Viability, Invariance and Applications


  • Ovidiu Carja, Al. I. Cuza University 700506 Iasi, Romania
  • Mihai Necula, Al. I. Cuza University 700506 Iasi, Romania
  • Ioan I. Vrabie, Al. I. Cuza University 700506 Iasi, Romania

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.
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Primary Markets:Graduate students, specialists and researchers in O.D.E., P.D.E., Differential Inclusions, Optimal ControlSecondary Markets:Physicists, Engineers, Chemists, Economists, Biologists.


Book information

  • Published: June 2007
  • Imprint: ELSEVIER
  • ISBN: 978-0-444-52761-5


"This book deals with a systematic treatment of those tangency conditions in connection with viability or invariance problems of increasing generality, together with some applications of the previously developed abstract theory. The material is presented in a very clear and well-organized way."--Zentralblatt MATH 2012-1239-34068

Table of Contents

PrefaceChapter 1. GeneralitiesChapter 2. Specific preliminary resultsOrdinary differential equations and inclusionsChapter 3. Nagumo type viability theoremsChapter 4. Problems of invarianceChapter 5. Viability under Carathéodory conditionsChapter 6. Viability for differential inclusionsChapter 7. ApplicationsPart 2 Evolution equations and inclusionsChapter 8. Viability for single-valued semilinear evolutions Chapter 9. Viability for multi-valued semilinear evolutionsChapter 10. Viability for single-valued fully nonlinear evolutionsChapter 11. Viability for multi-valued fully nonlinear evolutionsChapter 12. Carathéodory perturbations of m-dissipative operatorsChapter 13. Applications Solutions to the proposed problemsBibliographical notes and commentsBibliographyName IndexSubject IndexNotation