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 functionsProvides 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 solvedIllustrates the applications from theory into practice Very clear and elegant style

1. Generalities 2. Specific preliminary results

Ordinary differential equations and inclusions 3. Nagumo type viability theorems 4. Problems of invariance 5. Viability under Carathéodory conditions 6. Viability for differential inclusions 7. Applications

Part 2 Evolution equations and inclusions 8. Viability for single-valued semilinear evolutions  9. Viability for multi-valued semilinear evolutions 10. Viability for single-valued fully nonlinear evolutions 11. Viability for multi-valued fully nonlinear evolutions 12. Carathéodory perturbations of m-dissipative operators 13. Applications