problem is also provided.