In this paper we consider a class of hybrid systems, namely dynamical systems with piecewise-constant derivatives (PCD systems). Such systems consist of a partition of the Euclidean space into a finite set of polyhedral sets (regions). Within each region the dynamics is defined by a constant vector field, hence discrete transitions occur only on the boundaries between regions where the trajectories change their direction. With respect to such systems we investigate the reachability question: Given an effective description of the systems and of two polyhedral subsets P and Q of the state-space, is there a trajectory starting at some x 2 P and reaching some point in Q? Our main results are a decision procedure for two-dimensional systems, and an undecidability result for three or more dimensions. 1 Introduction 1.1 Motivation Hybrid systems (HS) are systems that combine intercommunicating discrete and continuous components. Most embedded systems belong to this class since they operate...

The purpose of this paper is twofold: (a) to provide a tutorial introduction to some key concepts from the theory of computational complexity, highlighting their relevance to systems and control theory, and (b) to survey the relatively recent research activity lying at the interface between these fields. We begin with a brief introduction to models of computation, the concepts of undecidability, polynomial time algorithms, NP-completeness, and the implications of intractability results. We then survey a number of problems that arise in systems and control theory, some of them classical, some of them related to current research. We discuss them from the point of view of computational complexity and also point out many open problems. In particular, we consider problems related to stability or stabilizability of linear systems with parametric uncertainty, robust control, time-varying linear systems, nonlinear and hybrid systems, and stochastic optimal control.

We define a class of recursive functions on the reals analogous to the classical recursive functions on the natural numbers, corresponding to a conceptual analog computer that operates in continuous time. This class turns out to be surprisingly large, and includes many functions which are uncomputable in the traditional sense.

We explore the simulation and computational capabilities of hybrid and continuous dynamical systems. The continuous dynamical systems considered are ordinary differential equations (ODEs). For hybrid systems we concentrate on models that combine ODEs and discrete dynamics (e.g., finite automata). We review and compare four such models from the literature. Notions of simulation of a discrete dynamical system by a continuous one are developed. We show that hybrid systems whose equations can describe a precise binary timing pulse (exact clock) can simulate arbitrary reversible discrete dynamical systems defined on closed subsets of R n . The simulations require continuous ODEs in R 2n with the exact clock as input. All four hybrid systems models studied here can implement exact clocks. We also prove that any discrete dynamical system in Z n can be simulated by continuous ODEs in R 2n+1 . We use this to show that smooth ODEs in R 3 can simulate arbitrary Turing machines, and henc...

We consider a model of analog computation which can recognize various languages in real time. We encode an input word as a point in R d by composing iterated maps, and then apply inequalities to the resulting point to test for membership in the language. Each class of maps and inequalities, such as quadratic functions with rational coefficients, is capable of recognizing a particular class of languages; for instance, linear and quadratic maps can have both stack-like and queue-like memories. We use methods equivalent to the VapnikChervonenkis dimension to separate some of our classes from each other, e.g. linear maps are less powerful than quadratic or piecewise-linear ones, polynomials are less powerful than elementary (trigonometric and exponential) maps, and deterministic polynomials of each degree are less powerful than their non-deterministic counterparts. Comparing these dynamical classes with various discrete language classes helps illuminate how iterated maps can...

. In this paper we define a precise notion of abstraction relation between continuous dynamical systems and discrete state-transition systems. Our main result states that every Turing Machine can be realized by a dynamical system with piecewise-constant derivatives in a 3-dimensional space and thus the reachability problem for such systems is undecidable for 3 dimensions. A decision procedure for 2-dimensional systems has been recently reported by Maler and Pnueli. On the other hand we show that some non-deterministic finite automata cannot be realized by any continuous dynamical system with less than 3 dimensions. 1 Introduction There has been recently an increasing interest in models of hybrid systems, i.e., systems that combine intercommunicating discrete and continuous components (see [9], [12], [3]). The introduction of these models is motivated by a real practical concern: more and more computers (discrete transition systems) are nowadays embedded within real-world control loops...

In this paper we studyproblJ: such as: given a discrete timedynamical system of the form x(t +1)=f(x(t)) where f : R n #R n is a piecewise a#ne function, decide whetheral trajectories converge to 0. We show in our main theorem that this AttractivityProblc isundecidabl as soon as n2. The same is true of tworelkMI problI+J Stabil+J (is thedynamical systemglJH #RI asymptotical# stablto andMortal#M (do al trajectories go through 0?). We then show that Attractivity andStabilI: becomedecidabl in dimension 1 for continuous functions. c # 2001El1/JkR Science B.V.Al rights reserved. Keywords: Discretedynamical systems; Piecewise a#ne systems; Piecewiselecew systems; Hybrid systems;Mortal/JM Stabil/JM Decidabilk: 1.IP141 In this paper we studyproblJ+ such as: given a discrete timedynamical system of the form x(t +1)=f(x(t)) where f : R n #R n is a(possibl discontinuous) piecewise # This research waspartl carried outwhil Bllkk was visitingTsitsiklJ at MIT (Cambridge) and Koiran at ENS (Lyon). This research was supported by the ARO under grant DAAL-03-92-G-0115, by the NATO under grant CRG-961115 and by the European Commission under the TMR(AlMkI;/z network contract ERBFMRXCT960074. # Corresponding author. E-mail addresses: blmCppCpA/J#JM:/zRkJ; (V.D.BlD./kIH Ol./kIH:J/zRkJ;/lkJ;/l (O. Bournez), pascal),/;MJMI/zRkJ;/ll (P. Koiran), christos@cs.berkel/ll (C.H. Papadimitriou), jnt@mit.edu (J.N. TsitsiklM#/ 0304-3975/01/$ - see front matter c # 2001El1/kRk Science B.V.Al rights reserved. PII: S0304-3975(00)00399-6 688 V.D. Blondel et al. / Theoretical Computer Science 255 (2001) 687--696 a#ne function, decide whetheral trajectories converge to 0. We show in our main theorem (Theorem 2) that this AttractivityProblc isundecidabl as soon as n2. The same is true of t...

It has become increasingly apparent this last decade that many problems in systems and control are NP-hard and, in some cases, undecidable. The inherent complexity of some of the most elementary problems in systems and control points to the necessity of using alternative approximate techniques to deal with problems that are unsolvable or intractable when exact solutions are sought. We survey

We investigate stability of some scheduling policies in queueing systems. To the day no algorithmic characterization exists for checking stability of a given policy in a given queueing system. In this paper we propose a certain generalized priority policy and prove that the stability of this policy is algorithmically undecidable. We also prove that stability of a homogeneous random walk in Z d + is undecidable. To the best of our knowledge this is the first undecidability result in the area of stability of queueing systems and random walks in Z d + . We conjecture that stability of other common policies like First-In-First-Out and priority policy is also an undecidable problem.

There exist several well known results concerning the computational complexity of training, design, etc. of (artificial) neural networks which demonstrated that these problems are at least NP complete. Extending these results we will prove here that the training of neural networks, i.e. the so called "loading problem", in general, is even "unsolvable". This implies in particular that there exists no general or universal training algorithm which for any given neural network architecture could determine a correct set of "weights" such that a certain desired input/output behavior of the neural network is achieved. 1 Introduction Artificial neural networks, like the Multi Layer Perceptron (MLP), can be used for approximating any function belonging to certain "reasonable" classes (e.g. continuous or measurable functions) as closely as desired. Related results can be found for example in [8] etc. These results are often celebrated as the theoretical justifications of what lead to a renaissa...