On Invariant Sets and the Theory of Barriers for Constrained Systems and their Applications

Set invariance provides a framework to formulate, analyze, and solve control problems related to dynamical systems and their continually increasing requirements on safe operation.
A common way to formulate safety conditions for mathematical system models are constraints, which define the allowed space of operation.
Constraint satisfaction is, in general, a challenging task in control and can be achieved using invariant sets.

This thesis is concerned with invariant sets, with the focus on robust invariance and control invariance, and presents their beneficial utilization to solve problems in control as well as application areas such as engineering, ecology and epidemiology.
In particular, the problem of constrained dynamical systems is considered and a novel set-based approach to safely operate a collection of linear systems without individual control signals is presented.
Furthermore, a set-based method is developed to determine the largest robust positively invariant set for nonlinear systems and applied to analyze problems in various application areas.