By V. I ArnolÊ¹d

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**Additional info for Contact geometry and wave propagation: Lectures given at the University of Oxford under the sponsorship of the International Mathematical Union **

**Example text**

Math. 1, 103–109 (1951) 15. : D´emonstration e´ l´ementaire du th´eor`eme de M. Borel sur les nombres absolument normaux et d´etermination effective d’un tel nombre. Bull. Soc. Math. France 45, 125–132 (1917) 16. UCSC Genome Browser. edu/goldenPath/hg19/chromosomes/ 17. : Normal numbers. D. thesis, University of California, Berkeley, CA (1949) Chapter 4 Optimality Conditions for Semivectorial Bilevel Convex Optimal Control Problems Henri Bonnel and Jacqueline Morgan Abstract We present optimality conditions for bilevel optimal control problems where the upper level is a scalar optimal control problem to be solved by a leader and the lower level is a multiobjective convex optimal control problem to be solved by several followers acting in a cooperative way inside the greatest coalition and choosing amongst efficient optimal controls.

2. Note that the terminal time t1 is fixed for the lower level problem, but it is a decision variable for the leader. e. when T = {t1 }. 3. (LL)(t1 ,ul ) may be also considered as the problem to be solved by the grand coalition of a p-player cooperative differential game (see [35] and its extensive references list) where the functional Ji and the control ui represent the payoff and the control of the player number i, i ∈ {1, . . , p}.

7) Then (x, u) ∈ (C × D) ∩ graA and xn , un → x, u . Proof. Set V = C − C, which is a closed linear subspace. Since xn − PC xn → 0, we have PC xn x and thus x ∈ C. Likewise, u ∈ D and hence C = x +V and D = u + V ⊥ . 8) It follows that PC : z → PV z + PV ⊥ x and PD : z → PV ⊥ z + PV u. H. 10h) = x, u . 4. 6. 5 generalizes [1, Theorem 2], which corresponds to the case when C is a closed linear subspace and D = C⊥ . 5 may be obtained from [1, Theorem 2] by a translation argument. 5 is different and much simpler than the proof of [1, Theorem 2].

### Contact geometry and wave propagation: Lectures given at the University of Oxford under the sponsorship of the International Mathematical Union by V. I ArnolÊ¹d

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