{'fraction_non_alphanumeric': 0.017825311942959002, 'fraction_numerical': 0.0, 'mean_word_length': 5.2444444444444445, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Optimality Conditions for Solutions to Hyperbolic Balance Laws

We study a class of optimization problems for solutions to a hyperbolic system of balance laws. The controller can affect the evolution of the system through the source terms, as well as the boundary conditions. Our main goal is to obtain necessary conditions in order that a piecewise regular solution, with finitely many shocks, be optimal. These necessary conditions are derived by constructing a family of “generalized cotangent vectors”, which can be transported backwards along a trajectory.