Continuity and computability of reachable sets

The computation of reachable sets of nonlinear dynamic and control systems is an important problem of systems theory. In this paper we consider the computability of reachable sets using Turing machines to perform approximate computations. We use Weihrauch's type-two theory of effectivity for computable analysis and topology, which provides a natural setting for performing computations on sets and maps. The main result is that the reachable set is lower-computable, but is only outer-computable if it equals the chain-reachable set. In the course of the analysis, we extend the computable topology theory to locally-compact Hausdorff spaces and semicontinuous set-valued maps, and provide a framework for computing approximations

Dependence on Initial Conditions of Attainable Sets of Control Systems with p-Integrable Controls

Quasi-linear systems governed by p-integrable controls, for 1 < p < ∞ with constraint ‖u(·)‖p ≤ µ0 are considered. Dependence on initial conditions of attainable sets are studied

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

Robust computations with dynamical systems

In this paper we discuss the computational power of Lipschitz dynamical systems which are robust to in nitesimal perturbations. Whereas the study in [1] was done only for not-so-natural systems from a classical mathematical point of view (discontinuous di erential equation systems, discontinuous piecewise a ne maps, or perturbed Turing machines), we prove that the results presented there can be generalized to Lipschitz and computable dynamical systems. In other words, we prove that the perturbed reachability problem (i.e. the reachability problem for systems which are subjected to in nitesimal perturbations) is co-recursively enumerable for this kind of systems. Using this result we show that if robustness to in nitesimal perturbations is also required, the reachability problem becomes decidable. This result can be interpreted in the following manner: undecidability of veri cation doesn't hold for Lipschitz, computable and robust systems. We also show that the perturbed reachability problem is co-r.e. complete even for C1-systems

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

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

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

Computability, Noncomputability, and Hyperbolic Systems

In this paper we study the computability of the stable and unstable manifolds of a hyperbolic equilibrium point. These manifolds are the essential feature which characterizes a hyperbolic system. We show that (i) locally these manifolds can be computed, but (ii) globally they cannot (though we prove they are semi-computable). We also show that Smale's horseshoe, the first example of a hyperbolic invariant set which is neither an equilibrium point nor a periodic orbit, is computable

Controllability and Falsification 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