
Previous Article
Exterior Problem of Boltzmann Equation with Temperature Difference
 CPAA Home
 This Issue

Next Article
Symmetry groups in nonlinear elasticity: an exercise in vintage mathematics
Identification of the class of initial data for the insensitizing control of the heat equation
1.  Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico 
2.  Basque Center for Applied Mathematics (BCAM), Gran Via 35, 48009 Bilbao 
But in the context of pure insensitization there are very few results identifying the class of initial data that can be insensitized. This is a delicate issue which is related to the fact that insensitization turns out to be equivalent to suitable observability estimates for a coupled system of heat equations, one being forward and the other one backward in time. The existing Carleman inequalities techniques can be applied but they only give interior information of the solutions, which hardly allows identifying the initial data because of the strong irreversibility of the equations involved in the system, one of them being an obstruction at the initial time $t=0$ and the other one at the final one $t=T$.
In this article we consider different geometric configurations in which the subdomains to be insensitized and the one in which the external control acts play a key role. We show that, under rather restrictive geometric restrictions, initial data in a class that can be characterized in terms of a summability condition of their Fourier coefficients with suitable weights, can be insensitized. But, the main result of the paper, which might seem surprising, shows that this fails to be true in general, so that even the first eigenfunction of the system can not be insensitized. This result is similar to those obtained in the context of the null controllability of the heat equation in unbounded domains in [14] where it is shown that smooth and compactly supported initial data may not be controlled.
Our proofs combine the existing observability results for heat equations obtained by means of Carleman inequalities, energy and gaussian estimates and Fourier expansions.
[1] 
Franck Boyer, Víctor HernándezSantamaría, Luz De Teresa. Insensitizing controls for a semilinear parabolic equation: A numerical approach. Mathematical Control & Related Fields, 2019, 9 (1) : 117158. doi: 10.3934/mcrf.2019007 
[2] 
Fredi Tröltzsch, Daniel Wachsmuth. On the switching behavior of sparse optimal controls for the onedimensional heat equation. Mathematical Control & Related Fields, 2018, 8 (1) : 135153. doi: 10.3934/mcrf.2018006 
[3] 
Víctor HernándezSantamaría, Liliana Peralta. Some remarks on the Robust Stackelberg controllability for the heat equation with controls on the boundary. Discrete & Continuous Dynamical Systems  B, 2020, 25 (1) : 161190. doi: 10.3934/dcdsb.2019177 
[4] 
Donghui Yang, Jie Zhong. Optimal actuator location of the minimum norm controls for stochastic heat equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 10811095. doi: 10.3934/mcrf.2018046 
[5] 
C. Brändle, E. Chasseigne, Raúl Ferreira. Unbounded solutions of the nonlocal heat equation. Communications on Pure & Applied Analysis, 2011, 10 (6) : 16631686. doi: 10.3934/cpaa.2011.10.1663 
[6] 
Arthur Ramiandrisoa. Nonlinear heat equation: the radial case. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 849870. doi: 10.3934/dcds.1999.5.849 
[7] 
Delio Mugnolo. Gaussian estimates for a heat equation on a network. Networks & Heterogeneous Media, 2007, 2 (1) : 5579. doi: 10.3934/nhm.2007.2.55 
[8] 
Sergei A. Avdonin, Sergei A. Ivanov, JunMin Wang. Inverse problems for the heat equation with memory. Inverse Problems & Imaging, 2019, 13 (1) : 3138. doi: 10.3934/ipi.2019002 
[9] 
Chulan Zeng. Time analyticity of the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations. Communications on Pure & Applied Analysis, , () : . doi: 10.3934/cpaa.2021197 
[10] 
Karl Kunisch, Lijuan Wang. The bangbang property of time optimal controls for the Burgers equation. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 36113637. doi: 10.3934/dcds.2014.34.3611 
[11] 
Nicolas Hegoburu. Time optimal internal controls for the LotkaMcKendrick equation with spatial diffusion. Mathematical Control & Related Fields, 2019, 9 (4) : 697718. doi: 10.3934/mcrf.2019047 
[12] 
Yunfei Peng, X. Xiang. A class of nonlinear impulsive differential equation and optimal controls on time scales. Discrete & Continuous Dynamical Systems  B, 2011, 16 (4) : 11371155. doi: 10.3934/dcdsb.2011.16.1137 
[13] 
Karl Kunisch, Lijuan Wang. Bangbang property of time optimal controls of semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 279302. doi: 10.3934/dcds.2016.36.279 
[14] 
U. Biccari, V. HernándezSantamaría, J. Vancostenoble. Existence and cost of boundary controls for a degenerate/singular parabolic equation. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021032 
[15] 
Laurent Bourgeois. Quantification of the unique continuation property for the heat equation. Mathematical Control & Related Fields, 2017, 7 (3) : 347367. doi: 10.3934/mcrf.2017012 
[16] 
Arturo de Pablo, Guillermo Reyes, Ariel Sánchez. The Cauchy problem for a nonhomogeneous heat equation with reaction. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 643662. doi: 10.3934/dcds.2013.33.643 
[17] 
Ovidiu Cârjă, Alina Lazu. On the minimal time null controllability of the heat equation. Conference Publications, 2009, 2009 (Special) : 143150. doi: 10.3934/proc.2009.2009.143 
[18] 
Thierry Cazenave, Flávio Dickstein, Fred B. Weissler. Universal solutions of the heat equation on $\mathbb R^N$. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 11051132. doi: 10.3934/dcds.2003.9.1105 
[19] 
Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 52215237. doi: 10.3934/dcds.2015.35.5221 
[20] 
Angkana Rüland, Mikko Salo. Quantitative approximation properties for the fractional heat equation. Mathematical Control & Related Fields, 2020, 10 (1) : 126. doi: 10.3934/mcrf.2019027 
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]