English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We consider an optimal control problem constrained by a parabolic partialdifferential equation (PDE) with Robin boundary conditions. We use a well-posedspace-time variational formulation in Lebesgue--Bochner spaces with minimalregularity. The abstract formulation of the optimal control problem yields theLagrange function and Karush--Kuhn--Tucker (KKT) conditions in a naturalmanner. This results in space-time variational formulations of the adjoint andgradient equation in Lebesgue--Bochner spaces with minimal regularity.Necessary and sufficient optimality conditions are formulated and theoptimality system is shown to be well-posed.  Next, we introduce a conforming uniformly stable simultaneous space-time(tensorproduct) discretization of the optimality system in theseLebesgue--Boch\-ner spaces. Using finite elements of appropriate orders inspace and time for trial and test spaces, this setting is known to beequivalent to a Crank--Nicolson time-stepping scheme for parabolic problems.Differences to existing methods are detailed.  We show numerical comparisons with time-stepping methods. The space-timemethod shows good stability properties and requires fewer degrees of freedom intime to reach the same accuracy.

A Space-Time Variational Method for Optimal Control Problems: Well-posedness, stability and numerical solution