The control problem concerns the field of economics but my problem is purely about mathematics. I hope my question fits for MathematicsSE.
I have a following system in which $c$ is a control and $k$ and $\triangle$ are state variables. Variables with dot denotes the variation by time ($\dot{k}=\frac{dk}{dt}$).
(the proporties of the function $f(k)$ and $h(k)$ are not relevant with the question, so I don't write them in order to take less place in this post.)
$$\underset{\left\{ c_{t}\right\} }{max}\int_{t=0}^{\infty}\left[u\left(c_{t}\right)\right]e^{-\triangle}dt$$
I note $\triangle=\int_{0}^{t} (\rho + h\left(k_{t}\right))dt$ where $\rho$ is a constant parameter.
State variables are
$$\begin{align} \dot{k_{t}}=f\left(k_{t}\right)-c_{t}\\ \dot{\triangle}_{t}=\rho + h\left(k_{t}\right) \end{align}$$
As it is not possible to have a term with integral on a standard Hamiltonian, I take the $\triangle$ as a state variable, in order to handle a standard control problem.
I write the present-value Hamiltonian as
$$\mathcal{H}=u\left(c\right)e^{-\triangle}+\tilde{\lambda_{1}}\left[f\left(k\right)-c\right]+\tilde{\lambda_{2}}\left[\rho + h\left(k\right)\right]$$
In this optimization program, 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 ;
$$\mathcal{H}=u\left(c\right) +\lambda_{1}\left[f\left(k\right)-c\right]+\lambda_{2}\left[\rho + h\left(k\right)\right]$$
I am really stucked 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.
Your system is autonomous!. Observe that once you have added the variable $\Delta$ with $\dot{\Delta}=\rho + h(k_t)$, your running cost (lagrangian) do not depends explicitly of time: $$ \max_{c_t} \int_0^\infty L(k_t,\Delta) dt $$