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Error estimates for control problems governed by semilinear equations in the absence of Thikonov regularization
Prof. Dr. Mariano Mateos | Universidad de Oviedo | Spain
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Prof. Dr. Mariano Mateos | Universidad de Oviedo | Spain
Error estimates for the numerical approximation of optimal control problems are well studied in the case of functionals with a Thikonov regularization term. Nevertheless, the techniques for the obtention of these results rely strongly on the positiveness of the Thikonov regularization parameter. For instance, thanks to this positiveness, we can write the control explicitly in terms of the adjoint state and second order sufficient conditions can be
expressed in terms of the L2 norm of the control.
In this talk we show some techniques to obtain error estimates in the absence of such regularization term. We first recall the concepts of weak and strong local optimality and provide adequate second order sufficient conditions for strong optimality, which are the key to obtain error estimates when the functional does not have a Thikonov regularization term. These conditions allow us to obtain error estimates for the state variable. When the optimal control is
bang-bang, we are also able to deduce error estimates for the control. Both the elliptic and parabolic cases are investigated, different discretization concepts are considered and the case of sparse controls is also discussed.