On the Turnpike Phenomenon for Optimal Boundary Control Problems with Hyperbolic Systems

Gugat, Martin; Hante, Falk M.

doi:10.1137/17m1134470

Cited by 39 publications

(42 citation statements)

References 30 publications

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“…Likewise, one could deduce turnpike results for general evolution equations with admissible boundary control or observation operators under stabilizability and detectability conditions, a setting for which, to the authors best knowledge, there are no turnpike results for space dimension 2 or 3. For boundary controlled hyperbolic equations in one space dimension, turnpike results are given in [17,16]. For parabolic equations with boundary control or observation, a turnpike result was shown in [25,Theorem 6].…”

Section: Discussionmentioning

confidence: 99%

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Sensitivity Analysis of Optimal Control for a Class of Parabolic PDEs Motivated by Model Predictive Control

Grüne¹,

Schaller²,

Schiela³

2019

SIAM J. Control Optim.

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We analyze the sensitivity of the extremal equations that arise from the first order optimality conditions for time dependent optimization problems. More specifically, we consider parabolic PDEs with distributed or boundary control and a linear quadratic performance criterion. We prove the solution's boundedness with respect to the right-hand side of the first order optimality condition which includes initial data. If the system fulfills a particular stabilizability and detectability assumption, the bound is independent of the time horizon. As a consequence, the influence of a perturbation of the right-hand side decreases exponentially backward in time. We use this property for the construction of efficient numerical discretizations in a Model Predictive Control scheme. Moreover, a quantitative turnpike theorem in the W ([0, T ])-norm is derived.

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Section: Discussionmentioning

confidence: 99%

“…There are several different turnpike characterizations and a wide literature thereon, cf. [2,6,18,30,31,14,9,17,16]. A particular kind of turnpike behavior is the so called exponential turnpike property, cf.…”

mentioning

confidence: 99%

Sensitivity Analysis of Optimal Control for a Class of Parabolic PDEs Motivated by Model Predictive Control

Grüne¹,

Schaller²,

Schiela³

2019

SIAM J. Control Optim.

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“…For optimal control problems with partial differential equations see [15] or [19], where distributed control is considered for linear-quadratic optimal control problems. Problems of optimal boundary control are studied in [7]. In [17], both integral-and measure-turnpike properties are considered.…”

Section: Introductionmentioning

confidence: 99%

On the turnpike property with interior decay for optimal control problems

Gugat

2021

Math. Control Signals Syst.

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In this paper the turnpike phenomenon is studied for problems of optimal control where both pointwise-in-time state and control constraints can appear. We assume that in the objective function, a tracking term appears that is given as an integral over the time-interval $$[0,\, T]$$ [ 0 , T ] and measures the distance to a desired stationary state. In the optimal control problem, both the initial and the desired terminal state are prescribed. We assume that the system is exactly controllable in an abstract sense if the time horizon is long enough. We show that that the corresponding optimal control problems on the time intervals $$[0, \, T]$$ [ 0 , T ] give rise to a turnpike structure in the sense that for natural numbers n if T is sufficiently large, the contribution of the objective function from subintervals of [0, T] of the form $$\begin{aligned} {[}t - t/2^n,\; t + (T-t)/2^n] \end{aligned}$$ [ t - t / 2 n , t + ( T - t ) / 2 n ] is of the order $$1/\min \{t^n, (T-t)^n\}$$ 1 / min { t n , ( T - t ) n } . We also show that a similar result holds for $$\epsilon $$ ϵ -optimal solutions of the optimal control problems if $$\epsilon >0$$ ϵ > 0 is chosen sufficiently small. At the end of the paper we present both systems that are governed by ordinary differential equations and systems governed by partial differential equations where the results can be applied.

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“…There is a large literature on turnpike behavior, cf. [1,7,17,27,28,12,9,16,15]. A particular kind of turnpike behavior is the so called exponential turnpike property, cf.…”

mentioning

confidence: 99%

Exponential sensitivity and turnpike analysis for linear quadratic optimal control of general evolution equations

Grüne

Schaller

Schiela

2020

Journal of Differential Equations

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We analyze the sensitivity of linear quadratic optimal control problems governed by general evolution equations with bounded or admissible control operator. We show, that if the problem is stabilizable and detectable, the solution of the extremal equation can be bounded by the right-hand side including initial data with the bound being independent of the time horizon. Consequently, the influence of perturbations of the extremal equations decays exponentially in time. This property can for example be used to construct efficient space and time discretizations for a Model Predictive Control scheme. Furthermore, a turnpike property for unbounded but admissible control of general semigroups can be deduced.

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On the Turnpike Phenomenon for Optimal Boundary Control Problems with Hyperbolic Systems

Cited by 39 publications

References 30 publications

Sensitivity Analysis of Optimal Control for a Class of Parabolic PDEs Motivated by Model Predictive Control

Sensitivity Analysis of Optimal Control for a Class of Parabolic PDEs Motivated by Model Predictive Control

On the turnpike property with interior decay for optimal control problems

Exponential sensitivity and turnpike analysis for linear quadratic optimal control of general evolution equations

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