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The main purpose of necessary conditions of optimality (NCO) is, given an optimization problem, to select a set of candidates to minimizer. However, for some optimization problems with constraints in particular for optimal control problems , it may happen that the NCO are unable to provide any useful information; that is, the set of candidates to minimizers that satisfy certain NCO coincides with the set of all admissible solutions. When this happens, we say that the degeneracy phenomenon occurs. To avoid this phenomenon, stronger forms of NCO are developed. These stronger forms can be applied when the problem satisfies additional hypotheses, known as constraint qualifications. In the case of optimal control problems with state constraints that have higher index (i.e. the first derivative of the state constraint with respect to time does not depend on the control), most constraint qualifications described in literature are not adequate. We note that control problems with higher index state constraints arise frequently in practice. An example, explored here, is a common mechanical systems for which there is a constraint on the position (an obstacle in the path, for example) and the control acts as a second derivative of the position (a force or acceleration) which is a typical case arising in the area of mobile robotics. So, there is a need to develop new constraint qualifications, involving higher derivatives of the state constraint. In this paper, we make an overview of recent constraint qualifications that allow strengthened forms of the NCO for optimal control with higher index state-constraints. |
