This paper addresses the regularization of pointwise state constraints in optimal
control problems. By analyzing the associated dual problem, it is shown that the regularized problems
admit Lagrange multipliers in L2-spaces. Under a certain boundedness assumption, the solution of
the regularized problem converges to the one of the original state constrained problem. The results
of our analysis are confirmed by numerical tests.
An optimal control problem for a 2-d elliptic equation is investigated with pointwise control constraints.
This paper is concerned with discretization of the control by piecewise constant functions. The state and
the adjoint state are discretized by linear finite elements. Approximations of the optimal solution of the continuous
optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these
approximations have convergence order h2.
A class of optimal control problem for a semilinear elliptic partial differential equation
with control constraints is considered. It is well known that
sufficient second-order conditions ensure the stability of optimal solutions, the convergence of
numerical methods. Otherwise, such conditions are very difficult to verify (analytically or numerically).
We will propose a new approach: Starting with a numerical solution for a fixed mesh we will
show the existence of a local minimizer of the continuous problem. Moreover, we will prove that
this minimizer satisfies the sufficient second-order conditions.
An optimal control problem for a 2-d elliptic equation is investigated with pointwise control constraints.
This paper is concerned with discretization of the control by piecewise linear functions. The state and the
adjoint state are discretized by linear finite elements. Approximation of order h in the L1-norm is proved in the
main result.
A class of optimal control problems for a semilinear parabolic partial differential equation
with control and mixed control-state constraints is considered.
For this problem, a projection formula is derived
that is equivalent to the necessary optimality
conditions. As main result, the superlinear convergence of a semi-smooth Newton method is shown.
Moreover we show the numerical treatment and several numerical experiments.