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We use a space\u2013time variational formulation in Lebesgue\u2013Bochner spaces yielding a boundedly invertible solution operator. The abstract formulation of the optimal control problem yields the Lagrange function and Karush\u2013Kuhn\u2013Tucker conditions in a natural manner. This results in space\u2013time variational formulations of the adjoint and gradient equation in Lebesgue\u2013Bochner spaces, which are proven to be boundedly invertible. Necessary and sufficient optimality conditions are formulated and the optimality system is shown to be boundedly invertible. Next, we introduce a conforming uniformly stable simultaneous space\u2013time (tensorproduct) discretization of the optimality system in these Lebesgue\u2013Bochner spaces. Using finite elements of appropriate orders in space and time for trial and test spaces, this setting is known to be equivalent to a Crank\u2013Nicolson time-stepping scheme for parabolic problems. Comparisons with existing methods are detailed. We show numerical comparisons with time-stepping methods. The space\u2013time method shows good stability properties and requires fewer degrees of freedom in time to reach the same accuracy.<\/jats:p>","DOI":"10.1007\/s10589-023-00507-x","type":"journal-article","created":{"date-parts":[[2023,7,25]],"date-time":"2023-07-25T19:01:43Z","timestamp":1690311703000},"page":"767-794","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A space\u2013time variational method for optimal control problems: well-posedness, stability and numerical solution"],"prefix":"10.1007","volume":"86","author":[{"given":"Nina","family":"Beranek","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7842-2516","authenticated-orcid":false,"given":"Martin Alexander","family":"Reinhold","sequence":"additional","affiliation":[]},{"given":"Karsten","family":"Urban","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,7,25]]},"reference":[{"key":"507_CR1","volume-title":"Optimal Control of Partial Differential Equations: Theory, Methods, and Applications","author":"F Tr\u00f6ltzsch","year":"2010","unstructured":"Tr\u00f6ltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods, and Applications. 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The authors have no competing interests to declare that are relevant to the content of this article. All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript. Furthermore the authors have no financial or proprietary interests in any material discussed in this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}