Null and approximate controllability for weakly blowing up semilinear heat equations
Null and approximate controllability for weakly blowing up semilinear heat equations
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DOI：
10.1016/s0294-1449(00)00117-7
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发表时间：
2000-09
影响因子：
1.9
通讯作者：
E. Fernández-Cara;E. Zuazua
中科院分区：
文献类型：
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作者：
E. Fernández-Cara;E. Zuazua
考虑Rd中有界区域上的半线性热方程，其子域上具有控制和齐次Dirichlet边界条件。我们证明了系统在任何时候都是零能控的，只要存在全局定义有界的轨迹，并且非线性项f（y）满足|f（秒）|增长慢于|S| log3/2（1+| S|）作为|S| →∞。例如，这个条件可以由任何在无穷远处增长的函数f满足，如|S| logp（1+| S|），其中1<p<3/2（在这种情况下，在不存在对照的情况下，发生爆破）。我们还证明了，对于某些函数f的行为在无穷大，如|S| logp（1+| S|当p>2时，零可控性不成立。当f在无穷大时，问题仍然是开放的，|S| logp（1+| S|），其中3/2≤p≤2。同类的结果证明在近似可控性的背景下。
We consider the semilinear heat equation in a bounded domain of Rd, 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|log3/2(1+|s|) as |s|→∞ . For instance, this condition is fulfilled by any function f growing at infinity like |s|logp(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 infinite like |s|logp(1+|s|) with p>2 , null controllability does not hold. The problem remains open when f behaves at infinity like |s|logp(1+|s|) , with 3/2≤p≤2 . Results of the same kind are proved in the context of approximate controllability.