Viability, Invariance and Applications

The book is an almost self-contained presentation of the most important concepts and results in viability and invariance. The viability of a set K with respect to a given function (or multi-function) F, defined on it, describes the property that, for each initial data in K, the differential equation (or inclusion) driven by that function or multi-function) to have at least one solution. The invariance of a set K with respect to a function (or multi-function) F, defined on a larger set D, is that property which says that each solution of the differential equation (or inclusion) driven by F and issuing in K remains in K, at least for a short time.

The book includes the most important necessary and sufficient conditions for viability starting with Nagumo's Viability Theorem for ordinary differential equations with continuous right-hand sides and continuing with the corresponding extensions either to differential inclusions or to semilinear or even fully nonlinear evolution equations, systems and inclusions. In the latter (i.e. multi-valued) cases, the results (based on two completely new tangency concepts), all due to the authors, are original and extend significantly, in several directions, their well-known classical counterparts.

- New concepts for multi-functions as the classical tangent vectors for functions
- Provides the very general and necessary conditions for viability in the case of differential inclusions, semilinear and fully nonlinear evolution inclusions
- Clarifying examples, illustrations and numerous problems, completely and carefully solved
- Illustrates the applications from theory into practice
- Very clear and elegant style

Preface
Chapter 1. Generalities
Chapter 2. Specific preliminary results
Ordinary differential equations and inclusions
Chapter 3. Nagumo type viability theorems
Chapter 4. Problems of invariance
Chapter 5. Viability under Carathéodory conditions
Chapter 6. Viability for differential inclusions
Chapter 7. Applications
Part 2 Evolution equations and inclusions
Chapter 8. Viability for single-valued semilinear evolutions
Chapter 9. Viability for multi-valued semilinear evolutions
Chapter 10. Viability for single-valued fully nonlinear evolutions
Chapter 11. Viability for multi-valued fully nonlinear evolutions
Chapter 12. Carathéodory perturbations of m-dissipative operators
Chapter 13. Applications
Solutions to the proposed problems
Bibliographical notes and comments
Bibliography
Name Index
Subject Index
Notation