By Aaron Strauss (auth.)

This paper is meant for the newbie. it isn't a country of-the-art paper for learn staff within the box of regulate idea. Its function is to introduce the reader to a few of the issues and ends up in keep watch over idea, 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, resembling the utmost precept, have lengthy and tough proofs. i've got passed over those proofs. in most cases i've got incorporated basically the proofs that are both (1) now not too tough or (2) really enlightening as to the character of the outcome. i've got, even though, frequently tried to attract the most powerful end from a given facts. for instance, many current proofs on top of things thought for compact ambitions and specialty of suggestions additionally carry for closed goals 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, although, as i've got usually been content material simply 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 overseas summer time Schoo1 on Mathematica1 structures and Economics held in Varenna, Ita1y, June 1967.

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12 are was proved by NEUSTADT [3] Another nonlinear generalization of the bang-bang princip1e has been fiven by DILIBERTO [1] • 59 6. 5) for optimality, neither of which is as general as those whieh will be presented in §7 and However, in the linear time optimal case, §8. we are ab1e to see some of the geometrie aspeets of eontro1 which are not at all obvious in the general ease. 1 . 00 ) . ) be a continuous Let t D(t) { Jo y(s) u(s) ds n x m matrix 60 Then for each ------- t > 0 D(t) is compact and convex, and The compactness of be10ngs to {Yk} [0 functions on u (0) * exists D(t) fo11ows easi1y from the convexity of is harder to prove.

Then for each ------and hence ~B = K. , = KBBPC K. ,same time. Hence if there exists a time optimal control, there exists a bang-bang time optimal control. 6. ,-positive real parts and if rank n-l (B , AB , ••• ,A then every point in Rn ~ be steered to = B) x =0 n. ~!. piecewise constant, bang-bang control. The application of Corollary Every point in R2 is immediate. 6 to the railroad train example can be steered to the origin by a con- + land - land switching between them finitely many times. 5 admits two types of generalizations.

Is in- be the unit cube n c Then by Remark ~ng If x = 0 n , then 0 . 5 . 4 K (t ) (LA) o n, then for every -- t 0 < 0 . depends on whether or A with positive real part. u =0 be interior to the bounded set target. 2) Q have 39 rank Then n. PROOF. Q. eigenvalue of A has positive We may assurne that A is in Jordan canonical form. To see this, let be the Jordan form of A, where x = Ty into carries (LA) y = T is non-singular. Jy + T- l B u for which the controllability matrix has rank K = Rn if and only if for The transformation (LA).