State-Constrained Control-Affine Parabolic Problems I: First and Second Order Necessary Optimality Conditions
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State-Constrained Control-Affine Parabolic Problems I: First and Second Order Necessary Optimality Conditions M. Soledad Aronna1
3 ´ eric ´ Bonnans2 · Axel Kroner ¨ · J. Fred
Received: 12 September 2019 / Accepted: 28 September 2020 / © Springer Nature B.V. 2020
Abstract In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order necessary conditions relying on the concept of alternative costates and quasi-radial critical directions. The appendix provides an example illustrating the applicability of our results. Keywords Optimal control of partial differential equations · Semilinear parabolic equations · State constraints · Second order analysis · Control-affine problems The first author was supported by FAPERJ, CNPq and CAPES (Brazil) and by the Alexander von Humboldt Foundation (Germany). The second author thanks the ‘Laboratoire de Finance pour les March`es de l’Energie’ for its support. The second and third authors were supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, in a joint call with Gaspard Monge Program for optimization, operations research and their interactions with data sciences. This is the first part of a work on optimality conditions for a control problem of a semilinear heat equation. More precisely, the full version, available at https://arxiv.org/abs/1906.00237v1, has been divided in two, resulting in the current manuscript (that corresponds to Part I) and https://arxiv.org/abs/1909.05056 (which is Part II). M. Soledad Aronna
[email protected] J. Fr´ed´eric Bonnans [email protected] Axel Kr¨oner [email protected] 1
Escola de Matem´atica Aplicada FGV EMAp, Rio de Janeiro 22250-900, Brazil
2
Inria Saclay and CMAP, Ecole Polytechnique, CNRS, Universit´e Paris Saclay, 91128 Palaiseau, France
3
Weierstrass Institute for Applied Analysis and Stochastics, 10117 Berlin, Germany
M.S. Aronna et al.
1 Introduction This is the first part of two papers on necessary and sufficient optimality conditions for an optimal control problem governed by a semilinear heat equation containing bilinear terms coupling the control and the state, and subject to constraints on the control and state. The control may have several components and enters in an affine way in the cost. In this first part we derive necessary optimality conditions of first and second order, in the second part [1] sufficient optimality conditions are shown. In the context of second order conditions for problems governed by control-affine ordinary differential equations we can mention several works, starting with the early papers [2] by Goh and [3] by Kelley, later [4] by Dmitruk, and recently [
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