Electron. J. Differential Equations, Vol. 2017 (2017), No. 189, pp. 1-40.

Galerkin approximations of nonlinear optimal control problems in Hilbert spaces

Mickael D. Chekroun, Axel Kroner, Honghu Liu

Nonlinear optimal control problems in Hilbert spaces are considered for which we derive approximation theorems for Galerkin approximations. Approximation theorems are available in the literature. The originality of our approach relies on the identification of a set of natural assumptions that allows us to deal with a broad class of nonlinear evolution equations and cost functionals for which we derive convergence of the value functions associated with the optimal control problem of the Galerkin approximations. This convergence result holds for a broad class of nonlinear control strategies as well. In particular, we show that the framework applies to the optimal control of semilinear heat equations posed on a general compact manifold without boundary. The framework is then shown to apply to geoengineering and mitigation of greenhouse gas emissions formulated here in terms of optimal control of energy balance climate models posed on the sphere $\mathbb{S}^2$.

Submitted April 2, 2017. Published July 28, 2017.
Math Subject Classifications: 35Q86, 35Q93, 35K58, 49J15, 49J20, 86-08.
Key Words: Nonlinear optimal control problems; Galerkin approximations; Greenhouse gas emissions; Energy balance models; Trotter-Kato approximations.

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Mickaël D. Chekroun
Department of Atmospheric & Oceanic Sciences
University of California
Los Angeles, CA 90095-1565, USA
email: mchekroun@atmos.ucla.edu
Axel Kröner
Institut für Mathematik
Humbolt-Universität Berlin
10099 Berlin, Germany
email: axel.kroener@inria.fr
Honghu Liu
Department of Mathematics
Virginia Polytechnic Institute and State University
Blacksburg, Virginia 24061, USA
email: hhliu@vt.edu

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