I'm considering an optimal control problem of form $$M=\underset{p(k)\in [0,V]}{\max}\int_{k\in K} p(k)h(k)\ dk,$$ where $h(k)$ is a given function and $p(k)$ must be (weakly) decreasing. I know that if $h(k)$ is also decreasing, then the bang-bang principle gives the optimal solution.
However, I am curious about the case where $h(k)$ is not decreasing. If we restrict to a decreasing step functions $p(k)=p_n (k)$ which at most takes $n$ different values. How well would $$M_n=\underset{p_n(k)\in [0,V]}{\max}\int_{k\in K} p_n(k)h(k)\ dk$$ approximate $M$? How would I approach this problem, and what role does $h(k)$ play here?