# Turnpike in a Nutshell

** Time: 17.10.2019, 10-12am
Room: K2-119 – Seminarraum (MHB-Building, U1.245)
**

**1. Turnpike theory: An introduction**

Introduction by Enrique Zuazua (FAU)

**Abstract:**

Turnpike properties have been established long time ago in finite-dimensional optimal control problems arising in econometry. They refer to the fact that, under quite general assumptions, the optimal solutions of a given optimal control problem settled in large time consist approximately of three pieces, the first and the last of which being transient short-time arcs, and the middle piece being a long-time arc staying exponentially close to the optimal steady-state solution of an associated static optimal control problem. This property, sometimes in an implicit manner, is often used in applications.

**2. The finite-time turnpike phenomenon for optimal control problems**

Talk of Martin Gugat (FAU)

**Abstract**

In this paper problems of optimal control are considered where in the objective function, in addition to the control cost there is a $L^1$-norm tracking term that measures the distance to a desired stationary state.

In the optimal control problem, the initial state is prescribed. We assume that the system is exactly controllable.

We show that both for systems that are governed by ordinary differential equations and for boundary control systems governed by the wave equation the optimal system state is steered exactly to the desired state after finite time, if the weight of the tracking term is sufficiently large.

**3. Turnpike theory and application**

Talk of Dario Pighin (Universidad Autónoma de Madrid, DeustoTech)

**Abstract:**

Under appropriate assumptions, the optima of a time-evolution control problem simplifies as the time horizon *T* goes to infinity, namely converge to the corresponding steady optima. When this occurs, we say the control problem enjoys the turnpike property. Some theoretical results both in ODE control and PDE control will presented. An industrial application to rotors imbalance suppression will be given. An open problem is posed: uniqueness of the optimal control in a control problem governed by a semilinear PDE.

**4. The theoretical and practical use of the turnpike property for Model Predictive Control**

Talk of Manuel Schaller (University of Bayreuth)

**Abstract**

Model Predictive Control (MPC) is a control method in which the solution of optimal control problems on infinite or indefinitely long horizons is split up into the successive solution of optimal control problems on relatively short finite time horizons. Only a first part with given length of this solution is implemented as a control for the longer, possibly infinite horizon.

For reasons of performance and stability of this scheme, the turnpike property as a feature of optimal control problems plays an important role. Moreover, the

turnpike property motivates an efficient way of discretizing the underlying optimal control problems. More specifically, one can use the stability of the system strongly linked to the turnpike property to show that discretization errors decay exponentially in time. This rigorously explains the behavior of goal oriented error estimation algorithms used in this context.

We present a stability property for a wide class of optimal control problems that one the one hand allows for keeping numerical effort low when using adaptive grid refinement, and on the other hand can be used to derive a quantitative turnpike property.

Chair in Applied Analysis (Alexander von Humboldt-Professorship)