By Aaron Strauss (auth.)

ISBN-10: 3540042520

ISBN-13: 9783540042525

ISBN-10: 3642510019

ISBN-13: 9783642510014

This paper is meant for the newbie. it isn't a country­ of-the-art paper for study staff within the box of regulate thought. Its objective is to introduce the reader to a few of the issues and leads to keep an eye on conception, to demonstrate the appliance of those re­ sults, and to supply a consultant for his additional examining in this topic. i've got attempted to encourage the implications with examples, especial­ ly with one canonical, uncomplicated instance defined in §3. Many effects, akin to the utmost precept, have lengthy and tough proofs. i've got passed over those proofs. generally i've got integrated purely the proofs that are both (1) now not too tough or (2) particularly enlightening as to the character of the end result. i've got, although, frequently tried to attract the most powerful end from a given facts. for instance, many current proofs on top of things concept for compact pursuits and distinctiveness of recommendations additionally carry for closed ambitions and non-uniqueness. eventually, on the finish of every part i've got given references to generalizations and origins of the consequences mentioned in that part. I make no declare of completeness within the references, even though, as i've got usually been content material in basic terms to refer the reader both to an exposition or to a paper which has an in depth bibliography. IV those 1ecture notes are revisions of notes I used for aseries of 9 1ectures on contro1 concept on the foreign summer time Schoo1 on Mathematica1 structures and Economics held in Varenna, Ita1y, June 1967.

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5). to equation But norm on in the same way as before. diction results, and thus Theorem Since ~C. 4. 2) has ~CCK, hence i f is not interior to have rank UpC we have UpCC~, we The same contra- is proved. 2. n peR x Q, A such that (y, u) in ~ A I x - y I P. Consider closed and let that f(x, u) . = (A) Then i f x = 0 and the zero target. Let is interior to ~C' Q it follows = K. PROOF. Let some interval [t 0' 0 N J. x (K . o Choose {x Let E uEUM > 0 steer so that x o to be x = 0 on 49 I u ( t) I ~ Le t P bu 0 Ix E p Then in - = X(t; p E P is compact.

We shall prove that m I b .. u. (s) ds 1J j=l J is such that k = 0, 1, x~K-(t). o 0 A has positive real part. If d th r b .. 2). j = 1, •.. , m, is composed entire- Hence 41 for some ,th d ~ i ~ d + r, u, component of J of J t a. u* = J 1 ~ j e 0 m. ~ Choose is non-zero. u* d+r Ad(to-S) L i=d 0 u* EU so that only the Then for a constant (t -s) u, J i-d 0 u. ds • b" ~J (i-d) ! J Now t J e 0 d+r Ad(to-S) L i=d 0 as t o - ~ to hold by choosing a. Then there exists x d from such that ds l~oo (i-d) !

8) denotes the first component of the 0 B}l o = Similar arguments for the rest of the components t 0 ~ s ~ with respect to t o ~ s ~ vB which contradicts A(t -s) 0 Setting 0 s ve proved. 8) for all B}l , B o = s =t vB 0 . Differentiating we have A(t -s) 0 0 , hence AB vAB = = 0 O. 6 . 2) t -- then there is ~ hyperplane of t 0 < 0 0 O. 2) has rank 1ess than +Rn . Let a1 so true. +Rm , 1 ~ ~ m} . 8. 5 shows that the fo110w- ~pe then there is ~ hyperplane of n Let ' , complet~ng t h e proo f . 7. 6, those contro1s defined on is contained in .

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