On normal forms of necessary conditions of optimality for dynamic optimization problems with constraints

In this work, we discuss normal forms of necessary conditions of optimality (NCO) for optimal control problems subject to pathwise state constraints and for problems in the calculus of variations with inequality constraints. It is known that standard forms of the NCO may fail to provide information that is useful to identify optimal solutions, namely when the multiplier associated with the objective function takes the value zero. The normal forms of the NCO guarantee that the conditions remain always informative, which is of importance in critical applications where decisions based on optimization are taken, such as autonomous systems. Based on a previous nondegenerate maximum principle for optimal control problems, we extend the strengthness of these conditions to normality while applying them to the particular case of calculus of variations problems. We compare our results with existent normal forms of NCO for dynamic optimization problems and conclude that, when applied to calculus of variations problems, we may say that, under similar conditions, we can apply such result to a wider class of problems, having less regularity on the data.