Game Theory

Download PDF by Greg Knowles (Eds.): An Introduction to Applied Optimal Control

By Greg Knowles (Eds.)

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OH qJz = - oS = -( -qJd = qJl' qJl(T) = qJz(T) = 0, as this is a free-end-point problem. Solving the adjoint equations gives C1 = -hT, so qJl(t) = ht - hT, ¢z(t) = qJl(t) = ht - hT, qJit) = ht Z 2 - (hT)t + c z ; qJz(T) = 0. (t), qJ~(t», l'sPsP for all t E (0,T). (t) - AqJ~(t». m. The Pontryagin MaximumPrinciple 46 the optimal control for 0 t ~ ~ T is given by p P*(t) = _upnknown { when W»O when when ~(t) = 0 ~(t) < 0; and from our remarks above, ht 2 hT2) W) = -c + ht - hT - A( 2: - (hT)t + -2­ Aht2 AhT 2 = --2- + (h + AhT)t - -2- - hT - c.

2. CLASSICAL CALCULUS OF VARIATIONS The oldest problem in the calculus of variations is probably the following: Minimize the integral g f(y(x), y'(x»dx over all differentiable func­ tions y passing through y~O) = Yo, y(T) = Yl' with y' piecewise con­ tinuous. We can solve this problem with the maximum principle. To do this, we just reformulate it as a control problem. Set y' = u, and since y' is not constrained, we take A, the set of admissible controls, to be all piecewise continuous functions.

8). Then we move - b seconds around the arc of the trajectory corresponding to u = + 1, which passes through the origin. At t = - b, sin(t + b) changes sign and we switch the control to u = -1. Since u = - 1, the optimal trajectory is circular with center at ( -1, 0),and passing through P i - We travel along this circle for rt seconds, in which time we traverse exactly a semicircle. (From Fig. ) After n seconds we shall reach the point P 2, which by symmetry is just the reflection of P 1 onto the circle with radius 1 and center (-1, 0).