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A Survey of Computational Complexity Results in Systems and Control
, 2000
"... 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 fi ..."
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Cited by 112 (20 self)
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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, NPcompleteness, 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, timevarying linear systems, nonlinear and hybrid systems, and stochastic optimal control.
Complexity of Stability and Controllability of Elementary Hybrid Systems
, 1997
"... this paper, weconsider simple classes of nonlinear systems and provethatbasic questions related to their stabilityandcontrollabilityare either undecidable or computationally intractable (NPhard). As a special case, weconsider a class of hybrid systems in which the state space is partitioned into tw ..."
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Cited by 34 (10 self)
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this paper, weconsider simple classes of nonlinear systems and provethatbasic questions related to their stabilityandcontrollabilityare either undecidable or computationally intractable (NPhard). As a special case, weconsider a class of hybrid systems in which the state space is partitioned into two halfspaces, and the dynamics in eachhalfspace correspond to a differentlinear system
Deciding Stability and Mortality of Piecewise Affine Dynamical Systems
, 2001
"... 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 ..."
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Cited by 22 (1 self)
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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 DAAL0392G0115, by the NATO under grant CRG961115 and by the European Commission under the TMR(AlMkI;/z network contract ERBFMRXCT960074. # Corresponding author. Email 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#/ 03043975/01/$  see front matter c # 2001El1/kRk Science B.V.Al rights reserved. PII: S03043975(00)003996 688 V.D. Blondel et al. / Theoretical Computer Science 255 (2001) 687696 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...
Complexity of ten decision problems in continuous time dynamical systems
"... Abstract — We show that for continuous time dynamical systems described by polynomial differential equations of modest degree (typically equal to three), the following decision problems which arise in numerous areas of systems and control theory cannot have a polynomial time (or even pseudopolynomi ..."
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Cited by 1 (1 self)
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Abstract — We show that for continuous time dynamical systems described by polynomial differential equations of modest degree (typically equal to three), the following decision problems which arise in numerous areas of systems and control theory cannot have a polynomial time (or even pseudopolynomial time) algorithm unless P=NP: local attractivity of an equilibrium point, stability of an equilibrium point in the sense of Lyapunov, boundedness of trajectories, convergence of all trajectories in a ball to a given equilibrium point, existence of a quadratic Lyapunov function, invariance of a ball, invariance of a quartic semialgebraic set under linear dynamics, local collision avoidance, and existence of a stabilizing control law. We also extend our earlier NPhardness proof of testing local asymptotic stability for polynomial vector fields to the case of trigonometric differential equations of degree four. I.