It is by now well known that the use of Carleman estimates allows to establish the controllability to trajectories of nonlinear parabolic equations. However, by this approach, it is not clear how to decide whether a given function is indeed reachable. In this paper, we pursue the study of the reachable states of parabolic equations based on a direct approach using control inputs in Gevrey spaces by considering a semilinear heat equation in dimension one. The nonlinear part is assumed to be an analytic function of the spatial variable x, the unknown y, and its derivative ${\partial }_{x}y$. By investigating carefully a nonlinear Cauchy problem in the spatial variable and the relationship between the jet of space derivatives and the jet of time derivatives, we derive an exact controllability result for small initial and final data that can be extended as analytic functions on some ball of the complex plane.

众所周知，使用Carleman估计可以建立对非线性抛物方程的轨迹的可控制性。但是，通过这种方法，尚不清楚如何确定给定功能是否确实可以实现。在本文中，我们通过考虑维数为1的半线性热方程，基于直接方法使用Gevrey空间中的控制输入来研究抛物方程的可达到状态。假定非线性部分是空间变量x，未知y及其导数的解析函数${\partial }_Xÿ$。通过仔细研究空间变量中的非线性柯西问题以及空间导数射流和时间导数射流之间的关系，我们得出了一些小的初始和最终数据的精确可控制性结果，该结果可以扩展为某些球的解析函数。复杂的飞机。