## Viability, invariance and applications.(English)Zbl 1239.34068

North-Holland Mathematics Studies 207. Amsterdam: Elsevier (ISBN 978-0-444-52761-5/hbk). xii, 344 p. (2007).
The book under review is devoted to the study of necessary and sufficient conditions for the existence of trajectories of differential equations or inclusions subject to state constraints. This topic, after J.-P. Aubin, goes by the name “viability and invariance”, while other authors prefer the denomination “weak and strong flow invariance”. The simplest problem considered here is the Cauchy problem for an ordinary differential equation in a possibly infinite-dimensional Banach space $$X$$, $u(t)= f(u(t)),\quad u(0)=\xi, \tag{1}$ with the further constraint that at least one solution (viability) or all solutions (invariance) must remain in a (locally) closed set $$K\subset X$$ in some nontrivial interval $$[0,T]$$. 0f course, $$f$$ is defined on $$K$$ and $$\xi\in{\mathcal K}$$. It is reasonable to expect that a necessary condition for this problem to have solutions will be based on a tangency relation involving both $$f$$ and $$K$$. This is exactly the case, and this condition is often also sufficient (as stated for example in the classical Nagumo theorem). 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. Several exercises (with solutions) and examples are provided, certainly helping the interested reader to understand the subject.
The book is divided into two parts. The first one deals with problems of type (1), with a possibly multivalued right-hand side $$F$$ in place of $$f$$, while the second one follows the same pattern, but with the addition of a possibly nonlinear operator $$A$$, which is the generator of a $$C_0$$-semigroup.
After two preliminary chapters (the second one introduces and studies in detail certain tangency conditions which are used throughout the whole book, together with measures of noncompactness), viability and invariance for problem (1) are studied. First the classical necessary condition is proved $$(f(\xi)\in T_K(\xi)$$ for all $$\xi\in K$$, where $T_K(\xi):=\{\eta\in X: \underset h\downarrow {0}{\lim\inf}\text{dist}(\xi+h\eta;K)=0\}$ is the Bouligand-Severi tangent cone to $$K$$ at $$\xi\in K)$$. Second, under suitable continuity and compactness conditions on $$f$$, the necessary condition is shown to be also sufficient. The main technical point of the proof is constructing a sequence of approximate solutions, which converges to a solution. This scheme remains the same throughout the whole book. Next, an extension to the nonautonomous case is presented, together with results concerning solutions defined on a maximal time interval. Chapter 4 deals with invariance, and the sufficient condition now reads as: there exists an open neighborhood $$V$$ of $$K$$ and a uniqueness function $$\omega$$ (i.e., providing uniqueness for the Cauchy problem $$r='\omega(r)r(0)=0$$ where $$\omega(0)=0$$ such that for all $$\xi\in V$$ $\underset h\downarrow {0}{\lim\inf}{\text{dist}(\xi+hf(\xi);K)-\text{dist}(\xi;K)\atop h}\leq\omega(\text{dist}(\xi;K)).$ Chapter 5 is devoted to nonautonomous differential equations, satisfying a Carathéodory type condition. Chapter 6 deals with viability and invariance for differential inclusions $$(u'\in F(u))$$, where $$F$$ is upper semicontinuous with compact and convex values. Here the condition for viability is essentially $$F(\xi)\cap T_K(\xi)\neq\emptyset$$ while the condition for invariance is $$F(\xi)\subseteq T_K(\xi)$$. The set-valued case is not pushed towards its maximal generality, as, for example, problems with a nonconvex right-hand side are not considered (for this topic see, e.g., A. Bressan [Funkc. Ekvacioj, Ser. Int. 31, No. 3, 459–470 (1988; Zbl 0676.34014)], for the lower semicontinuous case, as well as, e.g., [M. Bounkhel and T. Haddad, Electron. J. Differ. Equ. 2005, Paper No. 50, 10 p., electronic only (2005; Zbl 1075.34053)], for the upper semicontinuous case). Up to this chapter the content of the book is rather classical and can be found also in other books [e.g., J.-P. Aubin and A. Cellina, Differential inclusions. Set-valued maps and viability theory, Springer, Berlin (1984; Zbl 0538.34007); J.-P. Aubin, Viability theory, Birkhäuser Boston, Boston, MA (1991; Zbl 0755.93003)], possibly with equivalent conditions involving normal vectors [see, e.g., F.H. Clarke et al., Nonsmooth analysis and control theory, Springer, New York (1998; Zbl 1047.49500) (Chapter 4)]. Chapter 7 deals with several applications. The reviewer appreciates, in particular, an elegant extension of the Banach fixed point theorem (Theorem 7.4.1).
The core of the book is the second part, where problems of the type $u'\in Au+f(u),\;u(0)=\xi,\quad u(t)\in K,\;\forall t,$ or of the type $u'\in Au+F(u),\;u(0)=\xi,\quad u(t)\in K,\;\forall t,$ together with some nonautonomous (what concerns $$f$$ or $$F)$$ versions are considered. Here $$A$$ generates a $$C_0$$-semigroup $$S$$ and mild (i.e., integral) solutions are considered. The tangency condition now takes into account the semigroup $$S$$ and reads as $\underset h\downarrow {0}{\lim\inf}\text{dist}(S(h)\xi+h \eta f(\xi);K)=0$ for all $$\xi\in K$$. All other conditions for viability or invariance are modified accordingly. First the case where $$A$$ is linear is considered and next (Chapters 10–12) the fully nonlinear case is treated, with $$A$$ being $$m$$-dissipative. Chapter 13 is devoted to applications to some classes of PDEs. For example, existence of solutions satisfying some state constraints (e.g., positivity) for various classes of PDEs (semilinear/nonlinear diffusion, predator-prey, reaction-diffusion) is proved.
Finally, the reviewer points out that the interesting problem of viability or invariance for stochastic differential equations [see, e.g., J.-P. Aubin and G. Da Prato, Stochastic Anal. Appl. 16, No.1, 1-15 (1998; Zbl 0931.60059)]; M. Bardi and P. Goatin, in Stochastic analysis, control, optimization and applications, 191–208, Birkhäuser Boston, Boston, MA (1999; Zbl 0928.93067)], which is also connected with PDEs, has been treated only marginally in other books.

### MSC:

 34G20 Nonlinear differential equations in abstract spaces 34G25 Evolution inclusions 47H04 Set-valued operators 47J35 Nonlinear evolution equations 47N20 Applications of operator theory to differential and integral equations 49J21 Existence theories for optimal control problems involving relations other than differential equations
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