What is it about?
Mean field models provide a limit model for dynamics with a large number of interacting agents as the number N of agents goes to infinity. For each fixed N, for objective functionals with a certain structure the optimal states and optimal controls have a turnpike structure, that is they approach static states for large time horizons. The static states can be characterized as solutions of static optimal control problems. We show that also for the optimal controls and optimal states in the mean field limit, a turnpike structure is visible.
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Why is it important?
It is not easy to solve problems of dynamic optimal control. The turnpike phenomen provides valuable information about the structure of the solution that we can expect. This information is helpful for example to choose an initial guess for the optimal control to start an iterative scheme. Moreover, it implies that for large time horizons there is only a little loss in optimality if the first transient phase of the optimal control process is replaced by a control that is governed by an appropriate feedback law that steers the initial state towards the static equilibrium.
Perspectives
Our contribution will serve as a starting point for further analysis of the turnpike phenomenon for mean-field optimal control problems. For each N, an exponential turnpike property holds where the convergence towards the static equilibrium is exponentially fast. We expect that a result of a similar type also holds for the mean-field limit but up to now we do not have a proof.
Martin Gugat
Friedrich-Alexander-Universitat Erlangen-Nurnberg
Read the Original
This page is a summary of: The turnpike property for mean-field optimal control problems, European Journal of Applied Mathematics, February 2024, Cambridge University Press,
DOI: 10.1017/s0956792524000044.
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