Necessity of the CLF–CBF compatibility assumption for safe stabilization via the Zubov–HJB PDE framework

Determine whether the compatibility assumption—requiring that for every state x in the boundary layer {x ∈ R^n : 1 − ε ≤ h(x) < 1} there exists a control input u such that simultaneously (i) ⟨ζ, f(x)+g(x)u⟩ < −w_ε(x) for all proximal subgradients ζ ∈ ∂_P v(x) of a control Lyapunov function v, and (ii) ⟨ξ, f(x)+g(x)u⟩ ≤ α(1 − h(x)) for all proximal subgradients ξ ∈ ∂_P h(x) of the safe-set function h—is necessary for the existence of a control Lyapunov–barrier function produced by the Zubov–HJB PDE characterization that guarantees safe stabilization to the origin while avoiding the unsafe set U.

Background

The paper develops a Zubov–HJB PDE formulation for constructing control Lyapunov–barrier functions (CLBFs) that ensure safe stabilization for nonlinear control-affine systems. A central hypothesis used throughout is a compatibility assumption that, near the boundary of the safe set defined by h(x) < 1, there exists at each state a control input that simultaneously decreases a (possibly nonsmooth) control Lyapunov function v and satisfies the control barrier condition associated with h.

Formally, for states x with 1−ε ≤ h(x) < 1, the assumption requires the set Γ(x) of controls that satisfy both the CLF decrease inequality and the CBF safety inequality to be nonempty. This assumption rules out situations where stabilization and safety are individually achievable but cannot be achieved simultaneously by the same control, and it underpins results on the characterization of the safe null-controllability domain and the uniqueness of the viscosity solution to the Zubov–HJB PDE.

Although widely adopted and sufficient for the theoretical development in the paper, the authors point out that the necessity of this compatibility assumption is not established: no constructive counterexample is known, but a formal proof of necessity is lacking. Establishing whether this assumption is indeed necessary remains an explicit open question.