Automated and Sound Synthesis of Lyapunov Functions with SMT Solvers

Abstract

In this paper we employ SMT solvers to soundly synthesise Lyapunov functions that assert the stability of a given dynamical model. The search for a Lyapunov function is framed as the satisfiability of a second-order logical formula, asking whether there exists a function satisfying a desired specification (stability) for all possible initial conditions of the model. We synthesise Lyapunov functions for linear, non-linear (polynomial), and for parametric models. For non-linear models, the algorithm also determines a region of validity for the Lyapunov function. We exploit an inductive framework to synthesise Lyapunov functions, starting from parametric templates. The inductive framework comprises two elements: a learner proposes a Lyapunov function, and a verifier checks its validity-its lack is expressed via a counterexample (a point over the state space), for further use by the learner. Whilst the verifier uses the SMT solver Z3, thus ensuring the overall soundness of the procedure, we examine two alternatives for the learner: a numerical approach based on the optimisation tool Gurobi, and a sound approach based again on Z3. The overall technique is evaluated over a broad set of benchmarks, which shows that this methodology not only scales to 10-dimensional models within reasonable computational time, but also offers a novel soundness proof for the generated Lyapunov functions and their domains of validity.

Deductive Stability Proofs for Ordinary Differential Equations

“…Other Examples [1]. Stability for several ODEs with Lyapunov functions generated by an inductive synthesis technique [1, were successfully verified in KeYmaera X.…”

“…Lyapunov functions are the standard tool for showing stability of general, nonlinear ODEs [14,18,31] and finding suitable Lyapunov functions is an important problem in its own right [1,9,12,17,21,23,24,33,37]. This section shows how a candidate Lyapunov function, once found, can be used to rigorously prove stability.…”

“…2. Prior approaches [1,12,17,21,33] have emphasized the need to formally verify those arithmetical conditions in order to guarantee that a conjectured Lyapunov function correctly implies stability for a given system. This paper shows how deductive proofs of ODE stability can be carried out in differential dynamic logic (dL) [25,26,27], a logic for deductive verification of hybrid systems.…”

“…This paper shows how deductive proofs of ODE stability can be carried out in differential dynamic logic (dL) [25,26,27], a logic for deductive verification of hybrid systems. 1 The key insight is that stability properties can be specified by suitably nesting the dynamic modalities of dL with quantifiers of first-order logic. The resulting specifications are amenable to rigorous proof by combining dL's ODE safety [28] and liveness [36] proof principles with real arithmetic and first-order quantifier reasoning.…”

“…This makes it possible to syntactically derive stability for a given system from the small set of dL axioms which, in turn, enables trustworthy stability proofs in the KeYmaera X theorem prover for hybrid systems [11,26]. Notably, this approach directly verifies stability specifications, which goes beyond verifying arithmetic that imply those specifications [1,12,17,21,33]. This is crucial for advanced stability notions because those variations generally require subtle twists to the required arithmetical conditions on their Lyapunov functions [14]; proofs of stability specifications alleviate the onus on system designers to correctly pick and check the appropriate conditions for their applications.…”