Asymptotic Stability and the Turnpike Property in Some Simple Control Problems

1987 ◽  
pp. 36-49
Author(s):  
D. A. Carlson ◽  
A. Haurie
Keyword(s):  
Asymptotic Stability ◽  
Control Problems ◽  
Turnpike Property

Asymptotic Stability and the Turnpike Property in Some Simple Control Problems

1991 ◽  
pp. 32-43
Author(s):  
Dean A. Carlson ◽  
Alain B. Haurie ◽  
Arie Leizarowitz
Keyword(s):  
Asymptotic Stability ◽  
Control Problems ◽  
Turnpike Property

Turnpike properties of optimal relaxed control problems

2019 ◽  
Vol 25 ◽  
pp. 74
Author(s):  
Hongwei Lou ◽  
Weihan Wang
Keyword(s):  
Uniform Boundedness ◽  
Control Problems ◽  
Mean Square ◽  
Relaxed Control ◽  
Turnpike Property ◽  
The Mean ◽  
Adjoint Functions

In this paper, three kinds of turnpike properties for optimal relaxed control problems are considered. Under some convexity and controllability assumptions, we obtain the uniform boundedness of the optimal pairs and the adjoint functions. On the basis, we prove the integral turnpike property, the mean square turnpike property and the exponential turnpike property, respectively.

scholarly journals Local Turnpike Analysis Using Local Dissipativity for Discrete Time Discounted Optimal Control

2021 ◽  
Author(s):  
Lars Grüne ◽  
Lisa Krügel
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Local Equilibrium ◽  
Discount Factor ◽  
Control Problems ◽  
Asymptotically Stable ◽  
Local Behaviour ◽  
Turnpike Property ◽  
Global Equilibrium ◽  
Near Optimality

AbstractRecent results in the literature have provided connections between the so-called turnpike property, near optimality of closed-loop solutions, and strict dissipativity. Motivated by applications in economics, optimal control problems with discounted stage cost are of great interest. In contrast to non-discounted optimal control problems, it is more likely that several asymptotically stable optimal equilibria coexist. Due to the discounting and transition cost from a local to the global equilibrium, it may be more favourable staying in a local equilibrium than moving to the global—cheaper—equilibrium. In the literature, strict dissipativity was shown to provide criteria for global asymptotic stability of optimal equilibria and turnpike behaviour. In this paper, we propose a local notion of discounted strict dissipativity and a local turnpike property, both depending on the discount factor. Using these concepts, we investigate the local behaviour of (near-)optimal trajectories and develop conditions on the discount factor to ensure convergence to a local asymptotically stable optimal equilibrium.

scholarly journals Exponential turnpike property for fractional parabolic equations with non-zero exterior data

2020 ◽  
Author(s):  
Mahamadi WARMA ◽  
Sebastian Zamorano
Keyword(s):  
Optimal Control ◽  
Time Horizon ◽  
Optimal Control Problems ◽  
Control Problems ◽  
Heat Equations ◽  
Optimal Controls ◽  
Optimal Solutions ◽  
Turnpike Property ◽  
Final Destination ◽  
Time Averages

We consider averages convergence as the time-horizon goes to infinity of optimal solutions of time-dependent optimal control problems to optimal solutions of the corresponding stationary optimal control problems. Assuming that the controlled dynamics under consideration are stabilizable towards a stationary solution, the following natural question arises: Do time averages of optimal controls and trajectories converge to the stationary optimal controls and states as the time-horizon goes to infinity? This question is very closely related to the so-called turnpike property that shows that, often times, the optimal trajectory joining two points that are far apart, consists in, departing from the point of origin, rapidly getting close to the steady-state (the turnpike) to stay there most of the time, to quit it only very close to the final destination and time. In the present paper we deal with heat equations with non-zero exterior conditions (Dirichlet and nonlocal Robin) associated with the fractional Laplace operator $(-\Delta)^s$ ( $0<s<1$ ). We prove the turnpike property for the nonlocal Robin optimal control problem and the exponential turnpike property for both Dirichlet and nonlocal Robin optimal control problems.

scholarly journals Homotopic approach for turnpike and singularly perturbed optimal control problems

2021 ◽  
Vol 71 ◽  
pp. 43-53
Author(s):  
Olivier Cots ◽  
Joseph Gergaud ◽  
Boris Wembe
Keyword(s):  
Optimal Control ◽  
Singular Perturbations ◽  
Optimal Control Problems ◽  
Singularly Perturbed ◽  
Continuation Methods ◽  
Control Problems ◽  
General Control ◽  
Turnpike Property ◽  
Set Up ◽  
New Framework

The first aim of this article is to present the link between the turnpike property and the singular perturbations theory: the first one being a particular case of the second one. Then, thanks to this link, we set up a new framework based on continuation methods for the resolution of singularly perturbed optimal control problems. We consider first the turnpike case, then, we generalize the approach to general control problems with singular perturbations (that is with fast but also slow variables). We illustrate each step with an example.

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