joint work with Eduardo Casas Rentería
A problem of sparse optimal control for a semilinear parabolic equation is considered, where pointwise bounds on the control and mixed pointwise control-state constraints are given. A quadratic objective functional is to be minimized that includes a Tikhonov regularization term and the L1-norm of the control that accounts for sparsity. Special emphasis is laid on existence and regularity of Lagrange multipliers associated with the mixed control-state constraints. Since the interior of the underlying non-negative cone is empty, the duality theory for linear programming problems in Hilbert spaces is applied in a special way to prove the existence of Lagrange multipliers. First-order necessary optimality conditions are established and discussed up to the sparsity of optimal controls.
joint work with Irwin Yousept, Jun Zou
In this talk, I introduce an adaptive edge element method for a quasilinear H(curl)-elliptic problem in magnetostatics, based on a residual-type a posteriori error estimator and general marking strategies. The error estimator is shown to be reliable and efficient, and its resulting sequence of adaptively generated solutions converges strongly to the exact solution of the original quasilinear system. Numerical experiments are provided to verify the validity of the theoretical results.
Physical phenomena in electromagnetism can lead to hyperbolic variational inequalities with a Maxwell structure. They include electromagnetic processes arising in polarizable media, nonlinear Ohm's law, and high-temperature superconductivity (HTS). In this talk, we present well-posedness and regularity results for a class of hyperbolic Maxwell variational inequalities of the second kind, under different assumptions.