Legendre Pseudospectral Approximations of Optimal Control Problems

  • I. Michael Ross
  • Fariba Fahroo
Part III: Nonlinear Optimal Control
Part of the Lecture Notes in Control and Information Science book series (LNCIS, volume 295)


We consider nonlinear optimal control problems with mixed statecontrol constraints. A discretization of the Bolza problem by a Legendre pseudospectral method is considered. It is shown that the operations of discretization and dualization are not commutative. A set of Closure Conditions are introduced to commute these operations. An immediate consequence of this is a Covector Mapping Theorem (CMT) that provides an order-preserving transformation of the Lagrange multipliers associated with the discretized problem to the discrete covectors associated with the optimal control problem. A natural consequence of the CMT is that for pure state-constrained problems, the dual variables can be easily related to the D-form of the Lagrangian of the Hamiltonian. We demonstrate the practical advantage of our results by numerically solving a state-constrained optimal control problem without deriving the necessary conditions. The costates obtained by an application of our CMT show excellent agreement with the exact analytical solution.


Optimal Control Problem Closure Condition Jump Condition Pseudospectral Method Naval Postgraduate School 
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Authors and Affiliations

  • I. Michael Ross
    • 1
  • Fariba Fahroo
    • 2
  1. 1.Department of Aeronautics and Astronautics, Code AA/Ro, Naval Postgraduate School, Monterey, CA 93943 
  2. 2.Department of Applied Mathematics, Code MA/Ff, Naval Postgraduate School, Monterey, CA 93943 

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