论文标题
对线性和半线性热方程的不敏感控制与部分未知域
Insensitizing control for linear and semi-linear heat equations with partially unknown domain
论文作者
论文摘要
我们考虑一个具有差异的边界条件的半线性热方程，并且在r^n（$ n *中的n $ \）的有界域上构成了全球lipchitz非线性，假定是参考域的未知扰动。我们对一个不敏感的控制问题感兴趣，该问题在于找到分布式控制，使该状态的某些功能在一阶对域的扰动时不敏感。我们的第一个结果包括半线性热方程式上的近似不敏化特性。它基于线性化过程，并使用适当的固定点定理。对于线性案例，开发了适当的二元性理论，因此可以将问题视为众所周知的独特延续定理的结果。我们的第二个结果是针对线性案例的。我们显示了一个或两个参数给出的某些变形族的精确不敏感的属性。由于固有控制问题的非线性，因此没有二元性理论可用，因此我们的证明依赖于几何方法和直接计算。
We consider a semi-linear heat equation with Dirichlet boundary conditions and globally Lipschitz nonlinearity, posed on a bounded domain of R^N (N $\in$ N *), assumed to be an unknown perturbation of a reference domain. We are interested in an insensitizing control problem, which consists in finding a distributed control such that some functional of the state is insensitive at the first order to the perturbations of the domain. Our first result consists of an approximate insensitization property on the semi-linear heat equation. It rests upon a linearization procedure together with the use of an appropriate fixed point theorem. For the linear case, an appropriate duality theory is developed, so that the problem can be seen as a consequence of well-known unique continuation theorems. Our second result is specific to the linear case. We show a property of exact insensitization for some families of deformation given by one or two parameters. Due to the nonlinearity of the intrinsic control problem, no duality theory is available, so that our proof relies on a geometrical approach and direct computations.
