A posteriori error estimates for discontinuous Galerkin time-stepping method for optimal control problems governed by parabolic equations.

*(English)*Zbl 1085.65054A class of convex distributed optimal control problems governed by linear parabolic equations is considered. A control constraint is also considered using the usual idea of an abstract closed convex set \(K\). The widely used case of quadratic control problems is included in that abstract framework. The discontinuous Galerkin method is used to approximate the solution. The discontinuous polynomial base is used in time discretization and the conforming finite element method in space discretization. A posteriori error estimates are obtained for both the state and the control approximations, assuming only that the corresponding space mesh is nondegenerate. The results are applied to examples in which the constraint set \(K\) is defined by some inequality. Next improved error estimates are derived for problems with control constraints of obstacle type in which structure information is available for the constraint set \(K\).

Reviewer: Viorel Arnăutu (Iaşi)

##### MSC:

65K10 | Numerical optimization and variational techniques |

49J20 | Existence theories for optimal control problems involving partial differential equations |

49M15 | Newton-type methods |