In this chapter, the notion of geometric homogeneity is extended for differential inclusions. This kind of homogeneity provides the most advanced coordinate-free framework for analysis and synthesis of nonlinear discontinuous systems. The main qualitative properties of continuous homogeneous systems are extended to the discontinuous setting: the equivalence of the global asymptotic stability and the existence of a homogeneous Lyapunov function; the link between finite-time stability and negative degree of homogeneity; the equivalence between attractivity and asymptotic stability are among the proved results.

Chapter Contents:

• Abstract
• 2.1.1 Introduction
• 2.1.2 Preliminaries
• 2.1.2.1 Notations
• 2.1.2.2 Differential inclusions
• 2.1.2.3 Homogeneity
• 2.1.3 Homogeneous DIs
• 2.1.4 Qualitative results on homogeneous discontinuous systems
• 2.1.4.1 Converse Lyapunov theorem for homogeneous DIs
• 2.1.4.2 Application to FTS
• 2.1.4.3 Sufficient conditions for global asymptotic stability
• 2.1.5 Conclusion
• References

Inspec keywords: geometry; asymptotic stability; nonlinear systems; control system synthesis; Lyapunov methods; sampled data systems; continuous systems

Other keywords: geometric homogeneity; nonlinear discontinuous system analysis; global asymptotic stability; attractivity; differential inclusion homogeneity; continuous homogeneous systems; coordinate-free framework; nonlinear discontinuous system synthesis; discontinuous setting; finite-time stability; homogeneous Lyapunov function

Subjects: Combinatorial mathematics; Nonlinear control systems; Discrete control systems; Stability in control theory; Control system analysis and synthesis methods