# Stability analysis and dimension of a dynamic control system

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.

Differentiating $(3)$ with respect to time we get

$$u_{cc}\dot c = \dot \lambda \implies \frac {u_{cc}}{u_c} \dot c = \frac {\dot \lambda}{\lambda}$$

Inserting into $(6)$ we obtain

$$\dot M = -\frac {\eta_M}{\eta_{MM}}\frac {u_{cc}}{u_c} \dot c$$

So the fixed point of $M$ will happen under the same conditions that the fixed point of $c$ will. Also, the optimal time-evolution of $M$ is a scaled linear value of the optimal evolution of $c$. So it appears all aspects are captured if you "ignore" the $M$ variable.

A bit more formally, since your two decision variables are linearly dependent it means that the $3 \times 3$ Jacobian of the system will be singular at the fixed point. You will get a double eigenvalue and a single one, and you should be able to show what stability properties hold (or what is required for the desired stability notion to hold), in such a situation. Due to the existence of the double root, this requires a bit of a different mathematical approach.

• Thanks for the clear answer. I was also thinking, as you have mentionned, the dynamics of $M$ are governed by the dynamics of consumption. Thanks for the explanation on the singularity at the fixed point. I did not think about that. – optimal control Oct 30 '15 at 11:14