Counterexamples in Optimal Control Theory by S. Ia Serovaiskii, Semen Ya Serovaiskii

By S. Ia Serovaiskii, Semen Ya Serovaiskii

This monograph bargains with situations the place optimum keep watch over both doesn't exist or isn't precise, situations the place optimality stipulations are inadequate of degenerate, or the place extremum difficulties within the feel of Tikhonov and Hadamard are ill-posed, and different events. a proper program of classical optimisation equipment in such situations both results in incorrect effects or has no impact. The distinctive research of those examples should still supply a greater figuring out of the trendy idea of optimum keep watch over and the sensible problems of fixing extremum difficulties

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In this section, we study this phenomenon, and obtain results of Kiernan [Ki 2] and Kiernan-Kobayashi [K-K 2]. We follow both. For the rest of this section we let X be a complex subspace of a complex space Y. 32 HYPERBOLIC IMBED DINGS [II, §lJ We shall need to measure derivatives, so we suppose given a length function on Y. As far as we are concerned, the reader may assume that Y is a closed subspace of a complex manifold, and norms of derivatives can be measured in terms of a length function on this manifold.

A closed complex subspace of a complete hyperbolic space is complete hyperbolic. Let n: X ~ Y be a holomorphic map of complex spaces. Assume that Y is complete hyperbolic, and that to each y E Y there is a neighborhood V such that n-l(V) is complete hyperbolic. Then X is complete hyperbolic. Let n: X' ~ X be a covering. Then X is complete hyperbolic if and only if X' is complete hyperbolic. Proof. The first three assertions are immediate. 5, taking d~ = dy. Finally, we consider a covering as in (e).

Let x' E BH(x, s) n X and y' E BH(y, s) n X. By HI4 we know that dx(x', y') ~ dH(X ' , y') ~ s. This proves HIS. It is obvious that HI 5 implies HI 1. This concludes the proof of the equivalence between the five properties. Remarks. X is hyperbolic itself. This is obvious from HI 1. if and only if X is hyperbolically imbedded in [II, §1] DEFINITION BY EQUIVALENT PROPERTIES 35 If Xl is hyperbolically imbedded in Y1 and X 2 is hyperbolically imbedded in Y 2 then Xl X X 2 is hyperbolically imbedded in Y 1 x Y 2 • The proof is immediate, using the fact that the projection on each factor of the product is Kobayashi distance decreasing.

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