# Bifurcation of Extremals in Optimal Control by Jacob Kogan

By Jacob Kogan

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**Example text**

Then x(O) = x(s), and Eq. (30) reads as -J± = A(t)x, which is precisely system (1). 0 In other words, Vs is the dimension of Ker (R(s) - I), where R(t) is the matrizant of system (1). To say that qs is degenerate means that 1 is an eigenvalue of R(8). 7, this can happen only at isolated points. Definition 5. If Vs '" 0, we say that 8 is conjugate to 0 with multiplicity Vs. 0 We can also define conjugacy between any two points: 81 and 82 are conjugate ifthe equation ± = JA(t)x has a non-zero solution with x (81) = x (S2).

Definition 5. If Vs '" 0, we say that 8 is conjugate to 0 with multiplicity Vs. 0 We can also define conjugacy between any two points: 81 and 82 are conjugate ifthe equation ± = JA(t)x has a non-zero solution with x (81) = x (S2). It should be noted that this relation is not transitive: if (0,81) and (81, S2) are conjugate times, then 82 need not be conjugate to O. It should also be noted that the fact that 8 is conjugate to 0 does not imply that (8 + T) has the same property, even though A(t) is T-periodic.

Since is < 00, there must be finitely many 0'. 0 The next - and final - lemma studies more closely the discontinuities of the function s -+ is. Lemma 11. The function s -+ is is left continuous, and (41) Proof. We are dealing with a variable quadratic form qs on a variable space s). The first thing to do is to rescale everything to the fixed time interval L~(O, (0,1). Define a map p : L~(O, s) -+ L~(O, 1) by (PU)(t) = u(st). Then qs(u, u) = sq;(pu,pu), where q; is the quadratic form on L~(O, 1) defined by: (42) q;(v) = 111 2 0 [s (Jv, lllV) + (B(st)Jv, Jv)] dt .