Automatic Discovery of Analytical Control Laws for Viability Problems

The mathematical theory of viability, developed to formalize problems related to natural and social phenomena, investigates the evolution of dynamical systems under constraints. A main objective of this theory is to design control laws to keep systems inside viable domains. Examples include planning sustainable fishing activities in order to not exhaust fish populations; or regulating the waste disposal of different human activities in lakes, to avoid irreversible eutrophication. Control laws in viability theory are traditionally defined as rules, based on the current position in the state space with respect to the boundaries of the viability kernel. However, finding these boundaries is a computationally expensive procedure, feasible only for trivial systems. This talk describes an approach based on a white-box machine learning technique, Genetic Programming (GP), to discover control laws for viability problems in analytic form. Such laws can keep a system viable without the need of computing its viability kernel, facilitate communication with stakeholders, and improve explainability. A candidate set of control rules is encoded as GP trees describing equations. Evaluation is noisy, due to stochastic sampling: initial conditions are randomly drawn from the state space of the problem, and for each, a system of differential equations describing the system is solved, creating a trajectory. Candidate control laws are rewarded for keeping viable as many trajectories as possible, for as long as possible. The proposed approach is evaluated on established benchmarks for viability and delivers promising results.