×

Controls insensitizing the norm of the solution of a semilinear heat equation in unbounded domains. (English) Zbl 0895.93023

The author considers the following parabolic semilinear system: \[ \begin{cases} z_t-\Delta z+f(z)= \xi+ h\chi_\omega &\text{ in } Q=\Omega\times (0,T)\\ z=0 &\text{ on }\Sigma= \partial\Omega\times (0,T),\\ z(x,0)= y^0(x)+ \tau_0z^0 &\text{ in }\Omega, \end{cases} \tag{1} \] where \(\Omega\subset \mathbb{R}^n\) is an open unbounded set of class \(C^2\) uniformly, \(T>0\), \(\omega\) and \(\theta\) two open bounded subsets of \(\Omega\), \(f\) is a globally Lipschitz \(C^2\) function defined on \(\mathbb{R}\), with \(f(0)=0\) and with bounded second derivative. Here, the state equation (1) has incomplete data, i.e. \(\xi\) and \(y^0\) are given respectively in \(L^2(Q)\) and \(L^2(\Omega)\), \(z^0\in L^2(\Omega)\) is unknown and \(\| z^0\|_{L^2(\Omega)}=1\), \(\tau_0\in \mathbb{R}\) is unknown and small enough, \(h=h(x,t)\) is a control term to be determined.
First, the author recalls the notion of insensitizing control introduced by J. L. Lions and generalized by Bodart-Fabre in the following way: Let \(\varphi\) be a differentiable functional defined on the set of solutions of (1) and let \(\varepsilon>0\); we say that the control \(h\) \(\varepsilon\)-insensitizes \(\varphi(z)\) if \[ \Biggl| \frac{\partial\varphi(z(x,t;h,\tau_0))} {\partial \tau_0} \Biggr|_{\tau_0=0} \leq\varepsilon. \tag{2} \] The functional used in this paper is \[ \varphi(z)= \frac 12 \int_0^T \int_\theta z^2(x,t)dx dt. \tag{3} \] This is a case in which the condition (2) is equivalent to an approximate control problem.
The author generalizes the results of Bodart and Fabre to unbounded sets \(\Omega\) as follows: Let \(\Omega\subset\mathbb{R}^n\) be an open and unbounded set with boundary of class \(C^2\) uniformly, and suppose that \(\omega\cap\theta\neq \emptyset\) and the data \(\xi\in L^2(Q)\) and \(y^0\in L^2(\Omega)\) have compact support; then for every \(\varepsilon>0\), there exists a control \(h\in L^2(\omega\times (0,T))\), \(\varepsilon\)-insensitizing the functional (3). If \(1\leq n\leq 6\), the condition on the data \(\xi\) and \(y^0\) to have compact support can be eliminated. If \(n\geq 7\) the conclusion is true for data \(\xi\in L^1(0,T; L^2(\Omega))\cap L^{n/2}(\Omega)\) and \(y^0\in L^2(\Omega)\cap L^{n/2}(\Omega)\).

MSC:

93C20 Control/observation systems governed by partial differential equations
93B35 Sensitivity (robustness)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] S. Angenent: The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390, 1988, 79-96. Zbl0644.35050 MR953678 · Zbl 0644.35050 · doi:10.1515/crll.1988.390.79
[2] O. Bodart, C. Fabre: Controls Insensitizing the Norm of the Solution of a Semilinear Heat Equation, J. Math. An. and App., 195, 1995, 658-683. Zbl0852.35070 MR1356636 · Zbl 0852.35070 · doi:10.1006/jmaa.1995.1382
[3] F.E. Browder: Estimates and existence theorems for elliptic boundary value problems. Proc. N. A. S., 45, 1959, 365-375. Zbl0093.29402 MR132913 · Zbl 0093.29402 · doi:10.1073/pnas.45.3.365
[4] T. Cazenave, A. Haraux: Introduction aux problèmes d’évolution semi-linéaires, Collection S.M.A.I. Mathématiques et applications, Ellipses, Paris, 1980. Zbl0786.35070 · Zbl 0786.35070
[5] L. de Teresa, E. Zuazua: Approximate controllability of a semilinear heat equation in unbounded domains, Preprint. · Zbl 0946.93006 · doi:10.1016/S0362-546X(98)00085-6
[6] C. Fabre, J.P. Puel, E. Zuazua: Approximate controllability of the semilinear heat equation, Proc. Roy. Soc. Edinburgh, Sect. A, 125, 1995, 31-61. Zbl0818.93032 MR1318622 · Zbl 0818.93032 · doi:10.1017/S0308210500030742
[7] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural’ceva: Linear and Quasilinear Equations of Parabolic Type, A.M.S., Rhode Island, 1968. MR241822
[8] J.L. Lions: Remarques préliminaires sur le contrôle des systèmes à données incomplètes, in Actas del Congreso de Ecuaciones Diferenciales y Aplicaciones (CEDYA), Universidad de Málaga, 1989, 43-54. Zbl0724.35050 · Zbl 0724.35050
[9] J.L Lions, E. Magenes: Problèmes aux limites non homogènes et applications. Vol I & II, Dunod, Paris, 1968. Zbl0165.10801 · Zbl 0165.10801
[10] A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. Zbl0516.47023 MR710486 · Zbl 0516.47023
[11] J.C. Saut, B. Scheurer: Unique continuation for some evolution equations, J. Diff. Equations, 66(1), 1987, 118-139. Zbl0631.35044 MR871574 · Zbl 0631.35044 · doi:10.1016/0022-0396(87)90043-X
[12] J. Simon: Compact sets in the space Lp(0,T;B), Annali di Matematica Pura ed Applicata, CXLVI, (IV), 1987, 1173-1191. Zbl0629.46031 MR916688 · Zbl 0629.46031 · doi:10.1007/BF01762360
[13] H. Tanabe: Equations of Evolution, Pitman, Bath, 1979 Zbl0417.35003 MR533824 · Zbl 0417.35003
[14] H. Triebel: Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Berlin, 1978. Zbl0387.46032 MR503903 · Zbl 0387.46032
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.