I want to find the $u$ that minimizes \begin{align*} E\left(\exp\left(\int_0^Tu_t\,dt + V_T^2\right)\right) \end{align*} given $dV_t = (\alpha V_t +u_t)\,dt + \sigma\,dW_t$.
Without the $\exp$ in the expectation, I would use the Hamilton-Jacobi-Bellman equation here with the optimal value function \begin{align*} H(t,x) = \min_{u, V_t=x} E\left( \int_t^Tu_s^2\,ds + V_T^2 \, \vert\, \mathcal{F}_t \right), \end{align*} where $\mathcal{F}_t$ is the filtration generated by the Brownian motion $W_t$. But when I start with \begin{align*} H(t,x) = \min_{u, V_t=x} E\left(\exp\left( \int_t^Tu_s^2\,ds + V_T^2 \right)\, \vert\, \mathcal{F}_t \right), \end{align*} I get stuck at \begin{align*} H(t,x) = \min_{u, V_t=x} E\left(\exp\left( \int_t^{t+h} u_s^2\,ds \right) H(t+h, V_{t+h}) \, \vert\, \mathcal{F}_t \right), \end{align*} Applying the Ito formula on $H(t+h, V_{t+h})$ gives me \begin{align*} H(t+h, V_{t+h}) = H(t, V_{t}) + \int_t^{t+h} \frac{\partial H}{\partial s} + \frac{\partial H}{\partial x}(\alpha V_s+u_s)+\frac12 \frac{\partial^2 H}{\partial x^2}\sigma^2 \,ds + \int_t^{t+h} \frac{\partial H}{\partial x}\sigma\,dW_s \end{align*} and thus \begin{align*} E\left(H(t+h, V_{t+h})\,\vert\,\mathcal{F}_t\right) = H(t, V_{t}) + E\left(\int_t^{t+h} \frac{\partial H}{\partial s} + \frac{\partial H}{\partial x}(\alpha V_s+u_s)+\frac12 \frac{\partial^2 H}{\partial x^2}\sigma^2 \,ds\,\vert\,\mathcal{F}_t \right). \end{align*} Can some help me through the next steps to derive the Hamilton-Jacobi-Bellman equation for this problem?