ABSTRACT
This chapter presents a number of fundamental existence results for systems of ordinary differential equations in the setting of algebras of generalized functions. It recalls the definitions and results needed from the Colombeau theory of generalized functions. The chapter contains a representative sample of existence and uniqueness results, and hints at some applications in the calculus of variations and control theory. All algebras are invariant under composition with polynomially bounded smooth functions. A classical particle in a delta function potential is formally described by the Lagrangian. Hybrid control theory deals with systems involving both continuous and discrete controls. The natural Lipschitz or boundedness conditions or structural assumptions will then lead to existence globally in time.
