# American Institute of Mathematical Sciences

October  2014, 34(10): 4183-4210. doi: 10.3934/dcds.2014.34.4183

## Robust null controllability for heat equations with unknown switching control mode

 1 School of Mathematics, Sichuan University, Chengdu 610064, China 2 BCAM - Basque Center for Applied Mathematics, Mazarredo, 14, E-48009 Bilbao-Basque Country, Spain

Received  June 2012 Revised  August 2012 Published  April 2014

We analyze the null controllability for heat equations in the presence of switching controls. The switching pattern is a priori unknown so that the control has to be designed in a robust manner, based only on the past dynamics, so to fulfill the final control requirement, regardless of what the future dynamics is. We prove that such a robust control strategy actually exists when the switching controllers are located on two non trivial open subsets of the domain where the heat process evolves. Our strategy to construct these robust controls is based on earlier works by Lebeau and Robbiano on the null controllability of the heat equation. It is relevant to emphasize that our result is specific to the heat equation as an extension of a property of finite-dimensional systems that we fully characterize but that it may not hold for wave-like equations.
Citation: Qi Lü, Enrique Zuazua. Robust null controllability for heat equations with unknown switching control mode. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4183-4210. doi: 10.3934/dcds.2014.34.4183
##### References:
 [1] Y. Chitour and M. Sigalotti, On the stabilization of persistently excited linear systems,, SIAM J. Control Optim., 48 (2010), 4032. doi: 10.1137/080737812. [2] H. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension,, Arch. Rat. Mech. Anal., 43 (1971), 272. [3] E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing-up semilinear heat equations,, Annales Inst. Henri Poincaré, 17 (2000), 583. doi: 10.1016/S0294-1449(00)00117-7. [4] A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations,, Lecture Notes Series, (1996). [5] G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur,, Comm. Partial Differential Equations, 20 (1995), 335. doi: 10.1080/03605309508821097. [6] G. Lebeau and E. Zuazua, Null controllability of a system of linear thermoelasticity,, Arch. Rational Mech. Anal., 141 (1998), 297. doi: 10.1007/s002050050078. [7] Q. Lü, Bang-Bang principle of time optimal controls and null controllability of fractional order parabolic equations,, Acta Math. Sin. (Engl. Ser.), 26 (2010), 2377. doi: 10.1007/s10114-010-9051-1. [8] Q. Lü, A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators,, ESAIM Control Optim. Calc. Var., 19 (2013), 255. doi: 10.1051/cocv/2012008. [9] Q. Lü and G. Wang, On the existence of time optimal controls with constraints of the rectangular type for heat equations,, SIAM J. Control Optim., 49 (2011), 1124. doi: 10.1137/10081277X. [10] P. Martinez and J. Vancostenoble, Stabilisation et contrôle intermittent de l'équation des ondes,, C. R. Acad. Sci. Paris Sér. I Math, 218 (2005), 851. doi: 10.1016/S0764-4442(01)02128-0. [11] L. Miller, Controllability cost of conservative systems: Resolvent condition and transmutation,, J. Funct. Anal., 218 (2005), 425. doi: 10.1016/j.jfa.2004.02.001. [12] L. Miller, On the controllability of anomalous diffusions generated by the fractional laplacian,, Math. Control Signals Systems, 18 (2006), 260. doi: 10.1007/s00498-006-0003-3. [13] Yu. Netrusov and Yu. Safarov, Weyl asymptotic formula for the Laplacian on domains with rough boundaries,, Commun. Math. Phys., 253 (2005), 481. doi: 10.1007/s00220-004-1158-8. [14] D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations,, Studies in Appl. Math., 52 (1973), 189. [15] R. Shorten, F. Wirth, O. Mason, K. Wulff and Ch. King, Stability criteria for switched and hybrid systems,, SIAM Rev., 49 (2007), 545. doi: 10.1137/05063516X. [16] G. Wang, $L^\infty$-null controllability for the heat equation and its consequences for the time optimal control problem,, SIAM J. Control Optim., 47 (2008), 1701. doi: 10.1137/060678191. [17] E. Zuazua, Controllability and Observability of Partial Differential Equations: Some results and open problems,, in Handbook of Differential Equations: Evolutionary Differential Equations, (2006), 527. doi: 10.1016/S1874-5717(07)80010-7. [18] E. Zuazua, Switching control,, J. Eur. Math. Soc., 13 (2011), 85. doi: 10.4171/JEMS/245.

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##### References:
 [1] Y. Chitour and M. Sigalotti, On the stabilization of persistently excited linear systems,, SIAM J. Control Optim., 48 (2010), 4032. doi: 10.1137/080737812. [2] H. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension,, Arch. Rat. Mech. Anal., 43 (1971), 272. [3] E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing-up semilinear heat equations,, Annales Inst. Henri Poincaré, 17 (2000), 583. doi: 10.1016/S0294-1449(00)00117-7. [4] A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations,, Lecture Notes Series, (1996). [5] G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur,, Comm. Partial Differential Equations, 20 (1995), 335. doi: 10.1080/03605309508821097. [6] G. Lebeau and E. Zuazua, Null controllability of a system of linear thermoelasticity,, Arch. Rational Mech. Anal., 141 (1998), 297. doi: 10.1007/s002050050078. [7] Q. Lü, Bang-Bang principle of time optimal controls and null controllability of fractional order parabolic equations,, Acta Math. Sin. (Engl. Ser.), 26 (2010), 2377. doi: 10.1007/s10114-010-9051-1. [8] Q. Lü, A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators,, ESAIM Control Optim. Calc. Var., 19 (2013), 255. doi: 10.1051/cocv/2012008. [9] Q. Lü and G. Wang, On the existence of time optimal controls with constraints of the rectangular type for heat equations,, SIAM J. Control Optim., 49 (2011), 1124. doi: 10.1137/10081277X. [10] P. Martinez and J. Vancostenoble, Stabilisation et contrôle intermittent de l'équation des ondes,, C. R. Acad. Sci. Paris Sér. I Math, 218 (2005), 851. doi: 10.1016/S0764-4442(01)02128-0. [11] L. Miller, Controllability cost of conservative systems: Resolvent condition and transmutation,, J. Funct. Anal., 218 (2005), 425. doi: 10.1016/j.jfa.2004.02.001. [12] L. Miller, On the controllability of anomalous diffusions generated by the fractional laplacian,, Math. Control Signals Systems, 18 (2006), 260. doi: 10.1007/s00498-006-0003-3. [13] Yu. Netrusov and Yu. Safarov, Weyl asymptotic formula for the Laplacian on domains with rough boundaries,, Commun. Math. Phys., 253 (2005), 481. doi: 10.1007/s00220-004-1158-8. [14] D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations,, Studies in Appl. Math., 52 (1973), 189. [15] R. Shorten, F. Wirth, O. Mason, K. Wulff and Ch. King, Stability criteria for switched and hybrid systems,, SIAM Rev., 49 (2007), 545. doi: 10.1137/05063516X. [16] G. Wang, $L^\infty$-null controllability for the heat equation and its consequences for the time optimal control problem,, SIAM J. Control Optim., 47 (2008), 1701. doi: 10.1137/060678191. [17] E. Zuazua, Controllability and Observability of Partial Differential Equations: Some results and open problems,, in Handbook of Differential Equations: Evolutionary Differential Equations, (2006), 527. doi: 10.1016/S1874-5717(07)80010-7. [18] E. Zuazua, Switching control,, J. Eur. Math. Soc., 13 (2011), 85. doi: 10.4171/JEMS/245.
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