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__Optimal control of multiscale systems: an approach using logarithmic transformations__

C. Hartmann, W. Zhang, J. Latorre and G. Pavliotis

**Abstract**

Computing optimal controls of multiscale systems is often prohibitive, especially in high dimensions. A way out is to replace the equations of motion by reduced-order models that do not contain the microscales and which can be obtained by, e.g., homogenisation or averaging. But even though the reduced model may be a good approximation if the controls are known, the question is whether one can also compute an optimal control from it that approximates the optimal control of the original system. In most cases the answer is no, but for a certain class of multiscale diffusions this questions can be answered in the affirmative. The talk will be about this particular class of stochastic control problems that are linear quadratic in the control (though nonlinear in the states) and can be transformed to a control-free problem by a logarithmic transformation. We give a theoretical foundation of the method and discuss several numerical examples and applications.