Effective computation for nonlinear systems

Nonlinear dynamical and control systems are an important source of applications for theories of computation over the the real numbers, since these systems are usually to complicated to study analytically, but may be extremely sensitive to numerical error. Further, computerassisted proofs and verification problems require a rigorous treatment of numerical errors. In this paper we will describe how to provide a semantics for effective computations on sets and maps and show how these operations have been implemented in the tool Ariadne for the analysis, design and verification of nonlinear and hybrid systems

Hybrid trajectory spaces

In this paper, we present a general framework for describing and studying hybrid systems. We represent the trajectories of the system as functions on a hybrid time domain, and the system itself by its trajectory space, which is the set of all possible trajectories. The trajectory space is given a natural topology, the compact-open hybrid Skorohod topology, and we prove the existence of limiting trajectories under uniform equicontinuity assumptions. We give a compactness result for the trajectory space of impulse differential inclusions, a class of nondeterministic hybrid system, and discuss how to describe hybrid automata, a widely-used class of hybrid system, as impulse differential inclusions. For systems with compact trajectory space, we obtain results on Zeno properties, symbolic dynamics and invariant measures. We give examples showing the application of the results obtained using the trajectory space approac

Computability of controllers for discrete-time semicontinuous systems

In this paper we consider the computation of controllers for noisy nonlinear discrete-time systems described by upper-semicontinuous multivalued functions. We show that for the problem of controlling to a target set, if an open-loop solution exists, then a feedback controller with can be effectively computed in finite time from the problem data, and that the resulting system is robust with respect to perturbations. We extend the results for systems with partial observations and a dynamic output feedback law based on a finite automaton

Computing controllable sets of hybrid systems

In this paper we consider the controllability problem for hybrid systems, namely that of determining the set of states which can be driven into a given target set. We show that given a suitable definition of controllability, we can effectively compute arbitrarily accurate under-approximations to the controllable set using Turing machines. However, due to grazing or sliding along guard sets, we see that it may be able to demonstrate that an initial state can be controlled to the target set, without knowing any trajectory which solves the problem

Universal trellises

A flow in three-dimensions is universal if the periodic orbits contains all knots and links. Universal flows were shown to exist by Ghrist, and can be constructed by means of templates. Likewise, a planar diffeomorphism is universal if it has a suspension flow which is a universal flow. In this paper we prove the existence of a homoclinic trellis type for which any representative diffeomorphism is universal. This trellis type is remarkable in that it has zero entropy, and only two homoclinic intersection point

Forcing relations for homoclinic orbits of the Smale horseshoe map

An important problem in the dynamics of surface homeomorphisms is determining the forcing relation between orbits. The forcing relation between periodic orbits can be computed using existing algorithms. Here we consider forcing relations between homoclinic orbits. We outline a general procedure for computing the forcing relation, and apply this to compute the equivalence and forcing relations for homoclinic orbits of the Smale horseshoe ma

Computability and Representations of the Zero Set

In this note we give a new representation for closed sets under which the robust zero set of a function is computable. We call this representation the component cover representation. The computation of the zero set is based on topological index theory, the most powerful tool for finding robust solutions of equations

Optimal semicomputable approximations to reachable and invariant sets

In this paper we consider the computation of reachable, viable and invariant sets for discrete-time systems. We use the framework of type-two effectivity, in which computations are performed by Turing machines with infinite input and output tapes, with the representations of computable topology. We see that the reachable set is lower-semicomputable, and the viability and invariance kernels are upper-semicomputable. We then define an upper-semicomputable over-approximation to the reachable set, and lower-semicomputable under-approximations to the viability and invariance kernels, and show that these approximations are optimal

Optimal semicomputable approximations to reachable and invariant sets

In this paper we consider the computation of reachable, viable and invariant sets for discrete-time systems. We use the framework of type-two effectivity, in which computations are performed by Turing machines with infinite input and output tapes, with the representations of computable topology. We see that the reachable set is lower-semicomputable, and the viability and invariance kernels are upper-semicomputable. We then define an upper-semicomputable over-approximation to the reachable set, and lower-semicomputable under-approximations to the viability and invariance kernels, and show that these approximations are optima

Computable Types for Dynamic Systems

In this paper, we develop a theory of computable types suitable for the study of dynamic systems in discrete and continuous time. The theory uses type-two effectivity as the underlying computational model, but we quickly develop a type system which can be manipulated abstractly, but for which all allowable operations are guaranteed to be computable. We apply the theory to the study of differential inclusions, reachable sets and controllability