Error estimates for the numerical approximation of boundary semilinear elliptic control problems
(2004)
We study the numerical approximation of boundary optimal control problems governed
by semilinear elliptic partial differential equations with pointwise constraints on the control.
The analysis of the approximate control problems is carried out. The uniform convergence of discretized
controls to optimal controls is proven under natural assumptions by taking piecewise constant
controls. Finally, error estimates are established.
Optimality conditions for a class of optimal control problems with quasilinear elliptic equations
(2008)
A class of optimal control problems for quasilinear elliptic
equations is considered, where the coefficients of the elliptic
differential operator depend on the state function. First- and
second-order optimality conditions are discussed for an associated
control-constrained optimal control problem. In particular, the
Pontryagin maximum principle and second-order sufficient
optimality conditions are derived. One of the main difficulties is
the non-monotone character of the state equation.
Second-order sufficient optimality conditions are established for the optimal control
of semilinear elliptic and parabolic equations with pointwise constraints on the control and the state. In
contrast to former publications on this subject, the cone of critical directions is the smallest possible in the
sense that the second-order sufficient conditions are the closest to the associated necessary ones. The theory
is developed for elliptic distributed controls in domains up to dimension three. Moreover, problems of elliptic
boundary control and parabolic distributed control are discussed in spatial domains of dimension two and one,
respectively.
A theorem on error estimates for smooth nonlinear programming
problems in Banach spaces is proved that can be used to derive
optimal error estimates for optimal control problems. This theorem is applied
to a class of optimal control problems for quasilinear elliptic equations.
The state equation is approximated by a finite element scheme, while different
discretization methods are used for the control functions. The distance of
locally optimal controls to their discrete approximations is estimated.