The Lagrange principle and the Pontryagin maximum principle in ill-posed optimal control problems

We consider the regularization of the classical optimality conditions—the Lagrange principle and the Pontryagin maximum principle—in a convex optimal control problem for a parabolic equation with distributed and boundary controls, and also with a finite number functional equality constraints given by ‘`point’ functionals nondifferentiable in the Fréchet sense, which are the values of the solution of the third initial-boundary-value problem for the specified equation at preselected fixed (possibly boundary) points of the cylindrical domain of the independent variables.