The turnpike phenomenon refers to a similarity property in optimal control problems, i.e., for varying initial conditions of the dynamics and varyign horizon lengths the optimal solutions are structurally similar. Put differently, they stay close to the optimal steady state (a.k.a. the turnpike) in the middle part of the horizon and this part grows as the horizon increases.

Early observations of the phenomenon can be traced back to papers by John von Neumann and Frank P. Ramsey which appeared in the 1930s. The term turnpike was coined in the 1958 book on Linear Programming and Economic Analysis by Dorfman, Solow, and Samuelson. Therein, they coined the term turnpike in optimal control by writing:

"[...] It is exactly like a turnpike paralleled by a network of minor roads. There is a fastest route between any two points; and if the origin and destination are close together and far from the turnpike, the best route may not touch the turnpike. But if the origin and destination are far enough apart, it will always pay to get on to the turnpike and cover distance at the best rate of travel, even if this means adding a little mileage at either end. [...]"

Our research has contributed 

• to analyzing the turnpike using dissipativity concepts that can be traced back to Jan C. Willems and his seminalt 1971 and 1972 papers,
• to resolving the long standing issue surrounding the adjoint terminal conditions in infinite-horizon problems. (a.k.a. Halkin´s problem) by showing that under mild assumptions the turnpike phenomenon implies infinite-horizon stability,
• to the analysis of turnpike in mixed-integer optimal control problems,
• to extending the turnpipke concepts to generalized infinite horizon attractors (mainfolds and linear subspaces) in optimal control problems for thermodynamic systems and Euler-Lagrange systems, and 
• to exploiting the turnpike property for the closed-loop analysis of receding-horizon optimal control (a.k.a. model predictive control) with economic objective functions.

In recent research,

• we have extended turnpike and dissipaticity concepts to stochastic optimal control problems,
• we used turnpike concepts for closed-loop analysis of model predictive path-following control problems, and
• we used turnpike concepts to analyze the training of deep neural networks (a.k.a. deep learing) from an optimal control perspective. We also derive explicit depth bounds using turnpike concepts.

[1] Dorfman, R., Samuelson, P. A., & Solow, R. M. (2012). Linear programming and economic analysis. Courier Corporation.

[2] Willems, J. (1971). Least squares stationary optimal control and the algebraic Riccati equation. IEEE Transactions on automatic control, 16(6), 621-634.

[3] Willems, J. C. (1972). Dissipative dynamical systems part I: General theory. Archive for rational mechanics and analysis, 45(5), 321-351.

[4] Willems, J. C. (1972). Dissipative dynamical systems part II: Linear systems with quadratic supply rates. Archive for rational mechanics and analysis, 45, 352-393.

[5] Faulwasser, T., Korda, M., Jones, C. N., & Bonvin, D. (2014, December). Turnpike and dissipativity properties in dynamic real-time optimization and economic MPC. In 53rd ieee conference on decision and control (pp. 2734-2739). IEEE.

[6] Faulwasser, T., Korda, M., Jones, C. N., & Bonvin, D. (2017). On turnpike and dissipativity properties of continuous-time optimal control problems. Automatica, 81, 297-304.

[7] Faulwasser, T., Grüne, L., & Müller, M. A. (2018). Economic nonlinear model predictive control. Foundations and Trends® in Systems and Control, 5(1), 1-98.

[8] Faulwasser, T., & Murray, A. (2020). Turnpike properties in discrete-time mixed-integer optimal control. IEEE Control Systems Letters, 4(3), 704-709.

[9] Faulwasser, T., & Bonvin, D. (2017). Exact turnpike properties and economic NMPC. European Journal of Control, 35, 34-41.

[10] Faulwasser, T., Flaßkamp, K., Ober-Blöbaum, S., Schaller, M., & Worthmann, K. (2022). Manifold turnpikes, trims, and symmetries. Mathematics of Control, Signals, and Systems, 34(4), 759-788.

[11] Faulwasser, T., & Grüne, L. (2022). Turnpike properties in optimal control: An overview of discrete-time and continuous-time results. Handbook of numerical analysis, 23, 367-400.

[12]  Krügel, L., Faulwasser, T., & Grüne, L. (2023, December). Local Turnpike Properties in Finite Horizon Optimal Control. In 2023 62nd IEEE Conference on Decision and Control (CDC) (pp. 5273-5278). IEEE.

[12] Ou, R., Schießl, J., Baumann, M. H., Grüne, L., & Faulwasser, T. (2023). A Polynomial Chaos Approach to Stochastic LQ Optimal Control: Error Bounds and Infinite-Horizon Results. arXiv preprint arXiv:2311.17596.

[14] Schießl, J., Baumann, M. H., Faulwasser, T., & Grüne, L. (2024). On the relationship between stochastic turnpike and dissipativity notions. IEEE Transactions on Automatic Control.

[15] Faulwasser, T., Hempel, A. J., & Streif, S. (2024). On the turnpike to design of deep neural networks: Explicit depth bounds. IFAC Journal of Systems and Control, 30, 100290.

[16] Püttschneider, J., & Faulwasser, T. (2024). On Dissipativity of Cross-Entropy Loss in Training ResNets. arXiv preprint arXiv:2405.19013.