![]() ![]() In this paper, we formulate new inductive proof rules for continuous dynamical systems for establishing robust notions of stability and safety. To bridge this gap, we need to rethink about the fundamental definitions. Thus, there is a discrepancy between what the standard theory requires and what is needed in practice, or what can be achieved computationally. However, proving a system is stable within an arbitrarily tiny neighborhood around the origin is all we really need in practice. Since any numerical error blurs the difference between strict and non-strict inequality, one can conclude that numerically-driven methods are not suitable for verifying these strict constraints. A dynamical system is stable if there exists a function that vanishes exactly at the origin and its derivatives strictly decreases over time. Take the Lyapunov analysis of stability properties as an example. However, the fundamental challenge with using numerically-driven methods in inductive proofs is that numerical errors make it impossible to verify the induction steps in the standard sense. They have been used for many bounded-time verification and synthesis problems for highly nonlinear systems . Recent work on numerically-driven decision procedures provides a promising direction to bypass this difficulty . However, to check the inductive conditions for nonlinear dynamical systems, one has to solve nonlinear SMT problems over real numbers, which are highly intractable or undecidable . The standard approach of checking invariance conditions in program analysis is to use Satisfiability Modulo Theories (SMT) solvers . Most importantly, they can not handle systems with non-polynomial nonlinearity, and thus fall short of a general framework for verifying practical systems of significant complexity. ![]() However, these algorithms are either extremely expensive or numerically brittle. The standard approaches for the validation step use symbolic quantifier elimination or Sum-of-Squares techniques . In both cases, once a candidate certificate (Lyapunov or barrier functions) is proposed, the verification problem is reduced to checking the validity of a universally-quantified first-order formula over real-valued variables. ![]() Likewise, proving unbounded safety of a dynamical system requires one to find a barrier function (or differential invariant ) that separates the system’s initial state from the unsafe regions, and whenever the system states reach the barrier, the system dynamics always points towards the safe side of the barrier . A system is stable at the origin in the sense of Lyapunov, if one can find a Lyapunov function (essentially a ranking function) that is everywhere positive except for reaching exactly zero at the origin, and never increases over time along the direction of the system dynamics . For instance, proving stability of a dynamical system is similar to proving termination of a program. Infinite-time stability and safety properties of continuous dynamical systems are typically established via inductive arguments over continuous time. ![]()
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