Approximate controllability and its well-posedness for the semilinear reaction-diffusion equation with internal lumped controls

Alexander Khapalov

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 4, page 83-98
  • ISSN: 1292-8119

Abstract

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We consider the one dimensional semilinear reaction-diffusion equation, governed in Ω = (0,1) by controls, supported on any subinterval of (0, 1), which are the functions of time only. Using an asymptotic approach that we have previously introduced in [9], we show that such a system is approximately controllable at any time in both L2(0,1)( and C0[0,1], provided the nonlinear term f = f(x,t, u) grows at infinity no faster than certain power of log |u|. The latter depends on the regularity and structure of f (x, t, u) in x and t and the choice of the space for controllability. We also show that our results are well-posed in terms of the “actual steering” of the system at hand, even in the case when it admits non-unique solutions.

How to cite

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Khapalov, Alexander. "Approximate controllability and its well-posedness for the semilinear reaction-diffusion equation with internal lumped controls." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 83-98. <http://eudml.org/doc/197268>.

@article{Khapalov2010,
abstract = { We consider the one dimensional semilinear reaction-diffusion equation, governed in Ω = (0,1) by controls, supported on any subinterval of (0, 1), which are the functions of time only. Using an asymptotic approach that we have previously introduced in [9], we show that such a system is approximately controllable at any time in both L2(0,1)( and C0[0,1], provided the nonlinear term f = f(x,t, u) grows at infinity no faster than certain power of log |u|. The latter depends on the regularity and structure of f (x, t, u) in x and t and the choice of the space for controllability. We also show that our results are well-posed in terms of the “actual steering” of the system at hand, even in the case when it admits non-unique solutions. },
author = {Khapalov, Alexander},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {The semilinear reaction-diffusion equation; approximate controllability; internal lumped control multiple solutions.; semilinear reaction-diffusion equation; approximate controllability; internal lumped control; multiple solutions},
language = {eng},
month = {3},
pages = {83-98},
publisher = {EDP Sciences},
title = {Approximate controllability and its well-posedness for the semilinear reaction-diffusion equation with internal lumped controls},
url = {http://eudml.org/doc/197268},
volume = {4},
year = {2010},
}

TY - JOUR
AU - Khapalov, Alexander
TI - Approximate controllability and its well-posedness for the semilinear reaction-diffusion equation with internal lumped controls
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 4
SP - 83
EP - 98
AB - We consider the one dimensional semilinear reaction-diffusion equation, governed in Ω = (0,1) by controls, supported on any subinterval of (0, 1), which are the functions of time only. Using an asymptotic approach that we have previously introduced in [9], we show that such a system is approximately controllable at any time in both L2(0,1)( and C0[0,1], provided the nonlinear term f = f(x,t, u) grows at infinity no faster than certain power of log |u|. The latter depends on the regularity and structure of f (x, t, u) in x and t and the choice of the space for controllability. We also show that our results are well-posed in terms of the “actual steering” of the system at hand, even in the case when it admits non-unique solutions.
LA - eng
KW - The semilinear reaction-diffusion equation; approximate controllability; internal lumped control multiple solutions.; semilinear reaction-diffusion equation; approximate controllability; internal lumped control; multiple solutions
UR - http://eudml.org/doc/197268
ER -

References

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