Multiplier Methods for Nonlinear Optimal Control

Error estimates are derived for an augmented Lagrangian approximation to an optimal control problem. Convex control constraints are treated explicitly while a Lagrange multiplier is introduced for the nonlinear system dynamics. The nonlinear optimal control problem does not fit the classical theory for estimating the error in multiplier approximations, since the natural coercivity assumption is formulated in a Hilbert space where the cost is not differentiable. This discrepancy between the function space setting needed for coercivity and that needed for differentiability is compensated for by regularity results associated with the necessary conditions. The paper concludes with an analysis of the optimal penalty parameter corresponding to a given finite-element discretization.