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Controllability properties for the one-dimensional Heatequation under multiplicative or nonnegative additive controls with local mobilesupport∗∗∗

Published online by Cambridge University Press:  27 March 2012

Luis Alberto Fernández  and
Alexander Yuri Khapalov
Luis Alberto Fernández
Affiliation:
Departamento de Matemáticas, Estadística y Computación, Avda. de los Castros, s/n, Universidad de Cantabria, 39005 Santander, Spain. [email protected]
Alexander Yuri Khapalov
Affiliation:
Department of Mathematics, Washington State University, Pullman, 99164-3113 WA, USA; [email protected]
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Abstract

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

Keywords

Parabolic equationapproximate controllabilitymultiplicative controlsnonnegative locally distributed controls
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 4 , October 2012 , pp. 1207 - 1224
Copyright
© EDP Sciences, SMAI, 2012

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References

A. Baciotti, Local Stabilizability of Nonlinear Control Systems. World Scientific, Singapore (1992).
Ball, J.M. and Slemrod, M., Feedback stabilization of distributed semilinear control systems. Appl. Math. Optim. 5 (1979) 169–179. Google Scholar
Ball, J.M., Mardsen, J.E. and Slemrod, M., Controllability for distributed bilinear systems. SIAM J. Control Optim. 20 (1982) 575–597. Google Scholar
Baudouin, L. and Salomon, J., Constructive solution of a bilinear optimal control problem for a Schrödinger equation. Syst. Control Lett. 57 (2008) 453–464. Google Scholar
Beauchard, K. and Laurent, C., Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control. J. Math. Pures Appl. 94 (2010) 520–554. Google Scholar
Cannarsa, P. and Khapalov, A.Y., Multiplicative controllability for reaction-diffusion equations with target states admitting finitely many changes of sign. Discrete Contin. Dyn. Syst. Ser. B 14 (2010) 1293–1311. Google Scholar
Chambrion, T., Mason, P., Sigalotti, M., and Boscain, U., Controllability of the discrete-spectrum Schrödinger equation driven by an external field. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009) 329–349. Google Scholar
Coron, J.M., On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well. C. R. Math. Acad. Sci. Paris 342 (2006) 103–108. Google Scholar
Díaz, J.I., Henry, J. and Ramos, A.M., On the approximate controllability of some semilinear parabolic boundary-value problems. Appl. Math. Optim. 37 (1998) 71–97. Google Scholar
Ervedoza, S. and Puel, J.P., Approximate controllability for a system of Schrödinger equations modeling a single trapped ion. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009) 2111–2136. Google Scholar
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).
A. Friedman, Partial Differential Equations. Holt, Rinehart and Winston, New York (1969).
Khapalov, A.Y., Mobile point controls versus locally distributed ones for the controllability of the semilinear parabolic equation. SIAM J. Control Optim. 40 (2001) 231–252. Google Scholar
Khapalov, A.Y., 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. Google Scholar
Khapalov, A.Y., Controllability properties of a vibrating string with variable axial load. Discrete Contin. Dyn. Syst. 11 (2004) 311–324. Google Scholar
A.Y. Khapalov, Controllability of Partial Differential Equations Governed by Multiplicative Controls, edited by Springer Verlag. Lect. Notes Math. 1995 (2010).
Khapalov, A.Y. and Mohler, R.R., Reachable sets and controllability of bilinear time-invariant systems : A qualitative approach. IEEE Trans. Automat. Control 41 (1996) 1342–1346. Google Scholar
Kime, K., Simultaneous control of a rod equation and a simple Schrödinger equation. Syst. Control Lett. 24 (1995) 301–306. Google Scholar
O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type. Am. Math. Soc., Providence, RI (1968).
Lenhart, S. and Liang, M., Bilinear optimal control for a wave equation with viscous damping. Houston J. Math. 26 (2000) 575–595. Google Scholar
Liang, M., Bilinear optimal control for a wave equation. Math. Models Methods Appl. Sci. 9 (1999) 45–68. Google Scholar
J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag (1971).
Müller, S., Strong convergence and arbitrarily slow decay of energy for a class of bilinear control problems. J. Differ. Equ. 81 (1989) 50–67. Google Scholar
A.I. Prilepko, D.G. Orlovsky and I.A. Vasin, Methods for solving inverse problems in mathematical physics. Marcel Dekker Inc., New York (2000).
Rink, R. and Mohler, R.R., Completely controllable bilinear systems. SIAM J. Control 6 (1968) 477–486.Google Scholar