Null and approximate controllability for weakly blowing up semilinear heat equations. (English) Zbl 0970.93023

Summary: We consider the semilinear heat equation in a bounded domain of \(\mathbb{R}^d\), with control on a subdomain and homogeneous Dirichlet boundary conditions. We prove that the system is null-controllable at any time provided a globally defined and bounded trajectory exists and the nonlinear term \(f(y)\) is such that \(|f(s)|\) grows slower than \(|s|\log^{3/2}(1+|s|)\) as \(|s|\to\infty\). For instance, this condition is fulfilled by any function \(f\) growing at infinity like \(|s|\log^p(1+|s|)\) with \(1< p<3/2\) (in this case, in the absence of control, blow-up occurs). We also prove that, for some functions \(f\) that behave at infinity like \(|s|\log^p(1+|s|)\) with \(p> 2\), null controllability does not hold. The problem remains open when \(f\) behaves at infinity like \(|s|\log^p(1+|s|)\), with \(3/2\leq p\leq 2\). Results of the same kind are proved in the case of approximate controllability.


93C20 Control/observation systems governed by partial differential equations
93B05 Controllability
93C10 Nonlinear systems in control theory
35B37 PDE in connection with control problems (MSC2000)
Full Text: DOI Numdam EuDML


[1] Anita, S; Barbu, V, Internal null controllability of nonlinear heat equation, preprint, (1999)
[2] Barbu, V, Exact controllability of the superlinear heat equation, preprint, (1998)
[3] Brezis, H; Cazenave, Th; Martel, Y; Ramiandrisoa, A, Blow up for ut−δu=g(u) revisited, Adv. differential equations, Vol. 1, 1, 73-90, (1996) · Zbl 0855.35063
[4] Cannarsa P.M., Komornik V., Loreti P., Well posedness and control of the semilinear wave equation with iterated logarithms, ESAIM: COCV (to appear) · Zbl 0959.93026
[5] Cazenave, T; Haraux, A, Equations d’évolution avec nonlinéarité logarithmique, Ann. fac. sci. Toulouse, Vol. 2, 21-51, (1980) · Zbl 0411.35051
[6] Cazenave, T; Haraux, A; Barbu, V, Introduction aux problèmes d’évolution semilinéaires, Mathématiques & applications, (1989), Ellipses Paris
[7] Fabre, C; Puel, J.P; Zuazua, E, Approximate controllability of the semilinear heat equation, Proc. royal soc. Edinburgh, Vol. 125A, 31-61, (1995) · Zbl 0818.93032
[8] Fernández-Cara, E, Null controllability of the semilinear heat equation, Esaim: cocv, Vol. 2, 87-107, (1997) · Zbl 0897.93011
[9] Fernández-Cara E., Zuazua E., The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations (to appear) · Zbl 1007.93034
[10] Fursikov, A; Imanuvilov, O.Yu, Controllability of evolution equations, Lecture notes, Vol. 34, (1996), Seoul National University Korea · Zbl 0862.49004
[11] Galaktionov, V, On blow-up and degeneracy for the semilinear heat equation with source, Proc. royal soc. Edinburgh, Vol. 115A, 19-24, (1990) · Zbl 0715.35006
[12] Galaktionov, V; Vázquez, J.L, Regional blow-up in a semilinear heat equation with convergence to a hamilton – jacobi equation, SIAM J. math. anal, Vol. 24, 5, 1125-1276, (1993) · Zbl 0813.35033
[13] Glass, O, Sur la contrôlabilité des fluides parfaits incompressibles, ph.D. thesis, (January 2000), Université de Paris XI Orsay, France
[14] Henry, J, Contrôle d’un réacteur enzymatique à l’aide de modèles à paramètres distribués: quelques problèmes de contrôlabilité de systèmes paraboliques, ph.D. thesis, (1978), Université Paris VI
[15] Imanuvilov, O.Yu, Exact boundary controllability of the parabolic equation, Russian math. surveys, Vol. 48, 211-212, (1993)
[16] Khapalov, A, Some aspects of the asymptotic behavior of the solutions of the semilinear heat equation and approximate controllability, J. math. anal. appl., Vol. 194, 858-882, (1995) · Zbl 0846.35059
[17] Zuazua, E, Exact boundary controllability for the semilinear wave equation, (), 357-391
[18] Zuazua, E, Finite dimensional null-controllability of the semilinear heat equation, J. math. pures et appl., Vol. 76, 237-264, (1997) · Zbl 0872.93014
[19] Zuazua, E, Exact controllability for the semilinear wave equation in one space dimension, Ann. IHP, analyse non linéaire, Vol. 10, 109-129, (1996) · Zbl 0769.93017
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.