I am trying to solve an optimization problem which is probably reminiscent of optimal control theory but all of this is not exactly my field of expertize and I am a little bit lost in translation. If someone could please put me on the right tracks, it would most certainly save me a lifetime of useless wanderings.

In discrete time $t=[1,2,...,T]$, I would like to find the vector $\{b_t\}_{t \in T}$ such that $F(b_t):=\sum_{t=1}^{T} h(d_t-b_t)b_t$ is maximized, where:

- $\{d_t\}_{t \in T}$ is given and $d_t \geq0 \quad \forall t\in T$.
- $h(\cdot):\mathbb R^+ \to \mathbb R^+ $ is a strictly convex function.
- $d_t-b_t\geq0 \quad \forall t\in T$
- $\mid b_t \mid \leq \zeta \quad \zeta \in \mathbb R^+$
- $\sum_{t=1}^{t'} \big(\theta(b_t)+\eta\theta(-b_t)\big)\cdot b_t\leq 0 \quad \forall t'=[1,2,...,T]$ , where $0<\eta \leq1$ is fixed and $\theta(\cdot)$ is the Heaviside function.

I am pretty novice in optimization theory but from what I have understood this problem could maybe be solved using Karush-Khun-Tucker conditions, but then I don't clearly see how to proceed, notably with the heaviside's. Is there a better approach or is this problem simply unsolvable? Is there good (introductory) literature I should read about this?

Thank you for your help!