Optimization—Theory and Applications pp 206-232 | Cite as

# Examples and Exercises on Optimal Control

Chapter

## Abstract

A point where we seek the minimum of the functional

*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)

*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