2011
DOI: 10.2478/cmam-2011-0012
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Adaptive Finite Element Methods For Optimal Control Of Second Order Hyperbolic Equations

Abstract: In this paper we consider a posteriori error estimates for space-time finite element discretizations for optimal control of hyperbolic partial dierential equations of second order. It is an extension of Meidner and Vexler (2007), where optimal control problems of parabolic equations are analyzed. The state equation is formulated as a first order system in time and a posteriori error estimates are derived separating the in uences of time, space, and control discretization. Using this informa… Show more

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Cited by 37 publications
(30 citation statements)
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“…First, by implementing a procedure as in the numerical solution of the Black-Scholes PDE a set of equivalent nonlinear ordinary differential equations is obtained [Dragonescu and Soane (2013)], [Gerdts et al (2008)], [Guo and Billings (2007)], [Kröner (2011)], [Pinsky (1991)], [Winkler and Lohmann (2010)]. Next it is shown that the system of the nonlinear ODEs is a differentially flat one.…”
Section: Introductionmentioning
confidence: 98%
“…First, by implementing a procedure as in the numerical solution of the Black-Scholes PDE a set of equivalent nonlinear ordinary differential equations is obtained [Dragonescu and Soane (2013)], [Gerdts et al (2008)], [Guo and Billings (2007)], [Kröner (2011)], [Pinsky (1991)], [Winkler and Lohmann (2010)]. Next it is shown that the system of the nonlinear ODEs is a differentially flat one.…”
Section: Introductionmentioning
confidence: 98%
“…For instance, nonlinear wavetype differential equations is met in communication systems (transmission lines, optical fibers and electromagnetic waves propagation), in electronics (Josephson junctions), in fluid flow models, in structural engineering (dynamic analysis of buildings under seismic waves, mechanical structures subjected to vibrations, pendulum chains), in biomedical systems (voltage propagation and variations in neuron's membrane), etc. Solving nonlinear estimation and control problems for such systems is important for modifying their dynamics and for succeeding their functioning according to specifications [10][11][12][13][14][15][16][17].…”
Section: Introductionmentioning
confidence: 99%
“…The present paper applied semi-discretization for the numerical representation of the PDE dynamics, which is an approach also implemented in [12][13][14][15][16]. Moreover, this paper treats the problem of boundary control of the nonlinear Fokker-Planck PDE, which means that the boundary conditions are used as control inputs to modify this PDE dynamics.…”
Section: Introductionmentioning
confidence: 99%