I'm trying to set up a dynamic optimization problem as follows.
Let $\mathcal{W} := [\underline{w},\overline{w}]$, and $w:[0,1]\times\{0,\dots,N\}\to \mathcal{W}$
\begin{align} \min_{\phi: \mathcal{W}\to \mathbb{R}} &J[\phi] = \sum_{\xi=0}^N\int_{0}^1p(\xi)p(a)T\left(2\phi(w(a,\xi)) - \phi(\overline{w}) - \phi(\underline{w})\right)da,\\ &\text{s.t.: } \phi'(z)\geq g(z),\,\phi(1)=0,\,\phi'(1)=1,\\&\qquad\text{and } \phi(\overline{w}),\, \phi(\underline{w}) \text{ free}. \end{align}
We assume, of course, that $1 \in \mathcal{W}$. Moreover, the function $T$ is quasi-convex, and $g$ is known and (strictly) positive.
What I have tried:
My first idea was to change the variable $a$ to another variable $z$ and use a regular optimal control formulation, and I think this might work but there are two issues:
- $w$ is not 1-to-1, and the inverses can get very messy, after dividing into regions where it is indeed 1-to-1.
- I would like to understand the general case, since in my particular problem I might not know the form of $w$.
- even after changing the variable, the presence of the function $\phi$ evaluated at the endpoints inside the integrals makes it somewhat non-standard, if I'm not mistaken.
My second idea was to try to derive some sort of Euler-Lagrange equation directly from $\phi'$, by noting that: \begin{equation} 2\phi(w(a,\xi)) - \phi(\overline{w}) - \phi(\underline{w}) = \int_{\underline{w}}^{\overline{w}}\phi'(z)\text{Sgn}(w(a,\xi)-z)dz =: V_{w(a,\xi)}[\phi'], \end{equation} so that the optimization problem, in terms of $\phi'(z) := u(z)$ reads:
\begin{align} \max_{u: \mathcal{W}\to \mathbb{R},\\ \phi(\underline{w}) \in \mathbb{R}} &\tilde{J}[u] = \sum_{\xi=0}^N\int_{0}^1p(\xi)p(a)T\left(V_{w(a,\xi)}[u]\right)da,\,\text{ s.t.: }\\ &\quad u(z)\geq g(z),\,\int_{\underline{w}}^{1}u(z)dz + \phi(\underline{w})=0,\,u(1)=1. \end{align}
My questions:
a. Is this reformulation correct? What would the necessary conditions look like?