Sufficient conditions for uniform asymptotic stability and input-to-state stability using high-order control barrier functions

Control barrier functions (CBFs) ensure safety of controlled dynamical systems by enforcing forward invariance of safe subsets of the state space. First-order CBFs are applicable for systems where the control input appears in the first time derivative of the controlled output. High-order CBFs (HOCBFs) extend the notion of CBFs to systems of any order, following a procedure reminiscent of the recursive design of a control Lyapunov function in backstepping. Asymptotic stability of compact safe sets for Lipschitz continuous HOCBF-based controllers has recently been reported in literature. In this paper, we extend this result by establishing sufficient conditions for uniform asymptotic stability of closed, but not necessarily compact, safe sets. Moreover, we show that uniform asymptotic stability holds for differential inclusions that correspond to allowing the control input to take on arbitrary values that satisfy the HOCBF-induced input constraints. This result circumvents the need to establish continuity properties of optimization-based safeguarding control laws. Sufficient conditions for input-to-state stability are also established, by constructing a vector comparison system from the worst-case evolution of the HOCBF along the disturbed versions of the aforementioned differential inclusions. The theoretical results are illustrated by two case studies.