Systems governed by ordinary differential equations arise in many applications as, e. g., in astronautics, aeronautics, robotics, and economics. The task of optimizing these systems leads to the optimal control problems investigated in this paper.
The aim is to find a control vector and the final time that minimize the functional
subject to a system of nonlinear differential equations
and inequality constraints
Here, the vector of control variables is denoted by and the vector of state variables is denoted by . The functions , and are assumed to be continuously differentiable. The controls are assumed to be bounded and measureable and may be fixed or free.