Some estimates for the jump of the derivative of the Lagrange multiplier function in optimal control problems with second-order state constraints

The optimal control problem for a nonlinear dynamic system of a cascade type with endpoint and irregular pointwise state constraints (the so-called state constraints of depth 2) is studied. This problem admits a refined formulation of Pontryagin’s maximum principle in terms of a (non-standard) Hamilton-Pontryagin function of the second order. The question of estimating the jump of the derivative of the Lagrange multiplier corresponding to the state constraint is studied. Some sufficient conditions have been obtained under which the maximum principle implies uniform in time estimates for the jump of the specified function. In particular, sufficient conditions have been given for the absence of a jump (i.e., continuous differentiability) of the multiplier. These results are based on the concepts of 2-regularity of the state constraint and the so-called regularity zone of the problem. The obtained estimates are of interest for the theory of Pontryagin’s maximum principle and can be used in practice, including the implementation of the known shooting method within the framework of one of the standard approaches to the numerical interpretation of the necessary optimality condition.