Numerical Approximation of Exact Controls for Waves

Ervedoza, Sylvain; Zuazua, Enrique

doi:10.1007/978-1-4614-5808-1_1

Cited by 8 publications

(7 citation statements)

References 32 publications

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“…(superscripts indicate components of vectors); x 1 and x 2 represent, in suitable units, numbers of preys and predators respectively. The solid lines give, for 0 ≤ t ≤ 1, the unperturbed solution x(t) with initial condition x(0) = (15, 10) and a perturbed solutionx(t) withx(0) = x(0)+η = (16,10): an increase in the number of preys at t = 0 leads at t = 1 to a decrease in the number of preys and to an increase in the number of predators. The stars are the points x(t) + δ(t), t = 0, 0.05, 0.10, .…”

Section: The Continuous Problem: Lagrange Multipliersmentioning

confidence: 99%

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Symplectic Runge--Kutta Schemes for Adjoint Equations, Automatic Differentiation, Optimal Control, and More

Sanz‐Serna¹

2016

SIAM Rev.

130

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The study of the sensitivity of the solution of a system of differential equations with respect to changes in the initial conditions leads to the introduction of an adjoint system, whose discretisation is related to reverse accumulation in automatic differentiation. Similar adjoint systems arise in optimal control and other areas, including classical Mechanics. Adjoint systems are introduced in such a way that they exactly preserve a relevant quadratic invariant (more precisely an inner product). Symplectic Runge-Kutta and Partitioned Runge-Kutta methods are defined through the exact conservation of a differential geometric structure, but may be characterized by the fact that they preserve exactly quadratic invariants of the system being integrated. Therefore the symplecticness (or lack of symplecticness) of a Runge-Kutta or Partitioned Runge-Kutta integrator should be relevant to understand its performance when applied to the computation of sensitivities, to optimal control problems and in other applications requiring the use of adjoint systems. This paper examines the links between symplectic integration and those applications. The article presents in a new, unified way a number of results now scattered or implicit in the literature. In particular we show how some common procedures, such as the direct method in optimal control theory and the computation of sensitivities via reverse accumulation, imply, probably unbeknownst to the user, 'hidden' integrations with symplectic Partitioned Runge-Kutta schemes.

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Section: The Continuous Problem: Lagrange Multipliersmentioning

confidence: 99%

“…A discussion of the advantages of the direct and indirect approaches is not within our scope here, see e.g. [41,Chapter 9], [10].…”

Section: The Discrete Problem: Direct Approachmentioning

confidence: 99%

Symplectic Runge--Kutta Schemes for Adjoint Equations, Automatic Differentiation, Optimal Control, and More

Sanz‐Serna¹

2016

SIAM Rev.

130

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“…Indeed, A T i A i ≥ 0 by construction. To show that A T i A i > 0, take v 0 = r 0 = v 1 = r 1 = 0, then ε i = λy T i A T i A i y i since g i = 0 and c 1 = 0 in (42). Consider the value of ε i on vectors of standard basis in R N i s .…”

Section: Solution Of the One-dimensional Variational Problemmentioning

confidence: 99%

“…4, we present a solution algorithm reducing the OCP in two-dimensional time-space domain (Problem 2) to a one-dimensional variational problem. Whereas usually a solution to an OPC for PDEs in general and for the wave equation in particular can be obtained only approximately, for example, by means of Fourier [3] or finite difference [42] methods, we present a way to explicitly derive an analytical solution. Since we focus on a continuous system with finite-dimensional control inputs while solving the OCP rigorously, such a solution may serve as a benchmark in both theoretical and engineering studies employing distributed loads.…”

Section: Optimal Control Problemmentioning

confidence: 99%

Controllability and Optimal Control Design for an Elastic Rod Actuated by Piezoelements

Kostin

Gavrikov

2022

IFAC-PapersOnLine

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“…Several strategies have been proposed to face this problem: multigrid strategy, filtering of the high frequency modes, introduction of viscous terms, etc. We refer to [14,3,26,21,8,9,20,22] for some works about the numerical control of the wave equation, and to [13,27] for some surveys. See also [11,6] for the numerics corresponding to the wave equation with damping.…”

Section: Introductionmentioning

confidence: 99%

Numerical Control of a Semilinear Wave Equation on an Interval

Cavalcanti

Rosier

2022

Advances in Distributed Parameter Systems

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HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

show abstract

Numerical Approximation of Exact Controls for Waves

Cited by 8 publications

References 32 publications

Symplectic Runge--Kutta Schemes for Adjoint Equations, Automatic Differentiation, Optimal Control, and More

Symplectic Runge--Kutta Schemes for Adjoint Equations, Automatic Differentiation, Optimal Control, and More

Controllability and Optimal Control Design for an Elastic Rod Actuated by Piezoelements

Numerical Control of a Semilinear Wave Equation on an Interval

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