By Viorel Barbu

This paintings is a revised and enlarged variation of a booklet with an identical identify released in Romanian via the Publishing residence of the Romanian Academy in 1989. It grew out of lecture notes for a graduate path given by way of the writer on the college if Ia~i and was once before everything meant for college kids and readers basically attracted to purposes of optimum keep an eye on of standard differential equations. during this imaginative and prescient the booklet needed to comprise an basic description of the Pontryagin greatest precept and various examples and purposes from quite a few fields of technological know-how. The evolution of keep an eye on technology within the final many years has proven that its meth ods and instruments are drawn from a wide spectrum of mathematical effects which transcend the classical thought of standard differential equations and actual analy ses. Mathematical components akin to useful research, topology, partial differential equations and countless dimensional dynamical structures, geometry, performed and should proceed to play an expanding function within the improvement of the regulate sciences. however, keep an eye on difficulties is a wealthy resource of deep mathematical difficulties. Any presentation of keep an eye on conception which for the sake of accessibility ignores those evidence is incomplete and not able to achieve its targets. that is why we essential to widen the preliminary viewpoint of the ebook and to incorporate a rigorous mathematical therapy of optimum regulate conception of methods ruled by way of ordi nary differential equations and a few commonplace difficulties from thought of disbursed parameter systems.

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**Example text**

Since y*(T) E 8A(D) then for every E > there exists fJeEA(D) such that ° Ihe - y*(T)11 ::; E. 1. U is a complete metric space. Proof Let us prove first that d is a distance. Let Ul, U2, U3 E U. Then we have {t;Ul(t) =1= U2(tn c {t,Ul(t) =1= U3(tn u =1= U2(tn {t, U3(t) and therefore Now let {un} be a Cauchy sequence in U. To prove that {un} is convergent it suffices to prove that some subsequence of {un} is convergent. Let {Unj } C {Un} be such that d( unj , Unj+l) ::; 2- j . We set E j = Uk2j{ t; unk (t) =1= U nk + 1 (tn· We have Ej+l C Ej and m(Ej) = 2- j .

Proof We take Xo = x and define inductively the sequence {x n } as follows. 5) i= Xn-l such that F(u) ::; F(xn-d - cd(Xn-l,U). 6) and choose Xn E Sn such that F(xn) - inf{F(u);u E Sn} ::; ~ (F(Xn-l - inf{F(u);u E Sn}). 7) We shall prove that the sequence {xn} so defined is convergent. 8) The sequence {F(xn)} is bounded from below and monotonically decreasing. 8) it follows that so is {x n }. Hence limn->oo Xn = Xc exists. 4). ,. ;::: F(xn-d ;::: F(xn) ;::: ... 29 Generalized Gradients and Optimality we conclude that because F is lower semicontinuous.

Amer. Math. Soc. 205 (1975), 247-262. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. Clarke, F. , Optimization and Nonsmooth Optimization, John Wiley & Sons (1983). Ekeland, I. 'On the Variational Principle', J. Math. Anal. Appl. 47 (1974), 324-353. , 'Nonconvex Minimization Problems', Bull. Amer. Math. Soc. 1 (1979), 443-474. loffe, A. D. , 'Subdifferentials of Convex Functions' (in Russian), Trud. Mosc. Mat. Obsc. 26 (1972), 3-73. , Fonctionnelles Convexes, Seminaire sur les equations aux derivees partielles, College de France, Paris (1966-1967).