The present analysis deals with the regularity of solutions of bilinear control systems of the type $x'=(A+u(t)B)x$where the state $x$ belongs to some complex infinite dimensional Hilbert space, the (possibly unbounded) linear operators $A$ and $B$ are skew-adjoint and the control $u$ is a real valued function. Such systems arise, for instance, in quantum control with the bilinear Schrodinger equation. For the sake of the regularity analysis, we consider a more general framework where $A$ and $B$ are generators of contraction semi-groups.Under some hypotheses on the commutator of the operators $A$ and $B$, it is possible to extend the definition of solution for controls in the set of Radon measures to obtain precise a priori energy estimates on the solutions, leading to a natural extension of the celebrated noncontrollability result of Ball, Marsden, and Slemrod in 1982. Complementary material to this analysis can be found in [hal-01537743v1]

中文翻译：

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双线性量子系统的正则传播子

目前的分析涉及 $x'=(A+u(t)B)x$ 类型的双线性控制系统的解的规律性，其中状态 $x$ 属于某个复杂的无限维希尔伯特空间，（可能是无界的） ) 线性运算符 $A$ 和 $B$ 是倾斜伴随的，控件 $u$ 是实值函数。例如，这种系统出现在双线性薛定谔方程的量子控制中。为了正则性分析，我们考虑一个更一般的框架，其中 $A$ 和 $B$ 是收缩半群的生成器。 在算子 $A$ 和 $B$ 的交换子的一些假设下，有可能扩展氡测度集合中控制的解的定义，以获得解的精确先验能量估计，从而导致著名的 Ball 不可控性结果的自然扩展，