Certifying the LTL Formula p Until q in Hybrid Systems

In this article, we propose sufficient conditions to guarantee that a linear temporal logic formula of the form
$p$
Until
$q$
, denoted by
$p \,\mathcal {U} q$
, is satisfied for a hybrid system. Roughly speaking, the formula
$p \,\mathcal {U} q$
is satisfied means that the solutions, initially satisfying proposition
$p$
, keep satisfying this proposition until proposition
$q$
is satisfied. To certify such a formula, connections to invariance notions—specifically, conditional invariance and eventual conditional invariance—as well as finite-time convergence properties are established. As a result, sufficient conditions involving the data of the hybrid system and an appropriate choice of Lyapunov-like functions, such as barrier functions, are derived. Examples illustrate the results throughout the article.