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

Grüne, Lars; Schaller, Manuel; Schiela, Anton

doi:10.1016/j.jde.2019.11.064

Cited by 65 publications

(74 citation statements)

References 29 publications

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“…In this paper we provide an abstract framework for exponential sensitivity analysis of nonlinear optimal control problems with respect to perturbations of the right-hand side of the first-order necessary optimality conditions. We extend previous results regarding linear quadratic optimal control problems, where an exponential damping property was proven for problems governed by non-autonomous parabolic equations in [26] and for problems governed by autonomous general evolution equations in [27]. The main tool in these works is a bound on the solution operator of the first order optimality conditions that is independent of the time horizon, which can be deduced under stabilizability and detectability assumptions.…”

Section: Introductionsupporting

confidence: 64%

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Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs

2021

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We analyze the sensitivity of the extremal equations that arise from the first order necessary optimality conditions of nonlinear optimal control problems with respect to perturbations of the dynamics and of the initial data. To this end, we present an abstract implicit function approach with scaled spaces. We will apply this abstract approach to problems governed by semilinear PDEs. In that context, we prove an exponential turnpike result and show that perturbations of the extremal equation's dynamics, e.g., discretization errors decay exponentially in time. The latter can be used for very efficient discretization schemes in a Model Predictive Controller, where only a part of the solution needs to be computed accurately. We showcase the theoretical results by means of two examples with a nonlinear heat equation on a two-dimensional domain.

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Section: Introductionsupporting

confidence: 64%

“…We briefly recall some of the existing literature on turnpike analysis. The linear quadratic case for control of evolution equations was considered in [7,11,[26][27][28][29]. A turnpike property for shape optimization was introduced in [33,49].…”

Section: Introductionmentioning

confidence: 99%

Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs

2021

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“…For finite-dimensional systems, the exponential turnpike property has been studied in [18]. Exponential sensitivity and turnpike analysis for linear quadratic optimal control of general evolution equations have been studied in [6]. The results that we present here allow for general nonlinear infinite-dimensional dynamcis that satisfy (1.1).…”

Section: Remark 24mentioning

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 are several possible notions of turnpike properties, some of them being stronger than the others (see [37]). Exponential turnpike properties have been established in [17,26,27,33,34] for the optimal triple resulting of the application of Pontryagin's maximum principle, ensuring that the extremal solution (state, adjoint and control) remains exponentially close to an optimal solution of the corresponding static controlled problem, except at the beginning and at the end of the time interval, as soon as T is large enough. This follows from hyperbolicity properties of the Hamiltonian flow.…”

Section: Introductionmentioning

confidence: 99%

Turnpike in optimal shape design

Trélat

Zuazua

2019

IFAC-PapersOnLine

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We introduce and study the turnpike property for time-varying shapes, within the viewpoint of optimal control. We focus here on second-order linear parabolic equations where the shape acts as a source term and we seek the optimal time-varying shape that minimizes a quadratic criterion. We first establish existence of optimal solutions under some appropriate sufficient conditions. We then provide necessary conditions for optimality in terms of adjoint equations and, using the concept of strict dissipativity, we prove that state and adjoint satisfy the measure-turnpike property, meaning that the extremal time-varying solution remains essentially close to the optimal solution of an associated static problem. We show that the optimal shape enjoys the exponential turnpike property in term of Hausdorff distance for a Mayer quadratic cost. We illustrate the turnpike phenomenon in optimal shape design with several numerical simulations.

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

Cited by 65 publications

References 29 publications

Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs

Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs

On the turnpike property with interior decay for optimal control problems

Turnpike in optimal shape design

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