Time-varying vector fields

The stability notions expressed here are usually introduced in the time-varying setting ${\dot x}= f(x,t)$. Since the vast majority of planning problems in this book are time-invariant, the presentation was confined to time-invariant vector fields. There is, however, one fascinating peculiarity in the topic of finding a feedback plan that stabilizes a system. Brockett's condition implies that for some time-invariant systems for which continuous, time-varying feedback plans exist, there does not exist a continuous time-invariant feedback plan [143,156,996]. This includes the class of driftless control systems, such as the simple car and the unicycle. This implies that to maintain continuity of the vector field, a time dependency must be introduced to allow the vector field to vary as ${x_{G}}$ is approached! If continuity of the vector field is not important, then this concern vanishes.