Perturbation method, subdifferentials of nonsmooth analysis, and regularization of the Lagrange multiplier rule in nonlinear optimal control

We consider the regularization of classical optimality conditions — the Lagrange principle (LP) and the Pontryagin maximum principle (PMP) — in a regular parametric nonlinear (nonconvex) optimal control problem for a parabolic equation with boundary control and with an operator equality-constraint additively depending on the parameter (perturbation method). The set of admissible controls of the problem and the values of the operator defining the equality-constraint are embedded into the spaces of square-summable functions. The main purpose of the regularized LP and PMP is the stable generation of minimizing approximate solutions (MASs) in the sense of J. Warga in the problem under consideration. The regularized LP and PMP are formulated as existence theorems for MASs consisting of minimals (subminimals) of modified Lagrange functionals whose constructions are direct consequences of the subdifferential properties of a lower semicontinuous and, generally speaking, nonconvex value function as a function of the parameter of the problem. They “overcome” the ill-posedness properties of the LP and PMP, are regularizing algorithms, and serve as a theoretical basis for creating algorithms for the practical solution of an optimization problem.