# Control Norms for Large Control Times

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

- Volume: 4, page 405-418
- ISSN: 1292-8119

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topIvanov, Sergei. "Control Norms for Large Control Times." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 405-418. <http://eudml.org/doc/197305>.

@article{Ivanov2010,

abstract = {
A control system of the second order in time with control
$u=u(t) \in L^2([0,T];U)$ is considered. If the
system is controllable in a strong sense and
uT is the control
steering the system to the rest at time
T,
then the L2–norm of uT decreases as $1/\sqrt T$
while the $L^1([0,T];U)$–norm of uT is approximately constant.
The proof is based on the moment approach
and properties of the relevant exponential family. Results are
applied to the wave equation with boundary or distributed controls.
},

author = {Ivanov, Sergei},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Controllability; exponential families.; families of exponentials; biorthogonal functions; Riesz bases},

language = {eng},

month = {3},

pages = {405-418},

publisher = {EDP Sciences},

title = {Control Norms for Large Control Times},

url = {http://eudml.org/doc/197305},

volume = {4},

year = {2010},

}

TY - JOUR

AU - Ivanov, Sergei

TI - Control Norms for Large Control Times

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 4

SP - 405

EP - 418

AB -
A control system of the second order in time with control
$u=u(t) \in L^2([0,T];U)$ is considered. If the
system is controllable in a strong sense and
uT is the control
steering the system to the rest at time
T,
then the L2–norm of uT decreases as $1/\sqrt T$
while the $L^1([0,T];U)$–norm of uT is approximately constant.
The proof is based on the moment approach
and properties of the relevant exponential family. Results are
applied to the wave equation with boundary or distributed controls.

LA - eng

KW - Controllability; exponential families.; families of exponentials; biorthogonal functions; Riesz bases

UR - http://eudml.org/doc/197305

ER -

## References

top- M. Asch and G. Lebeau, Geometrical aspects of exact boundary controllability for the wave equation - a numerical study. ESAIM: Contr., Optim. Cal. Var.3 (1998) 163-212.
- S. Avdonin and S. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, N.Y. (1995).
- S.A. Avdonin, M.I. Belishev and S.A. Ivanov, Controllability in filled domain for the multidimensional wave equation with singular boundary control. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)210 (1994) 7-21.
- S.A. Avdonin, S.A. Ivanov and D.L. Russell, Exponential bases in Sobolev spaces in control and observation problems for the wave equation. Proc. Roy. Soc. Edinburgh (to be submitted).
- C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Theor. Appl.30 (1992) 1024-1095.
- H.O. Fattorini, Estimates for sequences biorthogonal to certain complex exponentials and boundary control of the wave equation, Springer, Lecture Notes in Control and Information Sciences2 (1979).
- R. Glowinski, C.-H. Li and J.-L. Lions, A numerical approach to the exact controllability of the wave equation. (I) Dirichlet controls: description of the numerical methods. Japan J. Appl. Math.7 (1990) 1-76.
- F. Gozzi and P. Loreti, Regularity of the minimum time function and minimum energy problems: the linear case. SIAM J. Control Optim. (to appear).
- W. Krabs, On Moment Theory and Controllability of one-dimensional vibrating Systems and Heating Processes, Springer, Lecture Notes in Control and Information Sciences173 (1992).
- W. Krabs, G. Leugering and T. Seidman, On boundary controllability of a vibrating plate. Appl. Math. Optim.13 (1985) 205-229.
- I. Lasiecka, J.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators. J. Math. Pures Appl.65 (1986) 149-192.
- J.-L. Lions, Contrôlabilité exacte, stabilisation et perturbation des systèmes distribués, Masson, Paris Collection RMA1 (1988).
- N.K. Nikol'skii, A Treatise on the Shift Operator, Springer, Berlin (1986).
- D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions. SIAM Rev.20 (1978) 639-739.
- T.I. Seidman, The coefficient map for certain exponential sums. Nederl. Akad. Wetensch. Proc. Ser. A89 (= Indag. Math. 48) (1986) 463-468.
- T.I. Seidman, S.A. Avdonin and S.A. Ivanov, The ``window problem'' for complex exponentials. Fourier Analysis and Applications (to appear).
- D. Tataru, Unique continuation for solutions of PDE's; between Hörmander's theorem and Holmgren's theorem. Comm. PDE20 (1995) 855-884.

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