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Converse Theorems for Certificates of Safety and Stability

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arxiv 2406.14823 v2 pith:OEMZOZHN submitted 2024-06-21 math.OC cs.SYeess.SY

Converse Theorems for Certificates of Safety and Stability

classification math.OC cs.SYeess.SY
keywords controlconditionspairsafesafetycbfsclbfclf-cbf
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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Motivated by the key role of control barrier functions (CBFs) in assessing safety and enabling the synthesis of safe controllers in nonlinear control systems, this paper presents a suite of converse results on CBFs. Given any safe set, we first identify a set of general sufficient conditions which guarantee the existence of a CBF. Our technical analysis also enables us to define an extended notion of CBF which is always guaranteed to exist if the set is safe. We next turn our attention to the problem of joint safety and stability, and give conditions under which the notions of control Lyapunov-barrier function (CLBF) and compatible control Lyapunov function (CLF) and CBF pair are guaranteed to exist. Finally, we identify conditions under which a CLBF and a compatible CLF-CBF pair can be constructed from a non-compatible CLF-CBF pair. Throughout the paper, we intersperse different examples and counterexamples to motivate our results and position them within the state of the art.

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Cited by 2 Pith papers

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    A convex-minimization controller for affine inequalities is approximated by neural networks that act as universal formulas independent of state dimension for bounded input and constraint sizes.

  2. Uniform Feasibility For Smoothed Backup Control Barrier Functions

    math.OC 2025-11 unverdicted novelty 5.0

    Log-sum-exp smoothing of nonsmooth safe sets defined by min of C1 functions produces valid CBFs for a range of smoothing parameters, ensuring uniform feasibility for smoothed backup CBFs when a compact backup set is s...