Parametric dual regularization for an optimal control problem with pointwise state constraints

The perturbation method is used in the dual regularization theory for a linear convex optimal control problem with a strongly convex objective functional and pointwise state constraints understood as ones in $L_2$. Primary attention is given to the qualitative properties of the dual regularization method, depending on the differential properties of the value function ($S$-function) in the optimization problem. It is shown that the convergence of the method is closely related to the Lagrange principle and the Pontryagin maximum principle. The dual regularization scheme is shown to provide a new method for proving the maximum principle in the problem with pointwise state constraints understood in $L_2$ or $C$. The regularized Lagrange principle in nondifferential form and the regularized Pontryagin maximum principle are discussed. Illustrative examples are presented.