Optimal A Priori Error Estimates of Parabolic Optimal Control Problems with Pointwise Control

Abstract

In this paper we consider a parabolic optimal control problem with a pointwise (Dirac type) control in space, but variable in time, in two space dimensions. To approximate the problem we use the standard continuous piecewise linear approximation in space and the piecewise constant discontinuous Galerkin method in time. Despite low regularity of the state equation, we show almost optimal h 2 + k convergence rate for the control in L 2 norm. This result improves almost twice the previously known estimate in [W.

On the Chernous'ko Time-Optimal Problem for the Equation of Heat Conductivity in a Rod

“…Despite 70 years of development, the solution of concrete non-trivial examples of time-optimal control still needs considerable effort [2,4,19]. The problem becomes even more difficult when a control system is described by a partial differential equation [11,24,25,34], particularly, for the heat conductivity equation [12,22,26,29,35,36]. In [1], the correctness of parabolic equations for heat propagation is discussed and for that purpose, a parabolic equation with time delay is considered.…”

Discrete Maximal Parabolic Regularity for Galerkin Finite Element Methods for Nonautonomous Parabolic Problems

“…For time-independent coefficients the above result is well understood [4,6,7,18,19], however for time-dependent coefficients it is still an active area of research [3,8,9,17]. The maximal parabolic regularity is an important analytical tool and has a number of applications, especially to nonlinear problems and optimal control problems when sharp regularity results are required (cf., e.g., [21,26,27,28,31]).…”

Finite Element Approximations of Parabolic Optimal Control Problems with Controls Acting on a Lower Dimensional Manifold

“…In [21] the authors studied the finite element approximations to elliptic control problems with controls acting on a lower dimensional manifold, here we would like to generalise the results to parabolic case where the manifold may evolve in the time horizon. As a special case, Gong, Hinze and Zhou studied in [19] the finite element approximations to pointwise control of parabolic equations with control acting on finitely many spatial points which are independent of the time, the error estimates presented there are subsequently improved by Leykekhman and Vexler in [30] for two dimensional case. For optimal controls with compact support and sparsity we should mention the work of Kunisch and coauthors, who studied in [7], [8] and [28] the elliptic and parabolic optimal control problems in measure space.…”

“…The controls for traditional control problems studied in the literature act on a subdomain of Ω, thus the error estimates involve only global errors. For the control problems acting on lower dimensional manifold, one needs to rely on some local error estimates to derive improved error estimates compared to traditional techniques (see [30]). As indicated in [21], when the dimension of manifold γ(t) is one order lower than that of Ω, we are able to derive optimal a priori error estimates up to a logarithmic factor.…”

“…Otherwise, we derive a priori error estimates for the control with reasonable reduced order. In this paper, we derive a priori error estimates for the optimal control problems in both two and three dimensions, but only optimal error estimates are obtained in two dimension following the idea of [30]. The main results of this paper are summarised as follows.…”