We study a containment control problem (CCP) and a shape control problem (SCP) for systems whose initial condition is a random variable with known distribution. The two control problems both require exponential convergence to a desired trajectory, which is complemented by either; (i) a required cumulative distribution over a prescribed containment set at a specific transient time for the CCP, or; (ii) a maximum distance between an attained and a desired probability density function of the state for the SCP. For the CCP, we obtain solutions for both linear and nonlinear systems by designing the closed-loop such that the initial pdf shrinks or contracts to a desired trajectory. For the SCP, we obtain solutions for linear systems and an admissible desired pdf, by designing the closed-loop such that the evolution of the pdf at the transient time is similar to the target pdf.

我们研究了初始条件是具有已知分布的随机变量的系统的遏制控制问题 (CCP) 和形状控制问题 (SCP)。这两个控制问题都需要指数收敛到一个期望的轨迹，这两者都得到了补充；(i) 在 CCP 的特定瞬态时间在规定的密闭装置上所需的累积分布，或；(ii) SCP 状态达到的和期望的概率密度函数之间的最大距离。对于 CCP，我们通过设计闭环使初始 pdf 收缩或收缩到所需轨迹来获得线性和非线性系统的解决方案。对于 SCP，我们获得线性系统的解和可接受的期望 pdf，