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Linear turnpike theorem

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Emmanuel Trélat
Emmanuel Trélat
Emmanuel Trélat
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Abstract and Figures

The turnpike phenomenon stipulates that the solution of an optimal control problem in large time remains essentially close to a steady-state of the dynamics, itself being the optimal solution of an associated static optimal control problem. Under general assumptions, it is known that not only the optimal state and the optimal control, but also the adjoint state coming from the application of the Pontryagin maximum principle, are exponentially close to that optimal steady-state, except at the beginning and at the end of the time frame. In such a result, the turnpike set is a singleton, which is a steady-state. In this paper, we establish a turnpike result for finite-dimensional optimal control problems in which some of the coordinates evolve in a monotone way, and some others are partial steady-states of the dynamics. We prove that the discrepancy between the optimal trajectory and the turnpike set is linear, but not exponential: we thus speak of a linear turnpike theorem.
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Mathematics of Control, Signals, and Systems (2023) 35:685–739
https://doi.org/10.1007/s00498-023-00354-5
ORIGINAL ARTICLE
Linear turnpike theorem
Emmanuel Trélat1
Received: 26 October 2020 / Accepted: 18 March 2023 / Published online: 3 April 2023
© The Author(s), under exclusive licence to Springer-Verlag London Ltd., part of Springer Nature 2023
Abstract
The turnpike phenomenon stipulates that the solution of an optimal control problem
in large time remains essentially close to a steady-state of the dynamics, itself being
the optimal solution of an associated static optimal control problem. Under general
assumptions, it is known that not only the optimal state and the optimal control, but also
the adjoint state coming from the application of the Pontryagin maximum principle,
are exponentially close to that optimal steady-state, except at the beginning and at
the end of the time frame. In such a result, the turnpike set is a singleton, which is a
steady-state. In this paper, we establish a turnpike result for finite-dimensional optimal
control problems in which some of the coordinates evolve in a monotone way, and
some others are partial steady-states of the dynamics. We prove that the discrepancy
between the optimal trajectory and the turnpike set is linear, but not exponential: we
thus speak of a linear turnpike theorem.
Keywords optimal control ·turnpike ·Pontryagin maximum principle
1 Introduction and main results
1.1 Reminders on the exponential turnpike phenomenon
Let n,m∈IN ∗and let T>0 be arbitrary. We consider a general nonlinear optimal
control problem in IRn, in fixed final time T:
˙x(t)=f(x(t), u(t)) (1)
x(0)∈M0,x(T)∈M1(2)
u(t)∈(3)
BEmmanuel Trélat
emmanuel.trelat@sorbonne-universite.fr
1Sorbonne Université, CNRS, Université de Paris, Inria, Laboratoire Jacques-Louis Lions (LJLL),
75005 Paris, France
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... However, for other important classes of OCPs, optimal solutions do not converge to a static point but toward a more complicated set in state space. Several recent works have introduced different notions of non-static turnpikes, for instance periodic turnpike [25], linear turnpike [27], trim turnpike [10], etc. In [10], the turnpike was set into relation with symmetries of the optimal control problem in the context of mechanical systems and for a particular class of symmetries. ...
... with the reconstruction equationġ(t) = h(y(t), u(t)), g(0) = g 0 for g ∈ G. The turnpike property for optimal control problems of this form has been studied in [27]. ...
... In [27], the author obtained a linear turnpike for a class of optimal control problems, which can be interpreted as problems admitting a cyclic symmetry, i.e., in the form (1), but with additional boundary conditions on the group variable. Notice first that the class of symmetric optimal control problems that we treat is more general and has a strong motivation in mechanical systems, which usually admit conservation laws. ...
Trim turnpikes for optimal control problems with symmetries
Article
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  • Jan 2025
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Motivated by mechanical systems with symmetries, we focus on optimal control problems possessing certain symmetries. Following recent works (Faulwasser in Math Control Signals Syst 34:759–788 2022; Trélat in Math Control Signals Syst 35:685–739 2023), which generalized the classical concept of static turnpike to manifold turnpike we extend the exponential turnpike property to the exponential trim turnpike for control systems with symmetries induced by abelian or non-abelian groups. Our analysis is mainly based on the geometric reduction of control systems with symmetries. More concretely, we first reduce the control system on the quotient space and state the turnpike theorem for the reduced problem. Then we use the group properties to obtain the trim turnpike theorem for the full problem. Finally, we illustrate our results on the Kepler problem and the rigid body problem.
View
... Motivated by mechanical systems with symmetries, we focus on optimal control problems possessing certain symmetries. Following recent works [10,28], which generalized the classical concept of static turnpike to manifold turnpike we extend the exponential turnpike property to the exponential trim turnpike for control systems with symmetries induced by abelian or non-abelian groups. Our analysis is mainly based on the geometric reduction of control systems with symmetries. ...
... However, for other important classes of OCPs, optimal solutions do not converge to a static point but toward a more complicated set in state space. Several recent works have introduced different notions of non-static turnpikes, as for instance periodic turnpike [26], linear turnpike [28], trim turnpike [10] etc. In [10], the turnpike was set into relation with symmetries of the optimal control problem in the context of mechanical systems and for a particular class of symmetries. ...
... with the reconstruction equationġ(t) = h(y(t), u(t)), g(0) = g 0 for g ∈ G. The turnpike property for optimal control problems of this form have been studied in [28]. ...
