Kolloquium: E. Trélat, On the turnpike property

On the turnpike property

Referent: Prof. Dr. Emmanuel Trélat, Sorbonne Université
Laboratoire Jacques-Louis Lions, CNRS, Paris, Frankreich
Veranstalter: Enrique Zuazua

Abstract:
The turnpike property was discovered in the 50’s by the Nobel prize Samuelson in econometry. It stipulates that the optimal trajectory of an optimal control problem in large time remains essentially close to a steady state, itself being the optimal solution of an associated static optimal control problem.
We have established the turnpike property for general nonlinear finite and infinite dimensional optimal control problems, showing that the optimal trajectory is, except at the beginning and the end of the time interval, exponentially close to some (optimal) stationary state, and that this property holds as well for the optimal control and for the adjoint vector coming from the Pontryagin maximum principle. We prove that the exponential turnpike property is due to an hyperbolicity phenomenon which is intrinsic to the symplectic feature of the extremal equations. We infer a simple and efficient numerical method to compute optimal trajectories in that framework, in particular an appropriate variant of the shooting method.
The turnpike property turns out to be ubiquitous and the turnpike set may be more general than a single steady-state, like for instance a periodic trajectory. We also show the property of shape turnpike for PDE models in which a subdomain evolves in time according to some optimization criterion.