By Greg Knowles (Eds.)

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**Example text**

6) Equations (5) and (6) are just a linear homogeneous ordinary differential equation for the function ~, with zero initial data, and, by the uniqueness of the solutions for such problems, we must have ~(t) = for all t E [0, t*J, which contradicts the definition of normality. • ° 4. FURTHER EXAMPLES OF TIME OPTIMAL CONTROL Example 1 Consider the control problem Xl = X2 = X 2, -Xl + U, We wish to reach the origin in minimum time. lui :$ 1. (1) 4. , A= [ 0 b=[~l IJ -1 0' Ab = [~l hence {b, Ab} are linearly independent and the system is normal.

A= [ 0 b=[~l IJ -1 0' Ab = [~l hence {b, Ab} are linearly independent and the system is normal. Next X(t)=eAI=[ c~st e-A1b= [-sintJ. cos t sintJ, cos t - SIn t By the maximum principle, optimal controls must be of the form u*(t) = sgn( -'It sin t + 112 cos t) (111' I1z) ;/- (0,0) or u*(t) = sgn(sin(t + 0)) for some -n ~ 0 ~ tt. That is, the optimal control is unique, bang-bang, and its switches occur exactly n seconds apart. When u = + 1, Xl = Xz, Xz = -Xl dX I X dX2 -Xl 1- Xl = X + 1, z +1 acost, z = asint, and (1 - xddx I X~ + (1 - XI)Z = XZdX2' = a2 , which is a circle centered at (1,0).

0, that is, x*(t) = xoe T- 1 since x* is fT (1 - u*)x*(t)dt = x oeT- 1 • JT-l 48 OI. The Pontryagin Maximum Principle Example 3 An Inventory Control Problem (Bensoussan et al. [IJ) Consider an inventory of items being controlled over a time period [0, TJ. If d(t) represents the demand rate for these items at time t, and u(t) the production rate at time t, then the rate of change of the inventory level x(t) will be just x(t) = u(t) - d(t), = Xo, x(o) 05,t5,T. (2) Suppose the firm chooses desired levels Ud(t), xit), 05, t 5, T, for the production and inventory.