On regularization of classical optimality conditions in convex optimal control

We discuss regularization of two classical optimality conditions—the Lagrange principle (PL) and the Pontryagin maximum principle (PMP)—in a convex optimal control problem for a parabolic equation with an operator equality constraint and distributed initial and boundary controls. The regularized Lagrange principle and the Pontryagin maximum principle are based on two regularization parameters. These regularized principles are formulated as existence theorems for the original problem of minimizing approximate solutions.