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.
- CONTRACTION ANALYSIS