Stability analysis of dynamical systems plays an important role in the study of control techniques. LaSalle's invariance principle is a result about the asymptotic stability of the solutions to a nonlinear system of differential equations and several extensions of this principle have been designed to fit different particular kinds of system. In this paper we present a formalization, in the Coq proof assistant, of a slightly improved version of the original principle. This is a step towards a formal verification of dynamical systems.

This paper discusses the design of a hierarchy of structures which combine linear algebra with concepts related to limits, like topology and norms, in dependent type theory. This hierarchy is the backbone of a new library of formalized classical analysis, for the Coq proof assistant. It extends the Mathematical Components library, geared towards algebra, with topics in analysis. Issues of a more general nature related to the inheritance of poorer structures from richer ones arise due to this combination. We present and discuss a solution, coined forgetful inheritance, based on packed classes and unification hints.

Control theory provides techniques to design controllers, or control functions, for dynamical systems with inputs, so as to grant a particular behaviour of such a system. The inverted pendulum is a classic system in control theory: it is used as a benchmark for nonlinear control techniques and is a model for several other systems with various applications. We formalized in the Coq proof assistant the proof of soundness of a control function for the inverted pendulum. This is a first step towards the formal verification of more complex systems for which safety may be critical. CCS Concepts • Mathematics of computing → Ordinary differential equations; Topology; • Computer systems organization → Robotic control; • Software and its engineering → Software verification;

Goal-directed proof search in first-order logic uses metavariables to delay the choice of witnesses; substitutions for such variables are produced when closing proof-tree branches, using first-order unification or a theory-specific background reasoner. This paper investigates a generalisation of such mechanisms whereby theory-specific constraints are produced instead of substitutions. In order to design modular proofsearch procedures over such mechanisms, we provide a sequent calculus with meta-variables, which manipulates such constraints abstractly. Proving soundness and completeness of the calculus leads to an axiomatisation that identifies the conditions under which abstract constraints can be generated and propagated in the same way unifiers usually are. We then extract from our abstract framework a component interface and a specification for concrete implementations of background reasoners.