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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 28 (0 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...
Algorithms for symbolic/numeric control of affine dynamical systems
 In Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation
, 2005
"... We consider a general linear dynamical system and want to control its behavior. The goal is to reach a given target by minimizing a cost function. We provide a new generic algorithm with together exact, symbolic and numerical modules. In particular new efficient methods computing a block Kalman cano ..."
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Cited by 4 (3 self)
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We consider a general linear dynamical system and want to control its behavior. The goal is to reach a given target by minimizing a cost function. We provide a new generic algorithm with together exact, symbolic and numerical modules. In particular new efficient methods computing a block Kalman
Lipschitzian regularity of minimizers for optimal control problems with controlaffine dynamics
 APPLIED MATHEMATICS AND OPTIMIZATION
, 2000
"... We study Lagrange Problem of Optimal Control with a functional L (t, x (t) , u (t)) dt and control affine dynamics x = f (t, x)+ g (t, x) u and (a priori) unconstrained control u . We obtain conditions under which the minimizing controls of the problem are bounded  the fact which is crucia ..."
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Cited by 18 (8 self)
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We study Lagrange Problem of Optimal Control with a functional L (t, x (t) , u (t)) dt and control affine dynamics x = f (t, x)+ g (t, x) u and (a priori) unconstrained control u . We obtain conditions under which the minimizing controls of the problem are bounded  the fact which
A control problem for affine dynamical systems on a fulldimensional polytope
 AUTOMATICA
, 2004
"... ..."
Markov Chain Approximations for Deterministic Control Problems with Affine Dynamics and Quadratic Cost in the Control
 SIAM J. Numer. Anal
, 1998
"... We consider the construction of Markov chain approximations for an important class of deterministic control problems. The emphasis is on the construction of schemes that can be easily implemented and which possess a number of highly desirable qualitative properties. The class of problems covered is ..."
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Cited by 54 (0 self)
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is that for which the control is affine in the dynamics and with quadratic running cost. This class covers a number of interesting application areas, including problems that arise in large deviations, risksensitive and robust control, robust filtering, and certain problems from computer vision. Examples are given
On the existence, uniqueness and nature of Carathéodory and Filippov solutions for bimodal piecewise affine dynamical systems
"... In this paper, we deal with the wellposedness (in the sense of existence and uniqueness of solutions) and nature of solutions for discontinuous bimodal piecewise affine systems in a differential inclusion setting. First, we show that the conditions guaranteeing uniqueness of Filippov solutions in t ..."
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In this paper, we deal with the wellposedness (in the sense of existence and uniqueness of solutions) and nature of solutions for discontinuous bimodal piecewise affine systems in a differential inclusion setting. First, we show that the conditions guaranteeing uniqueness of Filippov solutions
PHAVer: Algorithmic verification of hybrid systems past HyTech
, 2005
"... In 1995, HyTech broke new ground as a potentially powerful tool for verifying hybrid systems – yet it has remained severely limited in its applicability to more complex systems. We address the main problems of HyTech with PHAVer, a new tool for the exact verification of safety properties of hybrid ..."
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Cited by 217 (9 self)
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of hybrid systems with piecewise constant bounds on the derivatives. Affine dynamics are handled by onthefly overapproximation and by partitioning the state space based on userdefinable constraints and the dynamics of the system. PHAVer’s exact arithmetic is robust due to the use of the Parma Polyhedra
Observability and Controllability of Piecewise Affine and Hybrid Systems
 IEEE Transactions on Automatic Control
, 1999
"... In this pap e we prove in a constructive way, the ee ale b e we e pie a#ne syste and a broad class of hybridsyste de e d by inte line dynamics, automata, and propositional logic. By focusing our inveon the forme class, we show through countethat obse ability and controllability prope rtie cannot b ..."
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Cited by 145 (21 self)
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In this pap e we prove in a constructive way, the ee ale b e we e pie a#ne syste and a broad class of hybridsyste de e d by inte line dynamics, automata, and propositional logic. By focusing our inveon the forme class, we show through countethat obse ability and controllability prope rtie cannot
Scanning Polyhedra with DO Loops
, 1991
"... Supercompilers perform complex program transformations which often result in new loop bounds. This paper shows that, under the usual assumptions in automatic parallelization, most transformations on loop nests can be expressed as affine transformations on integer sets de ned by polyhedra and that th ..."
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Cited by 216 (5 self)
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Supercompilers perform complex program transformations which often result in new loop bounds. This paper shows that, under the usual assumptions in automatic parallelization, most transformations on loop nests can be expressed as affine transformations on integer sets de ned by polyhedra
Results 1  10
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