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**A survey of the maximum principles for optimal control problems with state constraints.**
*(English)*
Zbl 0832.49013

This paper provides a useful overview of the different approaches used to solve optimal control problems with phase constraints via the maximum principle. The methods differ on the way that the constraints are handled. The direct adjoining approach uses a Lagrange multiplier to attach the constraint to the Hamiltonian. The indirect adjoining approach initially finds the first time derivative of the phase constraint that explicitly contains a control term. It then joins this to the Hamiltonian by way of a Lagrange multiplier. Some other variants are also analysed and the necessary conditions that are obtained by each approach are compared.

Sufficient conditions are also discussed and the various approaches for the necessary conditions are highlighted through their application to several examples.

This is an important survey of methods for analysing optimal control problems with phase constraints in that it shows the relationships between the various methods. It also details the extent of the rigour of each approach. This provides a useful guide to researchers interested in placing these on a firmer foundation.

Sufficient conditions are also discussed and the various approaches for the necessary conditions are highlighted through their application to several examples.

This is an important survey of methods for analysing optimal control problems with phase constraints in that it shows the relationships between the various methods. It also details the extent of the rigour of each approach. This provides a useful guide to researchers interested in placing these on a firmer foundation.

Reviewer: J.Murray (Kensington)

### MSC:

49K15 | Optimality conditions for problems involving ordinary differential equations |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

49L99 | Hamilton-Jacobi theories |