Controllability properties for the one-dimensional Heat equation under multiplicative or nonnegative additive controls with local mobile support

Luis Alberto Fernández; Alexander Yuri Khapalov

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 4, page 1207-1224
  • ISSN: 1292-8119

Abstract

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We discuss several new results on nonnegative approximate controllability for the one-dimensional Heat equation governed by either multiplicative or nonnegative additive control, acting within a proper subset of the space domain at every moment of time. Our methods allow us to link these two types of controls to some extend. The main results include approximate controllability properties both for the static and mobile control supports.

How to cite

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Fernández, Luis Alberto, and Khapalov, Alexander Yuri. "Controllability properties for the one-dimensional Heat equation under multiplicative or nonnegative additive controls with local mobile support." ESAIM: Control, Optimisation and Calculus of Variations 18.4 (2012): 1207-1224. <http://eudml.org/doc/272942>.

@article{Fernández2012,
abstract = {We discuss several new results on nonnegative approximate controllability for the one-dimensional Heat equation governed by either multiplicative or nonnegative additive control, acting within a proper subset of the space domain at every moment of time. Our methods allow us to link these two types of controls to some extend. The main results include approximate controllability properties both for the static and mobile control supports.},
author = {Fernández, Luis Alberto, Khapalov, Alexander Yuri},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {parabolic equation; approximate controllability; multiplicative controls; nonnegative locally distributed controls; static and mobile control supports},
language = {eng},
number = {4},
pages = {1207-1224},
publisher = {EDP-Sciences},
title = {Controllability properties for the one-dimensional Heat equation under multiplicative or nonnegative additive controls with local mobile support},
url = {http://eudml.org/doc/272942},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Fernández, Luis Alberto
AU - Khapalov, Alexander Yuri
TI - Controllability properties for the one-dimensional Heat equation under multiplicative or nonnegative additive controls with local mobile support
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2012
PB - EDP-Sciences
VL - 18
IS - 4
SP - 1207
EP - 1224
AB - We discuss several new results on nonnegative approximate controllability for the one-dimensional Heat equation governed by either multiplicative or nonnegative additive control, acting within a proper subset of the space domain at every moment of time. Our methods allow us to link these two types of controls to some extend. The main results include approximate controllability properties both for the static and mobile control supports.
LA - eng
KW - parabolic equation; approximate controllability; multiplicative controls; nonnegative locally distributed controls; static and mobile control supports
UR - http://eudml.org/doc/272942
ER -

References

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  1. [1] A. Baciotti, Local Stabilizability of Nonlinear Control Systems. World Scientific, Singapore (1992). Zbl0757.93061MR1148363
  2. [2] J.M. Ball and M. Slemrod, Feedback stabilization of distributed semilinear control systems. Appl. Math. Optim.5 (1979) 169–179. Zbl0405.93030MR533618
  3. [3] J.M. Ball, J.E. Mardsen and M. Slemrod, Controllability for distributed bilinear systems. SIAM J. Control Optim.20 (1982) 575–597. Zbl0485.93015MR661034
  4. [4] L. Baudouin and J. Salomon, Constructive solution of a bilinear optimal control problem for a Schrödinger equation. Syst. Control Lett.57 (2008) 453–464. Zbl1153.49023MR2413741
  5. [5] K. Beauchard and C. Laurent, Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control. J. Math. Pures Appl.94 (2010) 520–554. Zbl1202.35332MR2732927
  6. [6] P. Cannarsa and A.Y. Khapalov, Multiplicative controllability for reaction-diffusion equations with target states admitting finitely many changes of sign. Discrete Contin. Dyn. Syst. Ser. B14 (2010) 1293–1311. Zbl1219.93010MR2679642
  7. [7] T. Chambrion, P. Mason, M. Sigalotti, and U. Boscain, Controllability of the discrete-spectrum Schrödinger equation driven by an external field. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009) 329–349. Zbl1161.35049MR2483824
  8. [8] J.M. Coron, On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well. C. R. Math. Acad. Sci. Paris342 (2006) 103–108. Zbl1082.93002MR2193655
  9. [9] J.I. Díaz, J. Henry and A.M. Ramos, On the approximate controllability of some semilinear parabolic boundary-value problems. Appl. Math. Optim.37 (1998) 71–97. Zbl0936.93010MR1485063
  10. [10] S. Ervedoza and J.P. Puel, Approximate controllability for a system of Schrödinger equations modeling a single trapped ion. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009) 2111–2136. Zbl1180.35437MR2569888
  11. [11] L.A. Fernández, Controllability of some semilinear parabolic problems with multiplicative control, presented at the Fifth SIAM Conference on Control and its applications, held in San Diego (2001). 
  12. [12] A. Friedman, Partial Differential Equations. Holt, Rinehart and Winston, New York (1969). Zbl0224.35002MR445088
  13. [13] A.Y. Khapalov, Mobile point controls versus locally distributed ones for the controllability of the semilinear parabolic equation. SIAM J. Control Optim.40 (2001) 231–252. Zbl0995.93038MR1855314
  14. [14] A.Y. Khapalov, Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term : A qualitative approach. SIAM J. Control. Optim.41 (2003) 1886–1900. Zbl1041.93026MR1972539
  15. [15] A.Y. Khapalov, Controllability properties of a vibrating string with variable axial load. Discrete Contin. Dyn. Syst.11 (2004) 311–324. Zbl1175.93032MR2083420
  16. [16] A.Y. Khapalov, Controllability of Partial Differential Equations Governed by Multiplicative Controls, edited by Springer Verlag. Lect. Notes Math. 1995 (2010). Zbl1210.93005MR2641453
  17. [17] A.Y. Khapalov and R.R. Mohler, Reachable sets and controllability of bilinear time-invariant systems : A qualitative approach. IEEE Trans. Automat. Control41 (1996) 1342–1346. Zbl0876.93011MR1409481
  18. [18] K. Kime, Simultaneous control of a rod equation and a simple Schrödinger equation. Syst. Control Lett.24 (1995) 301–306. Zbl0877.93003MR1321139
  19. [19] O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type. Am. Math. Soc., Providence, RI (1968). Zbl0174.15403
  20. [20] S. Lenhart and M. Liang, Bilinear optimal control for a wave equation with viscous damping. Houston J. Math.26 (2000) 575–595. Zbl0976.49005MR1811943
  21. [21] M. Liang, Bilinear optimal control for a wave equation. Math. Models Methods Appl. Sci.9 (1999) 45–68. Zbl0939.49016MR1671527
  22. [22] J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag (1971). Zbl0203.09001MR271512
  23. [23] S. Müller, Strong convergence and arbitrarily slow decay of energy for a class of bilinear control problems. J. Differ. Equ.81 (1989) 50–67. Zbl0711.35017MR1012199
  24. [24] A.I. Prilepko, D.G. Orlovsky and I.A. Vasin, Methods for solving inverse problems in mathematical physics. Marcel Dekker Inc., New York (2000). Zbl0947.35173MR1748236
  25. [25] R. Rink and R.R. Mohler, Completely controllable bilinear systems. SIAM J. Control6 (1968) 477–486. Zbl0159.13001MR255270

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