By Henning Tolle (auth.)

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Extra info for Optimization Methods

Sample text

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(- ).