 By Greg Knowles (Eds.)

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Extra resources for An Introduction to Applied Optimal Control

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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. 5. 31 Numerical Computation of the Switching Times Here we have and Hence Ax o = [ YOJ -x o and from Example 1, e-Atb = [-sintJ.

YEn Then the maximum principle can be stated. 35 + ... + CPnfn(x, u) 36 III. The Pontryagin Maximum Principle Tbeorem 1 (Pontryagin ) Suppose u* is an optimal control for the above problem and x* is the corresponding trajectory. Then there exists a nonvanishing function cp*(t) = (cpt(t), . . , cp:(t» and cp~ such that oH i = 1,2, ... n, (a) x:", = -OCPi = i(x* u*), 1 , (b)

In particular, can we find a function C(t), 0 ~ t ~ T, such that u*(t) = C(t)x*(t), o~ t s T? 1. The Maximum Principle 39 In attempting to do this, divide (4) and (5) by x*, x* x* = ex f32 q>* x*' +2 cfJ* q>* = 2a- ex­ x* x*' - and set d(t) = q>*(t)/x*(t). Then d= _(X*q>* - cfJ*x*) = _ x* q>* (X*)2 x* x* + cfJ* x* or • d= f32 2 d - 2exd 2 -- + 2a with d(T) = o. (6) To solve equation (6)(the Ricatti equation), make the change of variable then (6) becomes that is, ~ + 2exe - af32~ = 0. Since d(T) = 0, we have e(T) = 0, and since we are only interested in the ratio we can choose ~(T) = 1.