Generic solutions to controlled balance laws

The paper is concerned with a scalar balance law, where the source term depends on a control function [Formula: see text]. Given a control [Formula: see text], it is proved that, for generic initial data [Formula: see text], the solution has finitely many shocks, interacting at most two at a time. Moreover, at the terminal time [Formula: see text] 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 [Formula: see text]. 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.