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

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.

• Do you feel like you understand the derivations that you've seen when the governing PDE is parabolic? If so, the essential idea -- using a Lagrange multiplier field to enforce the PDE, then show that the multiplier solves the adjoint equation -- carries over exactly to the case where the PDE is hyperbolic too. Feb 26, 2022 at 17:46
• I will read the parabolic derivation and come back to you Feb 27, 2022 at 9:29
• @DanielShapero, what do you mean by "using a Lagrange multiplier field to enforce the PDE"? Do you have a reference? Feb 27, 2022 at 21:32
• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
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Feb 28, 2022 at 14:40
• @edamondo I think your question what do you mean by "using a Lagrange multiplier field to enforce the PDE"? implies that you don't understand the general theory of dealing with constrained optimization problems as suggested by Daniel Shapero. There is no fundamental difference between parabolic and hyperbolic problems. If you understand how it works for parabolic problems, you will understand how to do it for hyperbolic problems. There is plenty of literature about optimal control in general, and that I would start with. Feb 28, 2022 at 17:47

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.

• Thank you for the answer. In which of the mentioned books do they derive the PDE that $\lambda$ solves? Mar 1, 2022 at 9:05
• That would be in the data assimilation book by Asch, Bocquet, and Nodet. Mar 1, 2022 at 16:45