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By Edwin Hewitt, Karl Stromberg

Hewitt E., Stromberg ok. actual and summary research (Springer, 1965, 1975 printing)(ISBN 0387901388)

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K i ;t /t2N 2 RN . 1;t /t2N 2 RN , . . , . k e ;t /t2N 2 RN which satisfy the following conditions: . 0; 0/. 0. xO t ; uO t / xO tC1 i D 0 when a D i . For all t 2 N, for all ˛ 2 f1; : : : ; k i g, ˛;t 0. Out / D 0. xO t ; uO t /. For all t 2 N, ki ke P P ˇ ˛ ut / C ut / D 0. O (i) (ii) (iii) (iv) (v) (vi) (vii) ˛D1 ˇD1 Proof. T; Á; xO T /) for a 2 fe; i g. 4 hold when a D i . 1), we see that . 0; 0/ implies . 0; 0; : : : ; 0/. And so by contraposition we obtain the following relation. 0; 0; : : : ; 0/ H) .

Pan ), or of (Pas ), or of (Pao ), or of (Paw ) when a 2 fe; i g. We assume that the following conditions are fulfilled: (a) For all t 2 N, Xt is a nonempty open convex subset of Rn and Ut is a nonempty subset of Rd . (b) For all t 2 N, the functions t and ft are differentiable with respect to the first vector variable. xt ; xtC1 /. 36) holds. Rn /N which satisfy the following . 0; 0/. 0. xO t ; uO t / xO tC1 i D 0 when a D i . xO t ; uO t ; ptC1 ; 0 /. xO t ; u; ptC1 ; 0 /. u2Ut Proof. The case a D e.

7. Pin ), or of (Pis ), or of (Pio ), or of (Piw ). xO t ; uO t /. xO t ; uO t /. (c) For all t 2 N, Ut is closed and Clarke-regular at uO t . 14) holds. Rn /N which satisfy the following . 0; 0/. 0. xO t ; uO t xO tC1 i D 0. xO t ; uO t ; ptC1 ; 0 /. Out / is the normal cone of Ut at uO t . Proof. 3. 5, which t 0 t T implies (cf. 5. 7 we obtain the following result. 8. Pin ), or of (Pis ), or of (Pio ), or of (Piw ). xO t ; uO t /. xO t ; uO t /. (c) For all t 2 N, uO t 2 int Ut . 14) holds. 3 Strong Pontryagin Principles in Infinite Horizon (i) (ii) (iii) (iv) (v) .

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