Generic Solutions to Controlled Balance Laws

TL;DR

This paper studies controlled scalar balance laws, proving generic solutions have finitely many shocks with limited interactions, and explores optimal control problems where shocks can merge at the terminal time, revealing differences from standard solutions.

Contribution

It establishes generic properties of solutions to controlled balance laws and demonstrates the existence of optimal controls with shock merging at the final time, highlighting qualitative differences from classical solutions.

Findings

Solutions have finitely many shocks with limited interactions.

Optimal controls can produce shocks merging at the terminal time.

Generic solutions differ qualitatively from classical Cauchy problem solutions.

Abstract

The paper is concerned with a scalar balance law, where the source term depends on a control function $\alpha(t)$. Given a control $\alpha\in \mathbf{L}^\infty\bigl([0,T]\bigr)$, it is proved that, for generic initial data $\bar u \in \mathcal{C}^3(\mathbb{R})$, the solution has finitely many shocks, interacting at most two at a time. Moreover, at the terminal time $T$ no shock interaction occurs, and no new shock is formed. In addition, a family of optimal control problems is considered, including a running cost and a terminal cost. An example is constructed where the optimal solution contains two shocks merging exactly at the terminal time $T$. Such behavior persists under any suitably small perturbation of the flux, source, and cost functions, and of the initial data. This shows that generic solutions of optimization problems have different qualitative properties, compared with generic solutions to Cauchy problems.