Combinatorial Abstractions of Dynamical Systems (original) (raw)

Completeness of Lyapunov Abstraction

Electronic Proceedings in Theoretical Computer Science, 2013

In this work, we continue our study on discrete abstractions of dynamical systems. To this end, we use a family of partitioning functions to generate an abstraction. The intersection of sub-level sets of the partitioning functions defines cells, which are regarded as discrete objects. The union of cells makes up the state space of the dynamical systems. Our construction gives rise to a combinatorial object-a timed automaton. We examine sound and complete abstractions. An abstraction is said to be sound when the flow of the time automata covers the flow lines of the dynamical systems. If the dynamics of the dynamical system and the time automaton are equivalent, the abstraction is complete. The commonly accepted paradigm for partitioning functions is that they ought to be transversal to the studied vector field. We show that there is no complete partitioning with transversal functions, even for particular dynamical systems whose critical sets are isolated critical points. Therefore, we allow the directional derivative along the vector field to be non-positive in this work. This considerably complicates the abstraction technique. For understanding dynamical systems, it is vital to study stable and unstable manifolds and their intersections. These objects appear naturally in this work. Indeed, we show that for an abstraction to be complete, the set of critical points of an abstraction function shall contain either the stable or unstable manifold of the dynamical system.

Robust computations with dynamical systems

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2010

In this paper we discuss the computational power of Lipschitz dynamical systems which are robust to infinitesimal perturbations. Whereas the study in [1] was done only for not-so-natural systems from a classical mathematical point of view (discontinuous differential equation systems, discontinuous piecewise affine 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 infinitesimal perturbations) is co-recursively enumerable for this kind of systems. Using this result we show that if robustness to infinitesimal perturbations is also required, the reachability problem becomes decidable. This result can be interpreted in the following manner: undecidability of verification doesn't hold for Lipschitz, computable and robust systems. We also show that the perturbed reachability problem is co-r.e. complete even for C ∞ -systems.

An Approach to Dynamical Systems using Exterior Spaces

There are considered the so called exterior spaces for dynamical systems. If (X,𝐭) defines a topological space, an externology is a non-empty collection ε of open subsets which is closed under finite intersections and such that, if E∈ε, U∈𝐭 and E⊂U, then U∈ε. If an open subset is a member of ε, it is said to be an exterior subset. An exterior space (X,ε,𝐭) consists of a space (X,𝐭) together with an externology ε. A map f:(X,ε,𝐭)↦(X ' ,ε ' ,𝐭 ' ) is an exterior map if it is continuous and f -1 (E)∈ε for all E∈ε ' . The paper further defines the dynamical systems as flows, the exterior flows, the limit spaces of a flow via exterior flows. The results concern various other limit properties via these notions.

A Computational Analysis of the Reachability Problem for a Class of Hybrid Dynamical Systems

1997

Hybrid systems possess continuous dynamics defined within regions of state spaces and discrete transitions among the regions. Many practical control verification and synthesis tasks can be reduced to reach ability problems for these systems that decide if a particular state-space region is reachable from an initial operating region. In this paper, we present a computational analysis of the face reachability problem for a class of three-dimensional dynamical systems whose state spaces are defined by piecewise constant vector fields and whose trajectories never return to a state-space region once they exit the region. These systems represent a restricted class of control systems whose dynamics results from a juxtaposition of piecewise parameterized vector fields. We had previously developed a computational algorithm for synthesizing the desired dynamics of a system in phase space by piecing together vector fields geometrically. We demonstrate in this paper that the reachability problem for this class of systems is decidable while the computation is provably intractable (i.e., PSPACE-hard). We prove the intractability via a reduction of satisfiability of quantified boolean formulas to this reachability problem. This result sheds light on the computational complexity of phase-space based control synthesis methods and extends the work of Asarin, Maler, and Pnueli [2] that proves computational undecidability for three-dimensional constant-derivative systems.

A Computatuional Analysis of the Reachability Problem for a Class of Hybrid Dynamical Systems

1996

Hybrid systems possess continuous dynamics de ned within regions of state spaces and discrete transitions among the regions. Many practical control veri cation and synthesis tasks can be reduced to reachability problems for these systems that decide if a particular state-space region is reachable from an initial operating region. In this paper, we present a computational analysis of the face reachability problem for a class of three-dimensional dynamical systems whose state spaces are de ned by piecewise constant vector elds and whose trajectories never return to a state-space region once they exit the region. These systems represent a restricted class of control systems whose dynamics results from a juxtaposition of piecewise parameterized vector elds. We had previously developed a computational algorithm for synthesizing the desired dynamics of a system in phase space by piecing together vector elds geometrically. We demonstrate in this paper that the reachability problem for this class of systems is decidable while the computation is provably intractable (i.e., PSPACE-hard). We prove the intractability via a reduction of satis ability of quanti ed boolean formulas to this reachability problem. This result sheds light on the computational complexity of phase-space based control synthesis methods and extends the work of Asarin, Maler, and Pnueli 2] that proves computational undecidability for three-dimensional constant-derivative systems.

Dynamical Systems with Specification

Journal of the Chungcheong Mathematical Society, 2015

In this paper we prove that C 1-generically, if a diffeomorphism f on a closed C ∞ manifold M satisfies weak specification on a locally maximal set Λ ⊂ M then Λ is hyperbolic for f. As a corollary we obtain that C 1-generically, every diffeomorphism with weak specification is Anosov.

Guaranteed Error Bounds on Approximate Model Abstractions Through Reachability Analysis

Lecture Notes in Computer Science, 2018

It is well known that exact notions of model abstraction and reduction for dynamical systems may not be robust enough in practice because they are highly sensitive to the specific choice of parameters. In this paper we consider this problem for nonlinear ordinary differential equations (ODEs) with polynomial derivatives. We introduce approximate differential equivalence as a more permissive variant of a recently developed exact counterpart, allowing ODE variables to be related even when they are governed by nearby derivatives. We develop algorithms to (i) compute the largest approximate differential equivalence; (ii) construct an approximate quotient model from the original one via an appropriate parameter perturbation; and (iii) provide a formal certificate on the quality of the approximation as an error bound, computed as an over-approximation of the reachable set of the perturbed model. Finally, we apply approximate differential equivalences to study the effect of parametric tolerances in models of symmetric electric circuits.

Some remarks on set-valued dynamical systems

The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 1981

It is shown that under some conditions a collection of continuous mappings gives rise to a set-valued dynamical system. Using this it is further shown that under some other conditions the system ẋ(t) ∈ F(x(t)) is equivalent to a set-valued dynamical system.