A generalized gradient method for optimal control problems with inequality constraints and singular arcs

The steepest descent methods of Bryson and Ho [1] and Kelly [6] and the conjugate gradient method of Lasdon, Mitter, and Waren [3] use control variables as the independent variables in the search procedure. The inequality constraints are often handled via penalty functions which result in poor convergence. Special difficulties are encountered in handling state variable inequality constraints and singular arcs [1]. This paper shows that these difficulties arise due to the exclusive use of control variables as the independent variables in the search procedure. An algorithm based on the generalized reduced gradient (GRG) algorithm of Abadie and Carpentier [5] and Abadie [7] for nonlinear programming is proposed to solve these problems. The choice of the independent variables in this algorithm is dictated by the constraints on the problem and could result in different combinations of state and control variables as independent variables along different parts of the trajectory. The gradient of the cost function with respect to the independent variables, called the generalized gradient, is calculated by solving a set of equations similar to the Euler-Lagrange equations. The directions of search are determined using gradient projection and the conjugate gradient method. Two numerical examples involving state variable inequality constraints are solved [2]. The method is then applied to two examples containing singular arcs and it is shown that these problems can be handled as regular problems by choosing some of the state variables as the independent variables. The relationship of the method to the reduced gradient method of Wolfe [4] and the generalized reduced method of Abadie [7] for nonlinear programming is shown.