Viability Theory

This monograph by Jean-Pierre Aubin, a French mathematician known for founding viability theory, asks a basic question: how can a system governed by changing rules be kept within a set of acceptable states? The book opens with an overview of its structure, then develops the central tools, including viability kernels and the related notions of viability and capturability.

The text moves from abstract theory into applications. Its chapters take up the control of robots, intertemporal optimization that balances present and future decisions, the sustainable management of renewable resources, and the regulation of control systems and traffic flow. Several discussions examine how inertia and hysteresis affect a system's ability to stay viable, and how a system can be restored when it drifts outside its acceptable region. Later chapters connect the theory to partial differential equations, including Hamilton-Jacobi and conservation laws, and to Hamilton-Jacobi-Bellman equations. Illustrations drawn from finance and economics show how the same ideas apply to economic models.

The book closes with the mathematical background it relies on, a short introduction to set-valued analysis, and the convergence and viability theorems that underpin the results. It suits researchers and graduate students in applied mathematics, control theory, and systems analysis.