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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...
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... In Section 2, we establish the exponential turnpike property for linear-quadratic (LQ) optimal control problems, in a very elementary way. The main result is Theorem 1. Section 3 is devoted to showing how Theorem 1 can be extended to finite-dimensional nonlinear optimal control problems, obtaining a local exponential turnpike property (i.e., the main result of [104], improved with results of [98]). In Section 3.4, we illustrate in detail the local and global aspects on simple but meaningful examples. ...
... The next theorem is the main result of [104], improved with some results of [98]. ...
... Then, following the arguments of the proof of Theorem 1, one obtains a local turnpike result, under the above smallness condition. The extension done in [105] allows to drop this smallness assumption, by using sensitivity analysis and conjugate point theory (as in [98,Proposition 1]). ...
The turnpike principle is a fundamental concept in optimal control theory, stating that for a wide class of long-horizon optimal control problems, the optimal trajectory spends most of its time near a steady-state solution (the ''turnpike'') rather than being influenced by the initial or final conditions. In this article, we provide a survey on the turnpike property in optimal control, adding several recent and novel considerations. After some historical insights, we present an elementary proof of the exponential turnpike property for linear-quadratic optimal control problems in finite dimension. Next, we show an extension to nonlinear optimal control problems, with a local exponential turnpike property. On simple but meaningful examples, we illustrate the local and global aspects of the turnpike theory, clarifying the global picture and raising new questions. We discuss key generalizations, in infinite dimension and other various settings, and review several applications of the turnpike theory across different fields.
... 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. ...
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.
... 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. ...
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.
... 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. ...
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.
... [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. ...
In this paper, we introduce turnpike arguments in the context of optimal state estimation. In particular, we show that the optimal solution of the state estimation problem involving all available past data serves as turnpike for the solutions of truncated problems involving only a subset of the data. We consider two different mathematical characterizations of this phenomenon and provide corresponding sufficient conditions that rely on strict dissipativity and decaying sensitivity. As second contribution, we show how a specific turnpike property can be used to establish performance guarantees when approximating the optimal solution of the full problem by a sequence of truncated problems, and we show that the resulting performance (both averaged and non-averaged) is approximately optimal with error terms that can be made arbitrarily small by an appropriate choice of the horizon length. In addition, we discuss interesting implications of these results for the practically relevant case of moving horizon estimation and illustrate our results with a numerical example.
... 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. ...
... 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]. ...
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.
... 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. ...
The training of ResNets and neural ODEs can be formulated and analyzed from the perspective of optimal control. This paper proposes a dissipative formulation of the training of ResNets and neural ODEs for classification problems by including a variant of the cross-entropy as a regularization in the stage cost. Based on the dissipative formulation of the training, we prove that the trained ResNet exhibit the turnpike phenomenon. We then illustrate that the training exhibits the turnpike phenomenon by training on the two spirals and MNIST datasets. This can be used to find very shallow networks suitable for a given classification task.
... Remark 5.7. As proved in [36], if one makes the change of variable s " t T , in (27) ...
... Strict dissipativity is a rather strong assumption for ensuring the measure turnpike property, which, as mentioned above, is rather weak compared with the exponential turnpike property. But in fact, under the assumption of strict dissipativity, Trélat (2020) showed that for almost every s ∈ (0, 1), y T (sT) → y and u T (sT) → u as T → +∞, which is a significantly stronger result. In fact, it can be said that, in some sense, the turnpike property is inherent within the notion of strict dissipativity. ...
... As a general principle, the turnpike can be any trajectory or any invariant set of the system. For instance, in Trélat (2020), the turnpike is a monotonically increasing trajectory, exemplified in practical applications by the motion ...
... The above non-uniqueness result has since been extended to more abstract settings by means of techniques using convexity properties of Chebyshev sets (Christof and Hafemeyer 2022), and has also been explored for finite-dimensional control systems in Trélat (2020). We discuss the latter further in Section 15. ...
The turnpike property in contemporary macroeconomics asserts that if an economic planner seeks to move an economy from one level of capital to another, then the most efficient path, as long as the planner has enough time, is to rapidly move stock to a level close to the optimal stationary or constant path, then allow for capital to develop along that path until the desired term is nearly reached, at which point the stock ought to be moved to the final target. Motivated in part by its nature as a resource allocation strategy, over the past decade, the turnpike property has also been shown to hold for several classes of partial differential equations arising in mechanics. When formalized mathematically, the turnpike theory corroborates insights from economics: for an optimal control problem set in a finite-time horizon, optimal controls and corresponding states are close (often exponentially) most of the time, except near the initial and final times, to the optimal control and the corresponding state for the associated stationary optimal control problem. In particular, the former are mostly constant over time. This fact provides a rigorous meaning to the asymptotic simplification that some optimal control problems appear to enjoy over long time intervals, allowing the consideration of the corresponding stationary problem for computing and applications. We review a slice of the theory developed over the past decade – the controllability of the underlying system is an important ingredient, and can even be used to devise simple turnpike-like strategies which are nearly optimal – and present several novel applications, including, among many others, the characterization of Hamilton–Jacobi–Bellman asymptotics, and stability estimates in deep learning via residual neural networks.
