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7. J. P . McDanell, and W. F . P o w e r s , N e c e s s a r y conditions for joining optimal singular and nonsingular s u b a r c s , SIAM J o u r n a l of Control, 9 , (1971). 8. N. X. Vinh, Optimal Singular Control With Applications to T r a j e c t o r y Optimization, NASA CR-3087, 1979 . 9. C. M a r c h a l , T h e o r e t i c a l R e s e a r c h in D e t e r m i n i s t i c Optimization, Pub. No. 139, 19 71 . ONERA CHAPTER 3 EQUATIONS OF MOTION 3. 1 INTRODUCTION In this chapter, we shall d e r i v e the equations of motion for use throughout the text.
12) as generated by the 26 Hamiltonian H * and H * respectively. 14) , we can w r i t e the coefficient of λ * / 8H * 8H * / 8F _ 2 _ _ 8F 2 I - » - ^ 8x = - D 2 F -* ->· 8p 8p D " ! -+■ | \ 8F\ 8t 8x / /8F / I v 2' -♦ 8x in this equation 8H * 1 _ > 8F ' 8p - ► 8p _ 8H * \ 1 | 8F\ ► / 8x 8t^ F Hence, we can write Eq. (2. 15) F along an optimal t r a j e c t o r y . 9) coupled with Eq. (2. 16) This condition, f i r s t derived in [ 4 ] , is a g e n e r a l i z a t i o n to nonautonomous s y s t e m of the condition given in [ 5] .
8F\ 8t 8x / /8F / I v 2' -♦ 8x in this equation 8H * 1 _ > 8F ' 8p - ► 8p _ 8H * \ 1 | 8F\ ► / 8x 8t^ F Hence, we can write Eq. (2. 15) F along an optimal t r a j e c t o r y . 9) coupled with Eq. (2. 16) This condition, f i r s t derived in [ 4 ] , is a g e n e r a l i z a t i o n to nonautonomous s y s t e m of the condition given in [ 5] . that, for F = Dx Φ = By comparing the E q s . 18) In the case where ΌΦ - D. Φ = D Φ = 0 at the t i m e t of the switching, the direction of the switching is decided upon analyzing higher o r d e r d e r i v a t i v e s of the switching function.