Abstract
The aim of this work is to study the optimality conditions and the adaptive multi-mesh fully discrete finite-element schemes for quadratic nonlinear parabolic integro-differential optimal control problems. We derive a posteriori error estimates in L 2 (J; H 1 (Ω))-norm and L 2 (J; L 2 (Ω))-norm for both the coupled state and control approximation. Such estimates can be used to construct reliable adaptive finite-element approximation for nonlinear parabolic integro-differential optimal control problems.MSC:49J20, 65N30.
Highlights
1 Introduction Parabolic integro-differential optimal control problems are very important for modeling in science
The finiteelement approximation of an optimal control problem by piecewise constant functions has been investigated by Falk [ ] and Geveci [ ]
The discretization for semilinear elliptic optimal control problems is discussed by Arada et al in [ ]
Summary
Parabolic integro-differential optimal control problems are very important for modeling in science. They have various physical backgrounds in many practical applications such as population dynamics, visco-elasticity, and heat conduction in materials with memory. The finite-element approximation of parabolic integro-differential optimal control problems plays a very important role in the numerical methods for these problems. The discretization for semilinear elliptic optimal control problems is discussed by Arada et al in [ ]. In [ ], Brunner and Yan analyzed the finite-element Galerkin discretization for a class of optimal control problems governed by integral equations and integro-differential equations. Systematic introductions of the finite-element method for optimal control problems can be found in [ – ]
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