Download e-book for kindle: Optimization Methods by Henning Tolle (auth.)

Download e-book for kindle: Optimization Methods by Henning Tolle (auth.)

By Henning Tolle (auth.)

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The second generalization, the severing of the assumption of autonomous functions, can be derived from the fundamental theorem, if we introduce an additional variable by (20) dXm + 1 . -- - -xm + 1 = 1 dt This yields a further differential equation as constraint, and we now have to consider an autonomous system with m + 1 variables. We shall not go further into details here. They are found in [ 26 1. In the Mayer problem we can either change over to the new independent variable P == P[x/tE),tEl, thus SdP, a "time-optimal" problem with respect to P, or we can put ~ L = - .

A (where 'Yj stands for 'Yj, 'Yj' the Lagrange multipliers. fj ) and the n -dimensionality is taken into account. The With (62) we have, from the Euler equation, the set of Euler equations: i = 1,2 ... n . For the Legendre condition we have the requirement n (64) and for the Weierstrass condition we have with ~ j=l a-y, J ay~ dYi = 0 1 AQ, Aj are called 30 (65) where (64), (65) are valid for a minimum of (59), and the index l' is used to distinguish it from the summation index i. The Erdmann-Weierstrass corner conditions state: (66) (£ - n ~ i=l ) .

2, in the interior of the region or, as in Fig. 3, on the boundary of the region. Here H(-) = 0 automatically follows for the highest point, a normation which is an immediate consequence of our overall procedure. Fig. 2. Maximum in the interior of the region B of the control function at the instant t = t 1 . 39 Admissible Area ~ Fig. 3. Maxim~m on the boundary of the region B of the control function at the instant t = t 1 . The interpretation of H(-) as an area whose maximum value characterizes the sought minimum of tE P= ~ L(xi,uI) dt suggests that in the presence of an arbitrary, nonoptimal solution UI(O) (t) we should tA seek an improvement by moving in the direction of the largest variation of H(-), that means in the direction of the line of descent given by grad H(- ).

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