×

Interior controllability of the \(nD\) semilinear heat equation. (English) Zbl 1243.93020

Summary: In this paper, we prove the interior approximate controllability of the following semilinear heat equation \[ \begin{cases} z_{t}(t,x) = \Delta z(t,x) + 1_{\omega}u(t,x)+f(t,z,u(t,x)) & \text{in} \quad (0, \tau] \times \Omega,\\ z = 0, & \quad \text{on} \quad (0, \tau) \times \partial \Omega, \\ z(0,x) = z_{0}(x), & x \in\Omega, \end{cases} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^{N}(N\geq1), z_0 \in L^{2}(\Omega)\), \(\omega\) is an open nonempty subset of \(\Omega\), and \(1_{\omega}\) denotes the characteristic function of the set \(\omega\). The distributed control \(u\) belong to \(\in L^{2}([0,\tau]; L^{2}(\Omega;))\) and the nonlinear function \(f:[0, \tau] \times \mathbb R \times \mathbb R \rightarrow \mathbb R\) is smooth enough and there are \(a,b, c \in \mathbb R\), with \(c \neq -1\), such that \[ \sup_{(t,z,u) \in Q_{\tau}} |f(t,z,u) -az-cu-b | < \infty, \] where \(Q_{\tau}= [0, \tau] \times \mathbb R \times \mathbb R\). Under this condition, we prove the following statement: For all open nonempty subset \(\omega\) of \(\Omega\) the system is approximately controllable on \([0, \tau]\). Moreover, we could exhibit a sequence of controls steering the nonlinear system (1) from an initial state \(z_0\) to an \(\epsilon\) neighborhood of the final state \(z_1\) at time \(\tau >0\), which is very important from a practical and numerical point of view.

MSC:

93B05 Controllability
93C25 Control/observation systems in abstract spaces
35K05 Heat equation
PDFBibTeX XMLCite
Full Text: Euclid

References:

[1] J. APPELL, H. LEIVA, N. MERENTES ANDA A. VIGNOLI, Un espectro de compresión no lineal con aplicaciones a la controlabilidad aproximada de sistemas semilineales , preprint S. AXLER, P. BOURDON AND W. RAMEY, Harmonic Fucntion Theory. Graduate Texts in Math., 137 . Springer Verlag, New york (1992).
[2] D.BARCENAS, H. LEIVA AND Z. SIVOLI, A Broad Class of Evolution Equations are Approximately Controllable, but Never Exactly Controllable. IMA J. Math. Control Inform. 22 , no. 3 (2005), 310-320. · Zbl 1108.93014 · doi:10.1093/imamci/dni029
[3] D.BARCENAS, H. LEIVA AND W. URBINA, Controllability of the Ornstein-Uhlenbeck Equation. IMA J. Math. Control Inform. 23 no. 1, (2006), 1-9. · Zbl 1106.93007 · doi:10.1093/imamci/dni016
[4] R.F. CURTAIN, A.J. PRITCHARD, Infinite Dimensional Linear Systems. Lecture Notes in Control and Information Sciences, 8 . Springer Verlag, Berlin (1978). · Zbl 0389.93001
[5] R.F. CURTAIN, H.J. ZWART, An Introduction to Infinite Dimensional Linear Systems Theory. Text in Applied Mathematics, · Zbl 0839.93001
[6] Springer Verlag, New York (1995).
[7] J.I. DIAZ, J.HENRY AND A.M. RAMOS, “On the Approximate Controllability of Some Semilinear Parabolic Boundary-Value Problemas”, Appl. Math. Optim 37-71 (1998). · Zbl 0936.93010 · doi:10.1007/s002459900069
[8] E. FERNANDEZ-CARA, “ Remark on Approximate and Null Controllability of Semilinear Parabolic Equations” ESAIM:Proceeding OF CONTROLE ET EQUATIONS AUX DERIVEES PARTIELLES, Vol.4, 1998, 73-81. · Zbl 0908.93014 · doi:10.1051/proc:1998021
[9] E. FERNANDEZ-CARA AND E. ZUAZUA,“Controllability for Blowing up Semilinear Parabolic Equations”, C.R. Acad. Sci. Paris, t. 330, serie I, p. 199-204, 2000. · Zbl 0952.93061 · doi:10.1016/S0764-4442(00)00115-4
[10] LUIZ A. F. de OLIVEIRA “On Reaction-Diffusion Systems” E. Journal of Differential Equations, Vol. 1998(1998), N0. 24, pp. 1-10. · Zbl 0909.35070
[11] H. LEIVA, “A Lemma on \(C_{0}\)-Semigroups and Applications PDEs Systems” Quaestions Mathematicae, Vol. 26, pp. 247-265 (2003). · Zbl 1073.47046 · doi:10.2989/16073600309486057
[12] H. LEIVA“Controllability of a System of Parabolic equation with non-diagonal diffusion matrix”. IMA Journal of Mathematical Control and Information; Vol. 32, 2005, pp. 187-199. · Zbl 1079.93007 · doi:10.1093/imamci/dni023
[13] H. LEIVA and Y. QUINTANA, “Interior Controllability of a Broad Class of Reaction Diffusion Equations”, Mathematical Problems in Engineering, Vol. 2009, Article ID 708516, 8 pages, doi:10.1155/2009/708516. · Zbl 1206.93017 · doi:10.1155/2009/708516
[14] XU ZHANG, A Remark on Null Exact Controllability of the Heat Equation. IAM J. CONTROL OPTIM. Vol. 40, No. 1(2001), pp. 39-53. · Zbl 1002.93025 · doi:10.1137/S0363012900371691
[15] E. ZUAZUA, Controllability of a System of Linear Thermoelasticity , J. Math. Pures Appl., 74 , (1995), 291-315. · Zbl 0846.93008
[16] E. ZUAZUA, Control of Partial Differential Equations and its Semi-Discrete Approximation. Discrete and Continuous Dynamical Systems, vol. 8 , No. 2. April (2002), 469-513. · Zbl 1005.35019 · doi:10.3934/dcds.2002.8.469
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.