Continuity of DE solution in the _density functions_?

[Anja]

Hi there,
I'm an economics grad student and looking for a pointer to a theorem/paper that solves the problem below.

Here goes:

I have the system
$$\dot{B(i)} =- \int_0^J \alpha(i,t)(\pi(i,t)-B(i)-G(t))m(t) d t$$
$$\dot{G(j)} =- \int_0^I \alpha(t,j)(\pi(t,j)-B(t)-G(j)w(t) d t$$
with fixed boundary conditions on B(I) and G(J) (both strictly positive).

$$\pi$$ is twice cont. diff, m and w are from a uniformly equicontinuous set of functions on a closed interval, and $$\alpha$$ is the indicator function for the $$(\pi(i,j)-B(i)-G(j))$$ term being nonnegative. I can impose conditions on $$\pi$$ so that this expression is zero at only two points i or j for any given j or i (the functions are "strictly single-peaked" in each dimension).

I would need the solution (B*, G*) to this system to be continuous w.r.t. (m,w) (all with the sup norm). This is part of proving existence of an equilibrium (fixed point) for a larger system, in case you are wondering... if I forgot anything, please let me know. Any help is greatly appreciated. Honorable mention in my thesis/paper if successful!

 

[jvicens]

Hi there,

I am a scientist with a background in mathematical economics and I believe I can provide some insight into your problem. The system you have presented is a dynamic optimization problem with two control variables, B and G, and two state variables, m and w. The goal is to find the optimal values of B and G that satisfy the given system of equations and constraints.

To solve this problem, you can use the Pontryagin maximum principle, which is a well-known theorem in optimal control theory. This theorem states that for a dynamic optimization problem with continuous and differentiable functions, the optimal control variables can be found by solving a set of differential equations known as the Hamiltonian equations. These equations involve the Hamiltonian function, which is a combination of the objective function and the constraints of the problem.

In your case, the Hamiltonian function would be the sum of the two integrals in your system of equations. By solving the Hamiltonian equations, you can find the optimal values of B and G that satisfy the given system of equations and constraints. These values will also be continuous with respect to the control variables m and w, as desired.

I hope this helps and good luck with your research! If you need further assistance or have any other questions, please don't hesitate to reach out.