By Greg Knowles (Eds.)

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M, the vectors {bj,Abj,A 2bj , . . ,A"-lbj } are linearly in dependent. lfthe function t E [0, t*], (3) is zero at more than a finite number of points, it must be identically zero, "Te-Atb j = 0 for all t E [0, t*], as (3)is an analytic function. 1) is not normal for somej, we must have "Te-Atbj == O. Substituting t = 0 gives "Tb j = O. ==O J 20 II. The General Linear Time Optimal Problem Similarly we can show "TArbj = ° for all r = 0,1,2, ... ,n - 1, and so {bj' Ab j, . . , An-Ib j} must be linearly dependent, that is, linear independence implies normality.

34 Il, The General Linear Time OptimalProblem 10. Show that the maximum of the Hamiltonian defined in Section 1 is constant in time if the control system i = Ax + Bu is autonomous. 11. Prove that a point Yl E flt(t) is hit by a unique trajectory if and only if y, is an extreme point of 9l(t). References [lJ D. Bushaw, Optimal discontinuous forcing terms, in "Contributions to the Theory [2J [3J [4J [5J [6J [7J of Non-linear Oscillations," pp. 29-52. Princeton Univ. Press, Princeton, New Jersey, 1958.

E-Atb = n L j= 1 e-Ajt(b' X)Xj . So we can expand (2) as n L Aj(XO • Xj)Xj = L [(b' X)Xj - 2e- Ajt1(b • X)Xj j= 1 + 2e- AjI2(b' X)Xj + .. : + Aj(Xo'Xj) = (b' xj)(l - 2e- Ajt, + 2e- Ajt2 - 2e Ajl3 + ... +( -1)'e Ajtr) for j = 1, 2, ... ) + [b : x 1)(1 - 2e- A,t, + 2e- A,t 2 + ... + (_lYe-A,t r) = 0 -Aix o' Xn) + (h : xn)(l - 2e- Ant, + 2e- Ant2 + '" + (_l)'e- Antr) = O. Note. It has been shown by Feldbaum [6, Chap. 3, Theorem 10] that if {AI' ... , An} are real, then r ~ n - 1. Take as an example the harmonic oscillator, Example 1, Section 4, x= Ax + bu, where x(O) = G:J.