I am trying to solve the problem of a firm facing the possibility of a future tax, in continuous time.
The firm maximizes $V(k)=\int_{t=0}^{\infty}e^{-rt} \pi_t dt$ with $\pi_t=f(k_t)-i_t$ and $\dot{k}=i_t-\delta k_t$. There is a probability $\rho$ per unit of time to transit to a new state where a tax is imposed, and where the profit becomes $f(k_t)-i_t-\tau k_t$. This is a partial equilibrium problem and we assume that $r$, $\delta$ and $\rho$ are exogenous.
Once we reach the new state where the tax is imposed, there is no possibility to go back to the previous state. The only uncertainty is about when the tax will be imposed, i.e, when the uncertainty will be resolved.
I am trying to solve this problem using Hamiltonian-Jacobi-Bellman (HJB) equations. The problem can be described by the following system: $$\begin{align*} rV_1 &= \max_{i} \{f(k_t)-i_t+\rho (V_2-V_1)+\dot{V}_1\} \\ rV_2 &= \max_{i} \{f(k_t)-i-\tau k_{t}+\dot{V}_2\} \end{align*}$$
I know how to solve the second equation, following the method of Walde 2012 using dynamic programming. First, rewrite $$f(k_t)-i_t-\tau k_t + V'_2(k_t)\dot{k_t}=f(k_t)-i_t-\tau k_t+ V'_2(k_t)(i_t-\delta k_t)$$ Then, take the FOC with respect to $i_t$, which yields $$V'_2(k_t)=1$$ Then, use the envelope condition to find \begin{align*} rV'_2=f'(k_t)-\tau+V''_2(k_t)(i_t-\delta k_t)-\delta V'_2(k_t) \end{align*} This can be simplified, using the FOC, as$$\begin{align*} f'(k_t)=r+\delta+\tau \end{align*}$$ which yields the same solution as a simple Hamiltonian would.
However, I am unsure how to proceed next and how to solve for the solution before the tax is imposed and the uncertainty resolved.
Additionally, if you have some references regarding dynamic control in continuous time, I would be very interested, especially if they treat the case of HJB with additional constraints.
Thank you in advance for your help!
PS: the question was first posted in the Economics SE, but I thought it might be more appropriate to post it here. Please let me know if you think I should delete my original question.