Solving numerically an Optimal Control Problem subject to a conservation law (transport equation)

Question

I was wondering if there is some known way to solve the following optimal control problem \begin{align} \text{min }&\mathcal{J}(\rho,\nu)\\ \text{such that }&∂_t\rho(t, x) = − div_x(((K ∗ ρ)(t, x) + (H ∗ ν)(t, x))ρ(t, x)) \text{ (conservation law)} ,\end{align} where $\mathcal{J}$ is for example a quadratic functional that forces the density $\rho$ to evolve in a desired way and $\nu$ is a density of control agents which we control in order to achieve the desired dynamic ($K$ and $H$ are respectively the way the density $\rho$ interacts with itself and with the density of the control agents). I know nothing about the subject but in the textbooks or papers on numerical methods for variational problems I found, they often treat the case when the constraint is given by elliptic or parabolic PDEs and if I am not mistaking the conservation law is a hyperbolic PDE.

Answer 1

From your comment, it seems like you might want to do some reading about variational calculus and PDE-constrained optimization. Briefly, let's suppose that the solution $\rho$ of the PDE lives in some function space $X$, the controls $\nu$ live in some space $V$, and you want to solve a time-dependent PDE that can be written as

$$\partial_t\rho = F(\rho, \nu), \quad \rho(t = 0) = \rho_0$$

for some right-hand side $F : X \times V \to X^*$ where $X^*$ is the dual space of $X$. You want to find the minimizer of some functional $\mathscr J(\rho, \nu)$ subject to the constraint that $\rho$ solves the PDE written above. If we then define the functional

$$\mathscr{L}(\rho, \nu, \lambda) \equiv \int_0^T\langle\partial_t\rho - F(\rho ,\nu), \lambda\rangle dt + \langle \rho(t = 0) - \rho_0, \lambda\rangle + \mathscr J(\rho, \nu),$$

where we've introduced a Lagrange multiplier $\lambda$, then a critical point of this functional w.r.t. $\rho$, $\nu$, and $\lambda$ is a solution of the constrained optimization problem. A big part of variational calculus is learning how to calculate the derivative of the Lagrangian functional $\mathscr{L}$ with respect to $\rho$, $\nu$, and $\lambda$, and how to derive the PDE that $\lambda$ solves.

My favorite references on variational calculus are Weinstock, ch. 4 of Courant & Hilbert, or ch. 2 of Asch, Bocquet, and Nodet. You're likely to need a bit of functional analysis, especially things related to dual spaces; see ch. 4-6 of Kolmogorov or, if you want to hit it hard, Lax for the general stuff. Then see Treves for Sobolev spaces.