Examples and Exercises on Optimal Control

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


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:
$$dx/dt = y,dy/dt = u,0 \leqslant t \leqslant {t_2},u \in U = \left[ { - 1 \leqslant u \leqslant 1} \right],I\left[ {x,y,u} \right] = g = {t_2},{t_1} = 0,x\left( {{t_1}} \right) = a,y\left( {{t_1}} \right) = b,x\left( {{t_2}} \right) = 0,y\left( {{t_2}} \right) = 0,$$
where we seek the minimum of the functional I under the constraints.


Minimum Time Optimal Trajectory Balance Growth Path Navigation Problem Switch Line 
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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

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