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# Examples and Exercises on Optimal Control

• Lamberto Cesari
Part of the Applications of Mathematics book series (SMAP, volume 17)

## Abstract

A point P moves along the x-axis governed by the equation x″ = u with |u| ≤ 1. We are to take P from any given state x = a, x′ = b to rest at the origin x = 0, x′ = 0 in the shortest time. By introducing phase coordinates x = x, y = x′, we have the Mayer problem of minimum time:
$$dx/dt = y,dy/dt = u,0 \leqslant t \leqslant {t_2},u \in U = \left[ { - 1 \leqslant u \leqslant 1} \right],I\left[ {x,y,u} \right] = g = {t_2},{t_1} = 0,x\left( {{t_1}} \right) = a,y\left( {{t_1}} \right) = b,x\left( {{t_2}} \right) = 0,y\left( {{t_2}} \right) = 0,$$
(6.1.1)
where we seek the minimum of the functional I under the constraints.

## Keywords

Minimum Time Optimal Trajectory Balance Growth Path Navigation Problem Switch Line
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag New York Inc. 1983

## Authors and Affiliations

• Lamberto Cesari
• 1
1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA