Linear Programming

Juan A. Gomez-Fernandez, Francisco Guerra-Vazquez, Miguel A.'s Approximation and Optimization: Proceedings of the PDF

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By Juan A. Gomez-Fernandez, Francisco Guerra-Vazquez, Miguel A. Jimenez-Pozo, Guillermo Lopez-Lagomasino

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Read or Download Approximation and Optimization: Proceedings of the International Seminar, held in Havana, Cuba, January 12-16, 1987 PDF

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Extra info for Approximation and Optimization: Proceedings of the International Seminar, held in Havana, Cuba, January 12-16, 1987

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3. 0 60 Chapter 2. 3. 1 "( : t f---' x(t),'f] : t f---' y(t), x(t) > y(t), lal ~ A with A > 1. There is a constant M = M(A) such that Ix - yl ~ Mix - yl for all t E 1R. Proof. 4 assures that Let ~(t) = x(t) - y(t) > 0 in [-T, T]. It is enough to show I~(O)I < MI~(O)I because of the invariance of the problem with respect to time translation. Subtraction of the two Euler equations :tFp(t,x,x) - Fx(t,x,x) 0, ~ Fp(t, y, y) - 0, Fx(t, y, y) gives d· dt(Ao~ with Ao B C 11 11 11 + BO - . (y-x) d)". 2 , J , with)" = cOA 2 T- 1 , where Co is an F dependent constant 2: 1 and A 2: 1 is a bound for lal and Ix(T) - x( -T)I, ly(T) - y( -T)I.

I- 0, if Define for q -I- 0, 3 p,q = b : t ~ x(t) = ~t + ~(t) 1~ E W 1 ,2[t 1 , t2], ~(t + q) = ~(t) } with the vector space operations t + p6(t) E q P'Y1 'Y1 , Et +6(t) +6(t) , + 'Y2 q if 'Yj : t ~ ~t + ~j(t). The dot product makes 3 p ,q a Hilbert space. Definition. A minimal of the functional I(,,()= foqF(t,x,X)dt is called a periodic minimal of type (q, p) . We write M (q, p) for the set of periodic minimals of type (q, p). We will sometimes also abbreviate x E M (q, p) if 'Y E M (q, p) is given by 'Y : t ~ x(t).

If n = 1, the phase space is three dimensional. For a function h : (t, x) the graph ~ = {(t,x,y) E 0 x lRn I y = h(t,x) } 1--+ h( t, x) is a two-dimensional surface. Definition. The surface ~ is called invariant under the flow of H, if the vector field XH = is tangent to at + Hyox - Hxoy ~. 1. Let (n = 1). If x = 'lj;(t, x) is an extremal field for F, then ~ = {(t,x,y) EO x lR I y = Fp(t,x, 'lj;(t, x)) } is C 1 and invariant under the flow of H. On the other hand, if ~ is a surface which is invariant under the flow of H and has the form ~ = {(t,x,y) EO x lR I y = h(t,x)}, where h E C 1 (0), then the vector field x = 'lj;(t, x) defined by 'lj; is an extremal field.

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