# Pontryagin theory

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

## Abstract

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) = $$(x,u,\lambda ) = (\xi ,\eta ,\tilde \lambda )$$.

## Keywords

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.

## References

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.
3. Luenberger, D.G. (1969), Optimization by Vector Space Methods, Wiley, New York.Google Scholar