Optimal Control Problem with Regular Mixed Constraints via Penalty Functions

Below we derive necessary conditions of optimality for problems with mixed constraints (see Dmitruk in Control Cybern 38(4A):923–957, 2009) using the method of penalty functions similar to the one we previously used to solve optimization problems for control sweeping processes (see, e.g., De Pinho et al. in Optimization 71(11):3363–3381, 2022) and, more recently, to solve optimal control problems with pure state constraints (see De Pinho et al. in Syst Control Lett 188:105816, 2024). We intentionally consider a smooth case and the simplest boundary conditions; we consider global minimum and assume that the set of trajectories of the control system is compact. Based on our penalty functions approach we develop a numerical method admitting estimates for its parameters needed to achieve a given precision.