Using logarithmic penalties in the shooting algorithm for optimal control problems

The paper deals with optimal control problems of ordinary differential equations with bound control constraints. We analyse the logarithmic penalty method for converting the problem into an unconstrained one, the latter being solved by a shooting algorithm. Convergence of the value function and optimal controls is obtained for linear quadratic problems, and more generally when the control variable enters linearly in the state equation and in a quadratic way in the cost function. We display some numerical results on two examples: an aircraft maneuver, and the stabilization of an oscillating system.