Numerical optimal control for nonlinear dynamical systems involving mixed-valued inputs and joint probability path-constraints

This paper proposes a numerical method for optimal control of nonlinear dynamical systems (OCNDSs) involving mixed-valued inputs (MVIs) and joint probability path-constraints (PPCs). By utilizing a transformation technology and a relaxation method, the original problem is first written as relaxation problem with only continuous-value inputs (CVIs). It is theoretically proved that the relaxation problem converges to the original problem. Then, a double-level approximation is proposed for the joint PPCs to obtain an approximate constrained optimal control problem (COCP) of the resulting relaxation problem and the convergence result of this approximation is established. Further, the resulting approximate problem is converted to a finite-dimensional problem (FDP) with bound constraints by using a control parametrization method, smoothing approximation, and penalty function. Following that, an improved sample approximation is proposed for transforming FDP into a deterministic problem while reducing computational costs. Finally, an improved greedy search strategy-based hybrid optimization algorithm (IGSSBHOA) is designed for OCNDSs based on the conjugate gradient method (CGM) and an improved greedy search strategy (IGSS), and its global convergence result is established. Anti-cancer drug scheduling (ACDS) is further generalized to demonstrate the validity of the proposed approach by considering the state variable/control input joint probability constraints. Simulation results illustrate that comparison against other typical approaches, the proposed approach not only has a better performance, but also is stable and robust.