By R. V. Gamkrelidze (auth.)
In the past due 1950's, the crowd of Soviet mathematicians such as L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko made basic contributions to optimum regulate thought. a lot in their paintings used to be amassed of their monograph, The Mathematical idea of optimum approaches. consequently, Professor Gamkrelidze made extra very important contributions to the idea of invaluable stipulations for difficulties of optimum keep watch over and normal optimization difficulties. within the current monograph, Professor Gamkrelidze offers his present view of the basics of optimum keep an eye on concept. it truly is meant to be used in a one-semester graduate path or complex undergraduate path. we're now making those principles on hand in English to all these attracted to optimum regulate thought. West Lafayette, Indiana, united states Leonard D. Berkovitz Translation Editor Vll Preface This ebook relies on lectures I gave on the Tbilisi kingdom collage through the fall of 1974. It includes, in essence, the foundations of common keep watch over concept and proofs of the utmost precept and simple life theorems of optimum regulate idea. even if the proofs of the fundamental theorems offered listed here are faraway from being the shortest, i feel they're absolutely justified from the conceptual view aspect. at least, the notions we introduce and the tools built have one unquestionable virtue -they are regularly used all through keep an eye on concept, and never just for the proofs of the theorems offered during this book.
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Example text
T, x):J P(t, x), it remains only to prove the inclusion p. (t, x) C conv P(t, x). Let there be a point pf/=. conv P(t, x) and let there exist, nevertheless, a probability measure p, concentrated on U and such that (P,,J(t, x, u) = p. We draw in R" an (n-1)-dimensional hyperplane rn- 1 through the point p in such a way that the convex set conv P(t, x) lies on one side of it. 5) f(t,x,'): W--+R" by Xl(U), and the characteristic function of the complement by X2(U): for f(t, x, u) Ern-I, for f(t, x, u) f/=.
It can be seen from the proof that we can take for Van arbitrarily small neighborhood of the compact set K lying in O. Therefore, an arbitrarily small positive number can be taken for t:. , the infinitely differentiable function IX(U) is "arbitrarily close" to the characteristic function of the compact set K. 1. Let a compact set K C Rr be covered by a finite number of open sets 01> . , Or Then there exist infinitely differentiable functions IXl (u), ... , IXp(U) such that (1) O~IXi(U)~ 1 \;fuER r , i=1,2, ...
6) which is called the chattering control. At every instant of time, the measure Ilt is concentrated at the points Ul(t), ... , uq(t) with the corresponding "weight coefficients" Ill(t), ... , Ilq(t). The control Ilt is, so to speak, "distributed" among the q points Ul (t), ... , uq(t) in the proportions Ill(t), ... ,Ilq(t). 2. Weak Convergence of Generalized Controls The notion of weak convergence is introduced in the linear space of all Radon measures on Rm with m ~ 1. We shall now remind the reader of this notion, and we shall make use of it in order to define the notion of weak convergence in the class of generalized controls.
