By Piernicola Bettiol, Piermarco Cannarsa, Giovanni Colombo, Monica Motta, Franco Rampazzo

ISBN-10: 3319069160

ISBN-13: 9783319069166

Since the Nineteen Fifties regulate idea has tested itself as a huge mathematical self-discipline, quite compatible for program in a couple of learn fields, together with complex engineering layout, economics and the scientific sciences. notwithstanding, given that its emergence, there was a necessity to reconsider and expand fields akin to calculus of adaptations, differential geometry and nonsmooth research, that are heavily tied to investigate on purposes. this day regulate conception is a wealthy resource of easy summary difficulties coming up from purposes, and offers an immense body of reference for investigating simply mathematical matters. in lots of fields of arithmetic, the large and becoming scope of task has been observed by means of fragmentation right into a multitude of slim specialties. notwithstanding, awesome advances are frequently the results of the search for unifying topics and a synthesis of other ways. regulate conception and its purposes are not any exception. the following, the interplay among research and geometry has performed a very important function within the evolution of the sphere. This publication collects a few fresh effects, highlighting geometrical and analytical elements and the potential connections among them. purposes give you the history, within the classical spirit of mutual interaction among summary idea and problem-solving practice.

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T; ; x/ (59) at which arrive ! 1. d. /; x. T/ D x. 2. d. /; x. /; y. T/ D y. T; ; x/ X of average velocities of the micro evolutions (see (41), p. T; n! T/ ! n! x. T/ ! ; x/ D y. T; ;x;y/ where ! T/: (60) Generalized Lax-Hopf Formulas for Cournot Maps and Hamilton-Jacobi-. . t/. t//; i D 1; : : : ; c (65) governs evolutions t 7! t//. -P. Aubin and L. Chen at which arrive ! 1. d. /; x. T/ D x. 2. d. /; x. /; yi . T/ D yi . T; ; x/ D n! T/ ! x. T; ;x/ of average velocities of the micro evolutions (see (41), p.

F of is P-exhaustive. T; ; x/, p 2 P and x. T; ; x/ such ! T/ D . T; ; x/ (51) Generalized Lax-Hopf Formulas for Cournot Maps and Hamilton-Jacobi-. . T; ; x/. t//. 5, p. 470, of Viability Theory. 18 (Generalized Lax-Hopf Formula for Exhaustive Maps) Let us consider a closed convex cone P X. T; ; x// and p 2 P. T; ; x/ P/ : (55) ; x//. This property is useful because the properties of Cournot maps hold true when the set-valued map the set-valued map F is not Marchaud but F D F C P is. 1 Lax-Hopf Formula for Micro-Macro Systems Micro-macro systems5 are particular systems of differential inclusions where the variables are split in two classes, the micro and macro ones, the evolutions of microvariables being independent of the evolution of macro-variables.

However, the role of the associated characteristic system was recognized, at least for defining the method of characteristics. It appeared later on that the characteristic system is the unifying feature of the problems dealing with Hamilton-JacobiBellman equations and intertemporal (or optimal control) problems, as well as for the construction of feedbacks regulating optimal evolutions. The Fenchel transform of a convex lower semicontinuous Lagrangian u 2 X 7! u/ 2 R [ fC1g is the convex lower semicontinuous function p 2 X ?