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. Control laws 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. We propose an approach based on Genetic Programming (GP) to discover control laws for viability problems in analytic form. Such laws could 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.

This paper addresses the issue of supporting the end-user of a classifier, when it is used as a decision support system, to classify new cases. We consider several kinds of classifiers: Statistical or machine learning classifiers, which are built on data, but also direct model-based classifiers that are built to solve a particular problem (like in viability or control problems). The end-user relies mainly on global information (like error rates or global sensitivity analysis) to assess the quality of the result given by the system. Class membership probability, if available, describes certainly the local statistical viewpoint. But it doesn't take into account other contextual information: Cases with high value of class membership probability can also be close to the decision boundary. In the case of numerical state space, we propose to use the decision boundary of the classifier (which always exists, even implicitly), to describe the situation of a particular case: The distance of a case to the decision boundary measures the robustness of the decision to a change in the input data. Other geometric concepts, such as the maximal maximal ball, can present a precise picture of the situation to the end-user. We show the interest of such a geometric study on different examples.