Indirect Solution of Inequality Constrained and Singular Optimal Control Problems Via a Simple Continuation Method

This paper develops a simple continuation method for the approximate solution of optimal control problems. The class of optimal control problems considered include (i) problems with bounded controls, (ii) problems with state variable inequality constraints (SVIC), and (iii) singular control problems. The method used here is based on transforming the state variable inequality constraints into equality constraints using nonnegative slack variables. The resultant equality constraints are satisfied approximately using a quadratic loss penalty function. Similarly, singular control problems are made nonsingular using a quadratic loss penalty function based on the control. The solution of the original problem is obtained by solving the transformed problem with a sequence of penalty weights that tends to zero. The penalty weight is treated as the continuation parameter. The paper shows that the transformed problem yields necessary conditions for a minimum that can be written as a boundary value problem involving index-1 differential–algebraic equations (BVP-DAE). The BVP-DAE includes the complementarity conditions associated with the inequality constraints. It is also shown that the necessary conditions for optimality of the original problem and the transformed problem differ by a term that depends linearly on the algebraic variables in the DAE. Numerical examples are presented to illustrate the efficacy of the proposed technique.