Autonomous or non-autonomous optimal control system? I have a following system with endogeneous discounting.
$c,k$ and $h(k)$ are consumption, capital and endogeneous discount function based on physical capital. (the properties of this function are not relevant with the question, so I don't write them in order to take less space in this post. The production function is increasing and concave as usual.)

$$\underset{\left\{ c_{t}\right\} }{max}\int_{t=0}^{\infty}\left[u\left(c_{t}\right)\right]e^{-\triangle}dt$$

My state variables are 

$$\begin{align}
\dot{k_{t}}=f\left(k_{t}\right)-c_{t}\\
\dot{\triangle}_{t}=\rho + h\left(k_{t}\right)
\end{align}$$

I note $\triangle=\int_{0}^{t} (\rho + h\left(k_{t}\right))dt$

I write the present-value Hamiltonian as 

$$\mathcal{H}^{present}=u\left(c\right)e^{-\triangle}+\tilde{\lambda_{1}}\left[f\left(k\right)-c\right]+\tilde{\lambda_{2}}\left[\rho + h\left(k\right)\right]$$


  In this model, my doubt is that if the system is autonomous or not because when I integrate $\dot{\triangle}$, I have $\triangle=
\rho t+ \int_{0}^{t}h\left(k_{t}\right)dt$, which depends explicitly on time $t$. So, evidently, this differential equation is non-autonomous.


Is this system really non-autonomous ? 

What I try to do, in order to have an autonomous system is to define variables. Let's say $\lambda_{1}=\tilde{\lambda_{1}}e^{\triangle}$ and $\lambda_{2}=\tilde{\lambda_{2}}e^{\triangle}$. In this way, I write the current-value Hamiltonian ;

$$\mathcal{H}^{current}=u\left(c\right) +\lambda_{1}\left[f\left(k\right)-c\right]+\lambda_{2}\left[\rho + h\left(k\right)\right]$$

I am really stuck at this point. For the moment, the system seems to be autonomous as the Hamiltonian does not depend explicitly on time here but I can not really be sure.

Hints or suggestions are all welcome.
What about rewriting the problem in the following way?

$$\underset{\left\{ c_{t}\right\} }{max}\int_{t=0}^{\infty}\left[u\left(c_{t}\right) e^{y_t}\right]e^{-\rho t}dt$$

with the new state variables defined as

$$\begin{align}
\dot{k_{t}}=f\left(k_{t}\right)-c_{t}\\
\dot{y}_{t}=-h\left(k_{t}\right)
\end{align}$$

given the initial conditions $(k(0),y(0))=(k_0,0)$.