Pontryagin theory

Part of the Chapman and Hall Mathematics Series book series (CHMS)


For the discrete optimal control problem (DOC1) of 4.6, define the function
$$\matrix{ {H\left( {x,u,\lambda } \right) = \sum\limits_{k = 0}^N {[\psi (\mathop x\nolimits_k ,\mathop u\nolimits_k ,k) + \mathop \lambda \nolimits_k \phi (\mathop x\nolimits_k ,\mathop u\nolimits_k ,k)]} } \cr { = \sum\limits_{k = 0}^N {\left[ {{1 \over 2}\mathop x\nolimits_k^T \mathop A\nolimits_k \mathop x\nolimits_k + {1 \over 2}\mathop u\nolimits_k^T \mathop B\nolimits_k \mathop u\nolimits_k + \mathop \lambda \nolimits_k \left( {\mathop a\nolimits_k \mathop x\nolimits_k + \mathop b\nolimits_k \mathop u\nolimits_k } \right)} \right]} } \cr { \equiv \sum\limits_k {\mathop h\nolimits_k (\mathop x\nolimits_k ,\mathop u\nolimits_k ,\mathop \lambda \nolimits_k )} } \cr } $$
where x is the column vector of x 0, x 1, … ,x N , and similarly u of u 0, …, u N and λ of λ0, …, λ N . Suppose that the minimum of (DOC1) is reached at (x, u, λ) = \( (x,u,\lambda ) = (\xi ,\eta ,\tilde \lambda ) \).


Mathematical Program Convex Cone Positive Semidefinite Adjoint Equation Local Solvability 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Berkovitz, L.D. (1974), Optimal Control Theory, Springer, New York.Google Scholar
  2. Craven, B.D. (1977), Lagrangean conditions and quasiduality, Bull. Austral. Math. Soc., 16, 325–339.CrossRefGoogle Scholar
  3. Luenberger, D.G. (1969), Optimization by Vector Space Methods, Wiley, New York.Google Scholar

Copyright information

© B. D. Craven 1978

Authors and Affiliations

  1. 1.University of MelbourneAustralia

Personalised recommendations