A transformation technique for optimal control problems with a state variable inequality constraint

Abstract

A slack variable is used to transform an optimal control problem with a scalar control and a scalar inequality constraint on the state variables into an unconstrained problem of higher dimension. It is shown that, for a p th order constraint, the p th time derivative of the slack variable becomes the new control variable. The usual Pontryagin principle or Lagrange multiplier rule gives necessary conditions of optimality. There are no discontinuities in the adjoint variables. A feature of the transformed problem is that any nominal control function produces a feasible trajectory. The optimal trajectory of the transformed problem exhibits singular arcs which correspond, in the original constrained problem, to arcs which lie along the constraint boundary.