$$\text{fun}:=\int_0^\infty e^{-rt}\left[u(c'(t))+w-p\,c'(t)\right]e^{-b\,c(t)}\,dt. $$
I am maximizing this objective function with some function c(t). As an extension, I want to get a solution when w changes from w1 to w2 at t=100 as well. Then, the objective function will be divided into two functions; 0~100 and 100~.
I am trying to find a continuous function (not necessarily differentiable). Then, is it enough to solve each part separately setting an initial condition of the second part equal to c*(t=100)? (Here c* is the solution for the first part).
The first that I can remind is that max{f(x)+g(x)} <= max{f(x)}+max{g(x)}. However, here, I think I am adding one more constraint for max{g(x)} compared to the LHS problem.
I was wondering if there is a reference that I can look at. Thank you very much in advance.