Calculation of Extremals in an Optimal Control Problem with a Higher-Order State Constraint

The problem of controlling the kth derivative of an object state under a linear state constraint, where $$k$$ is an arbitrary natural number, is studied. According to the existing terminology in literature, this is a so-called state-constrained control problem of order $$k$$ (the term “of depth $$k$$ ” is also used). This paper applies Pontryagin’s maximum principle to the problem under study and conducts a theoretical analysis of the resulting optimality conditions. Based on this analysis, a computational scheme for finding extremals is proposed.