Set invariance is a fundamental concept for the construction of controllers for constrained dynamical systems subject to state and control constraints, and bounded disturbances. Constraint satisfaction can be guaranteed for all time instances (and for bounded disturbances) if and only if the state is contained inside a (robust) controlled invariant set.
Set invariance methods can be used to derive optimal controllers for linear systems with atypical (specifically, non-quadratic) cost criteria. Moreover, robust controlled invariant sets operating in regions that can be enforced through feedback control to perpetually satisfy constraints in the presence of disturbances, can be used for the robust control of linear uncertain systems. Related applications include safety set verification, explicit MPC, Interpolation-based Control (IC), viability theory, etc.
Set invariance is strictly connected with Lyapunov stability.