I have an optimal control problem where I have two control and one state variable.
$$max\int_{0}^{\infty}\left(u\left(c\right)-P_{M}M\right)e^{-\rho t}dt\tag{1}$$
where $P_{M}$ is the unit price of carbon abatement activity ($CO_2$ abatement at atmosphere) and $M$ is the control variable, which is the level of abatement that we choose optimally.
The state variable is
$$\dot{S}=R\left(S\right)-c+\eta\left(M\right)\tag{2}$$
where $S$ is the environmental quality, $c$ is consumption, $R(S)$ is the regeneration of environment (like the production function in models with capital accumulation) and $\eta (M)$ is the carbon abatement function, which is supposed to be increasing and concave function. As it is logical, this function contributes to the increase of environmental quality.
I write the Hamiltonian ;
$$\mathcal{H}=u\left(c\right)-P_{M}M+\lambda\left[R\left(S\right)-c+\eta\left(M\right)\right]$$
The FOC are ;
$$u_{c}=\lambda\tag{3}$$
$$P_{M}=\lambda\eta_{M}\left(M\right)\tag{4}$$
$$\dot{\lambda}=\rho\lambda-\lambda\left(R_{S}\left(S\right)\right)\tag{5}$$
So, in this model, I have 2 control variables, consumption $c$ and abatement $M$ and a state variable $S$.
I have a doubt about writing the Jacobian Matrix, If I did not have the second control variable $M$, I would write a 2 dimension differential system of $\dot{c}$ and $\dot{S}$ but as I have a second control variable $M$, I am not sure if I can always describe the whole dynamics of the economy by two differential equations.
When I try to see the dynamics of the control variable $M$, with differentiating equation $(4)$ according to time, I have ;
$$\frac{\dot{\lambda}}{\lambda}+\frac{\eta_{MM}}{\eta_{M}}\dot{M}=0\tag{6}$$
According to the equation $(6)$ the dynamics of carbon abatement $M$ is represented by the dynamics of costate variable.
In this case, can I represent all dynamics describing this economy by only two differential equations which are $\dot{c}$ and $\dot{S}$ ?
Thanks in advance for suggestions and hints.
