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

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

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

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topKhapalov, 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 -

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