By H. W. Knobloch
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Additional resources for Higher Order Necessary Conditions in Optimal Control Theory
Control) and ~ ( I ) for t~[~(1),t] = ~. 3) (~,a : 0 ~ o , on the interval the collection property. ;~,a) ~ - a of the C~-function II a-Xol! ~e} x ( t ; X , x o) = x ( ~ ) expansion + xSp + integer Furthermore, t6[to,te] ~ for any having the following ~ function of Xs + l ) , (which may depend upon p). , we denote by property. the collection There exists a [to,to] _ - which may depend upon ~ -~ = ~ on ~ such that teO,p(t) p(') of subinter- - and a continuous and p ( t ) 6 ~ for t 6 3. Remarks. 6) vall This follows by standard such that the solution and admits the asymptotic all (= be t h ~ initial value of the trajectory.
1 - or rather - dependent set of the form holds in the form a representation from Lemma parameter x(~;l,Xo) It is easy to see that the above = f(t,x;u(t)),fi(t,x,l) therefore = a can also be written side admits fo(t,x,l) by X(to) fo(',l) ..... 7). 1 The Taylor-expansion ~t I = 0) of and arrive x(t;A,x o) is given by the formal power series C (t,x,zl (~),.. vZN(1), where the sequences (at t--~) u(t,t; Ui(t,A) Uo (t, I) ..... 1 that these Uo(t) respectively. 18) for of some ui(t,X) however at U(t).
UN) x). N=I, then Uo , UI) = x + ~. h(~)(t, x; U O, U1)z ~ I ~=1 (cf. 1). 16) that - because of the hypothesis of the lemma - the n-th of h (v) is zero, for v = 1,2, .... component This proves the lemma in case N=I. 8) for M = N-1 . 11), hence the n-th component is equal to the n-th component of x'. 12) Here x = f(t,x;u), x n+l x n+S = 1. is an additional state varlabel. 13) C*(t,x ,Zl, .... ZN; which is a tJo, Ul,... , U N ) (n+1)-dimensional vector. 2 holds for f* is C* (with xJa+1 xn). 12) of the system in question gives rise to a stronger statement.