Trim Turnpikes for Optimal Control Problems with Symmetries
Preprint
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  • Jun 2024
Motivated by mechanical systems with symmetries, we focus on optimal control problems possessing symmetries. Following recent works, which generalized the classical concept of static turnpike to manifold turnpike, we extend the exponential turnpike property to the exponential trim turnpike for control systems with symmetries induced by abelian or non-abelian groups. Our analysis is mainly based on the geometric reduction of control systems with symmetries. More concretely, we first reduce the control system on the quotient space and state the turnpike theorem for the reduced problem. Then we use the group properties to obtain the trim turnpike theorem for the full problem. Finally, we illustrate our results on the Kepler problem and the Rigid body problem.
View
... Such additional information (which is crucially required in the context of state estimation, as will be clear in Sections IV and V) is provided by exponential (or polynomial) turnpike characterizations that involve an explicit time-dependent bound on the difference of optimal trajectories and the turnpike, cf. e.g., [19], [21], [28], [29]. To cover arbitrary decay rates, we propose the following unified turnpike property involving general KL-functions. ...
... In contrast, global turnpike properties of optimal control problems could be established by combining assumptions of global nature (such as strict dissipativity) with assumptions of local nature that involve the linearizations at the turnpike (an optimal equilibrium), cf. [19], [21]. Extending these results to the more general case considered here is an interesting topic for future research. ...
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In this paper, we develop novel accuracy and performance guarantees for optimal state estimation of general nonlinear systems (in particular, moving horizon estimation, MHE). Our results rely on a turnpike property of the optimal state estimation problem, which essentially states that the omniscient infinite-horizon solution involving all past and future data serves as turnpike for the solutions of finite-horizon estimation problems involving a subset of the data. This leads to the surprising observation that MHE problems naturally exhibit a leaving arc, which may have a strong negative impact on the estimation accuracy. To address this, we propose a delayed MHE scheme, and we show that the resulting performance (both averaged and non-averaged) is approximately optimal and achieves bounded dynamic regret with respect to the infinite-horizon solution, with error terms that can be made arbitrarily small by an appropriate choice of the delay. In various simulation examples, we observe that already a very small delay in the MHE scheme is sufficient to significantly improve the overall estimation error by 20-25 % compared to standard MHE (without delay). This finding is of great importance for practical applications (especially for monitoring, fault detection, and parameter estimation) where a small delay in the estimation is rather irrelevant but may significantly improve the estimation results.
View
... Result on the turnpike property for infinite dimensional linear systems with distributed control can be found in [7], [8]. Turnpike results for finite-dimensional nonlinear optimal control problems are discussed for example in [9], [10]. ...
... For T ≥ 2, the system is exactly controllable (see Remark 2). For t ∈ [0, T ], define u exact (t) as in (10). Then the state that is generated by the control u exact reaches the desired state zero for position and velocity at the time t = 2. Afterwards, it remains there. ...
View
... The turnpike phenomenon has been widely described and analyzed in the literature (see Section 2 for introduction and references). The partial turnpike is a variant of the latter which involves some coordinates of the state but not all of them (see [33] for an example). This observation is crucial with respect to the continuation strategy we implement in section 10. ...
Numerical solving of an optimal control problem in large time horizon: the aerial vehicle guidance
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In this paper we consider an optimal control problem in large time horizon and solve it numerically. More precisely, we are interested in an aerial vehicle guidance problem: launched from a ground platform, the vehicle aims at reaching a ground/sea target under specified terminal conditions while minimizing a cost modelling some performance and constraint criteria. Our goal is to implement the indirect method based on the Pontryagin maximum principle (PMP) in order to solve such a problem. After modeling the problem, we implement continuations in order to ''connect'' a simple problem to the original one. Particularly, we exploit the turnpike property in order to enhance the efficiency of the shooting.
View
... [10]- [12]. Necessary and sufficient conditions for the presence of the turnpike phenomenon in optimal control problems are discussed in, e.g., [13], [14] and are usually based on dissipativity, controllability, and suitable optimality conditions. ...
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View
... Remark 5.7. As proved in [36], if one makes the change of variable s " t T , in (27) ...
View
... Interestingly the dissipativity route is linked to the foundational work of Willems, 1971 on infinite-horizon least-squares optimal control but it also generalizes to non-quadratic objectives and nonlinear systems (Faulwasser and Kellett, 2021). The turnpike can be regarded as the attractor of the infinite horizon optimal solutions (Trélat, 2023;Faulwasser and Kellett, 2021). We refer to Grüne, 2022 andGrüne, 2022 for recent literature overviews. ...
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The turnpike property refers to the phenomenon that in many optimal control problems, the solutions for different initial conditions and varying horizons approach a neighborhood of a specific steady state, then stay in this neighborhood for the major part of the time horizon, until they may finally depart. While early observations of the phenomenon can be traced back to works of Ramsey and von Neumann on problems in economics in 1928 and 1938, the turnpike property received continuous interest in economics since the 1960s and recent interest in systems and control. The present chapter provides an introductory overview of discrete-time and continuous-time results in finite and infinite-dimensions. We comment on dissipativity-based approaches and infinite-horizon results, which enable the exploitation of turnpike properties for the numerical solution of problems with long and infinite horizons. After drawing upon numerical examples, the chapter concludes with an outlook on time-varying, discounted, and open problems.
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