By Iu.P. Petrov (Eds.)

ISBN-10: 0125528507

ISBN-13: 9780125528504

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**Additional resources for Variational Methods in Optimum Control Theory**

**Sample text**

Let us fix all functions except one, y, = y , ( x ) , say, to which we shall assign the increment 6y, ( x ) . Then the variation of the functional (18) will depend only on one function, and exactly as in Section 5, the Euler equation for the function y, ( x ) aF d aF ay, dxdy,‘ will follow from the condition 6J = 0. - 0. 28 11. ;y , = y,(x) to yield the extremum of the functional (18), it is necessary that the functions y,(x) satisfy a system of differential equations, Euler equations : Fyn- d Fyn* =0 .

In principle, z may be expressed in terms of y and x from the coupling equation (40), be substituted into the integral (39), and the minimum of the usual functional of one variable might be sought. However, such elimination of variables often leads to very complex computations, and it is more convenient to utilize another method of solution, such as the Lagrange method of undetermined multipliers. The following simple mnemonic rule (theorem) exists: In order to find the extremum of the functional J = l X 1 F ( x ;y ; y ’ ; z ; z’) dx xo under the condition q ( x ; y ; z ) = 0, (41) (42) it is necessary to introduce an intermediate function H =F + A ( x ) V ( X ; y ; z), (43) where A(x)is a function of x as yet unknown, and to seek the extremum of the functional J1 = [ XI J xo H dx (44) by customary methods.

A curve passing through the points A and B and yielding the minimum distance has been found. This curve may be only the extremal. In fact, if this curve is not the extremal, then another curve may be drawn between these same points A and B on which the value of the functional will be less. But the integral term in (3) vanishes for an extremal and 6J = Fy4x=xl 6y1 + ( F - y’Fy*)lx=xl 6x1 - F y ’ l x = x o 6Yo - ( F - Y f ~ y 4 x = x6x0 o ’ Since to the accuracy of higher-order infinitesimals 6yo = CP’ (46x0 ; 6 y , = +‘(x) 6x1, the condition for the extremum 6J = 0 may then be written as 6J = (Fy# + F - f F y , ) l x = x ,6x1 - (Fyq’ + F - yfFyt)~x=xo 6x0.

### Variational Methods in Optimum Control Theory by Iu.P. Petrov (Eds.)

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