Optimal distributed control of the wave equation subject to state constraints

Gugat, Martin; Keimer, Alexander; Leugering, Günter

doi:10.1002/zamm.200800196

Cited by 22 publications

(13 citation statements)

References 18 publications

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“…For further publications on optimal control of second order hyperbolic equations see, e.g., the recent papers on time optimal control [24], adaptive finite element methods [10], and control problems with state constraints [7]. For results in the context of controllability for the the dynamical Lamé system we refer to [1] and for the wave equation to [4], where an overview about some recent results is presented.…”

Section: Introductionmentioning

confidence: 99%

Semismooth Newton Methods for Optimal Control of the Wave Equation with Control Constraints

Kröner¹,

Kunisch²,

Vexler³

2011

SIAM J. Control Optim.

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In this paper semi-smooth Newton methods for optimal control problems governed by the dynamical Lamé system are considered and their convergence behavior with respect to superlinear convergence is analyzed. Techniques from Kröner, Kunisch, Vexler (2011), where semi-smooth Newton methods for optimal control of the classical wave equation are considered, are transferred to control of the dynamical Lamé system. Three different types of control actions are examined: distributed control, Neumann boundary control and Dirichlet boundary control. The problems are discretized by finite elements and numerical examples are presented. 2 U , subject to y = S(u), y ∈ Y, u ∈ U ad ⊂ U ω with control space U ω , state space Y and α > 0. The control-to-state operator S : U ω → Y is assumed to be affine-linear, the functional G : Y → R to be quadratic. The control and state space and the operators are defined in more detail in the next section. The choice of the control-to-state operator incorporates distributed as well as Neumann and Dirichlet boundary control problems of the dynamical Lamé system which we will consider later. The set of admissible controls is defined by U ad = { u ∈ U ω | u a ≤ u ≤ u b } for given u a , u b ∈ U ω. To specify the control-to-state operator we introduce the dynamical Lamé system. Let Ω ⊂ R d , d = 1, 2, 3, be a bounded domain with C 2-boundary (bounded

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Section: Introductionmentioning

confidence: 99%

Semismooth Newton Methods for Optimal Control of the Wave Equation with Control Constraints

Kröner¹,

Kunisch²,

Vexler³

2011

SIAM J. Control Optim.

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“…From [8,18], it is known that the necessary and sufficient optimal condition of (1.1)-(1.4) leads to a state equation, an adjoint-state equation and a variational inequality, that is, we seek y(t, ·), p(t, ·) ∈ H 1 0 ( ) and u(t, ·) ∈ U ad such that…”

Section: The Crank-nicolson Finite Volume Schemementioning

confidence: 99%

A Priori Error Estimates of C rank–Nicolson Finite Volume Element Method for a Hyperbolic Optimal Control Problem

Luo

2016

Numer. Methods Partial Differential Eq.

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In this article, a Crank-Nicolson linear finite volume element scheme is developed to solve a hyperbolic optimal control problem. We use the variational discretization technique for the approximation of the control variable. The optimal convergent order O(h 2 + k 2 ) is proved for the numerical solution of the control, state and adjoint-state in a discrete L 2 -norm. To derive this result, we also get the error estimate (convergent order O(h 2 + k 2 )) of Crank-Nicolson finite volume element approximation for the second-order hyperbolic initial boundary value problem. Numerical experiments are presented to verify the theoretical results.

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“…The semi-smooth Newton methods are investigated in Kroner et al, 19 where the control is restricted by pointwise lower and upper bounds. For state constrained optimal distributed control of the wave equation, the Lavrentiev regularization is considered in Gugat et al 20 In this work, we propose another method on the basis of the shooting method and proper orthogonal decomposition method for solving the distributed optimal control problem (1). Our main aim is to develop an efficient algorithm with low computational complexity for solving (1).…”

Section: Introductionmentioning

confidence: 99%

An indirect shooting method based on the POD/DEIM technique for distributed optimal control of the wave equation

Sabeh

Shamsi

Navon

2017

Numerical Methods in Fluids

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Summary This paper presents a fast numerical method, based on the indirect shooting method and Proper Orthogonal Decomposition (POD) technique, for solving distributed optimal control of the wave equation. To solve this problem, we consider the first‐order optimality conditions and then by using finite element spatial discretization and shooting strategy, the solution of the optimality conditions is reduced to the solution of a series of initial value problems (IVPs). Generally, these IVPs are high‐order and thus their solution is time‐consuming. To overcome this drawback, we present a POD indirect shooting method, which uses the POD technique to approximate IVPs with smaller ones and faster run times. Moreover, in the presence of the nonlinear term, to reduce the order of the nonlinear calculations, a discrete empirical interpolation method (DEIM) is applied and a POD/DEIM indirect shooting method is developed. We investigate the performance and accuracy of the proposed methods by means of 4 numerical experiments. We show that the presented POD and POD/DEIM indirect shooting methods dramatically reduce the CPU time compared to the full indirect shooting method, whereas there is no significant difference between the accuracy of the reduced and full indirect shooting methods.

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Optimal distributed control of the wave equation subject to state constraints

Cited by 22 publications

References 18 publications

Semismooth Newton Methods for Optimal Control of the Wave Equation with Control Constraints

Semismooth Newton Methods for Optimal Control of the Wave Equation with Control Constraints

A Priori Error Estimates of C rank–Nicolson Finite Volume Element Method for a Hyperbolic Optimal Control Problem

An indirect shooting method based on the POD/DEIM technique for distributed optimal control of the wave equation

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