We consider the controllability problem for finite-dimensional linear autonomous control systems with nonnegative controls. Despite the Kalman condition, the unilateral nonnegativity control constraint may cause a positive minimal controllability time. When this happens, we prove that, if the matrix of the system has a real eigenvalue, then there is a minimal time control in the space of Radon measures, which consists of a finite sum of Dirac impulses. When all eigenvalues are real, this control is unique and the number of impulses is less than half the dimension of the space. We also focus on the control system corresponding to a finite-difference spatial discretization of the one-dimensional heat equation with Dirichlet boundary controls, and we provide numerical simulations.

我们考虑具有非负控制的有限维线性自治控制系统的可控性问题。尽管有卡尔曼条件，单边非负控制约束可能会导致正的最小可控制时间。发生这种情况时，我们证明，如果系统的矩阵具有真实的特征值，则在Radon测度的空间中将存在一个最小的时间控制，该时间控制由Dirac脉冲的有限和组成。当所有特征值都是真实的时，此控制是唯一的，并且脉冲数小于空间尺寸的一半。我们还关注与Dirichlet边界控制的一维热方程的有限差分空间离散化对应的控制系统，并提供了数值模拟。