dimension of a optimal control problem with one state and two control variables

Question

I have a optimal control problem where I have a two control and one state variable. (The field is economics but my question is purely on mathematics) The maximization program is ;

$$max\int_{0}^{\infty}\left(f\left(x\right)-P_{M}M\right)dt$$

where $x$ and $M$ are control variables and $P_{M}$ is the price given for $M$, it is a constant variable.

The state variable is

$$\dot{a}=R\left(a\right)-x+\eta\left(M\right)$$

Note that variables with dot represents the time variation of the variables. $\eta (M)$ is supposed to be increasing and concave function. The Hamiltonian of the problem is :

$$\mathcal{H}=f\left(x\right)-P_{M}M+\lambda\left[R\left(a\right)-x+\eta\left(M\right)\right]$$

I write the first order conditions in the following way ;

$$u_{x}=\lambda$$

$$P_{M}=\eta_{M}\left(M\right)$$

$$\dot{\lambda}=-\lambda\left(R_{a}\left(a\right)\right)$$

My question is : Can I represent the whole dynamics of this system by a two differential equations system of $\dot{\lambda}$ and $\dot{a}$ ?

Because when I differenciate equation $P_{M}=\eta_{M}\left(M\right)$, I have ;

$$\frac{\dot{\lambda}}{\lambda}+\frac{\eta_{MM}}{\eta_{M}}\dot{M}=0$$

from which I observe that the dynamics of the control variable $M$ is totally governed by the dynamics of $\lambda$

Thanks advance for hints and suggestions.

Answer 1

1. You do not need to differentiate the expression for $M$ to find the dependence. Just solve $H'_M=-P_M+\lambda\eta'(M)=0$ (notice that you missed $\lambda$ in your expression) to get the optimal control $M^*$. Obviously, it will depend on $\lambda$.

In general, this is the idea of the maximum principle that we express controls in terms of state and adjoin variables and then solve the resulting DEs.

Note however, that if $M\in [M_{min}, M_{max}]$, the optimal control $M^*$ may lie on the border.