Linear turnpike theorem

Trélat, Emmanuel

doi:10.48550/arxiv.2010.13605

Cited by 5 publications

(8 citation statements)

References 25 publications

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“…[46,Theorem 1]). The passage from local to global estimates seems to be possible [44,43] under some extra assumptions, such as the existence of a storage function, but this comes at the price of losing quite some information on the multiplicative constants appearing in (1.20). Moreover, the condition T > 1 κ log C d,α,β,γ + 2δ of Theorem 1.7 should be replaced with a condition of the form T > T 0 with T 0 depending on the initial conditions and potentially very large.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Entropic turnpike estimates for the kinetic Schrödinger problem

Chiarini¹,

Conforti²,

Greco³

et al. 2021

Preprint

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We investigate the kinetic Schrödinger problem, obtained considering Langevin dynamics instead of Brownian motion in Schrödinger's thought experiment. Under a quasilinearity assumption we establish exponential entropic turnpike estimates for the corresponding Schrödinger bridges and exponentially fast convergence of the entropic cost to the sum of the marginal entropies in the long-time regime, which provides as a corollary an entropic Talagrand inequality. In order to do so, we profit from recent advances in the understanding of classical Schrödinger bridges and adaptations of Bakry-Émery formalism to the kinetic setting. Our quantitative results are complemented by basic structural results such as dual representation of the entropic cost and the existence of Schrödinger potentials. T 0 |α P | 2 dt] < +∞ and the canonical process satisfies

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Entropic turnpike estimates for the kinetic Schrödinger problem

Chiarini¹,

Conforti²,

Greco³

et al. 2021

Preprint

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“…Strict dissipativity is a rather strong assumption for ensuring the measure turnpike property, which, as said above, is rather weak when compared to the exponential turnpike property. But actually, under the assumption of strict dissipativity, in [169] it is shown 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 engraved within the notion of strict dissipativity.…”

Section: P =mentioning

confidence: 98%

“…Therefore, necessarily, u 1 < 0 and u 2 > 0, whence the constraints in (9. The above non-uniqueness result has since been extended to more abstract settings by means of techniques using convexity properties of Chebychev sets ( [36]), and has also been explored for finite-dimensional control systems in [169]. We further discuss the latter in Section 15.…”

Section: Theorem 810 ([145]mentioning

confidence: 99%

“…As a general principle, the turnpike can be any trajectory or any invariant set of the system. For instance, in [169], the turnpike is a monotonically increasing trajectory, exemplified in practical applications by the motion of a medium (400m) distance runner ( [1]). In such applications, the velocity of the runner is essentially constant from beginning to end, but the position of the runner evolves in a monotonic fashion.…”

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confidence: 99%

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Turnpike in optimal control of PDEs, ResNets, and beyond

Geshkovski¹,

Zuazua²

2022

Preprint

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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 the insights from economics: for an optimal control problem set in a finite-time horizon, optimal controls and corresponding states, are close (often exponentially), during most of the time, except near the initial and final time, to the optimal control and 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.

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“…Indeed beginning with [1] there has been widespread interest in dissipativity notions of OCPs in context of model predictive control, see [5,9,7]. The driving force behind these investigations is the close relation between disspativity and turnpike properties of OCPs [10,7,25]…”

Section: Dissipativity Turnpike and Steady Statesmentioning

confidence: 99%

Control of port-Hamiltonian systems with minimal energy supply

Schaller¹,

Philipp²,

Faulwasser³

et al. 2020

Preprint

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We investigate optimal control of linear port-Hamiltonian systems with control constraints, in which one aims to perform a state transition with minimal energy supply. Decomposing the state space into dissipative and non-dissipative (i.e. conservative) subspaces, we show that the set of reachable states is bounded in dissipative directions. We analyze the corresponding steady-state optimization problem and prove that all optimal steady states lie in a non-dissipative subspace, which is induced by the kernel of the dissipation operator. We prove that all solutions of the optimal control problem exhibit the turnpike phenomenon with respect to this subspace, i.e., for varying initial conditions and varying horizon length the optimal state trajectories evolve close to the conservative subspace for the majority of the time. We conclude this paper by illustrating our finding by a numerical example.

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Linear turnpike theorem

Cited by 5 publications

References 25 publications

Entropic turnpike estimates for the kinetic Schrödinger problem

Entropic turnpike estimates for the kinetic Schrödinger problem

Turnpike in optimal control of PDEs, ResNets, and beyond

Control of port-Hamiltonian systems with minimal energy supply

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