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25
Small gain theorems for large scale systems and construction of ISS Lyapunov functions
 SIAM JOURNAL ON CONTROL AND OPTIMIZATION
, 2009
"... We consider interconnections of n nonlinear subsystems in the inputtostate stability (ISS) framework. For each subsystem an ISS Lyapunov function is given that treats the other subsystems as independent inputs. A gain matrix is used to encode the mutual dependencies of the systems in the network. ..."
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Cited by 23 (18 self)
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We consider interconnections of n nonlinear subsystems in the inputtostate stability (ISS) framework. For each subsystem an ISS Lyapunov function is given that treats the other subsystems as independent inputs. A gain matrix is used to encode the mutual dependencies of the systems in the network. Under a small gain assumption on the monotone operator induced by the gain matrix, a locally Lipschitz continuous ISS Lyapunov function is obtained constructively for the entire network by appropriately scaling the individual Lyapunov functions for the subsystems. The results are obtained in a general formulation of ISS, the cases of summation, maximization and separation with respect to external gains are obtained as corollaries.
A spectral theorem for convex monotone homogeneous maps
 In Proceedings of the Satellite Workshop on MaxPlus Algebras, IFAC SSSC’01
, 2001
"... Abstract. We consider convex maps f: R n → R n that are monotone (i.e., that preserve the product ordering of R n), and nonexpansive for the supnorm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point ..."
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Cited by 18 (9 self)
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Abstract. We consider convex maps f: R n → R n that are monotone (i.e., that preserve the product ordering of R n), and nonexpansive for the supnorm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point set of f, when it is nonempty, is isomorphic to a convex infsubsemilattice of R n, whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f. This yields in particular an uniqueness result for the bias vector of ergodic control problems. This generalizes results obtained previously by Lanery, Romanovsky, and Schweitzer and Federgruen, for ergodic control problems with finite state and action spaces, which correspond to the special case of piecewise affine maps f. We also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph, which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group
Maxplus convex geometry
 of Lecture Notes in Comput. Sci
, 2006
"... Abstract. Maxplus analogues of linear spaces, convex sets, and polyhedra have appeared in several works. We survey their main geometrical properties, including maxplus versions of the separation theorem, existence of linear and nonlinear projectors, maxplus analogues of the MinkowskiWeyl theore ..."
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Cited by 8 (6 self)
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Abstract. Maxplus analogues of linear spaces, convex sets, and polyhedra have appeared in several works. We survey their main geometrical properties, including maxplus versions of the separation theorem, existence of linear and nonlinear projectors, maxplus analogues of the MinkowskiWeyl theorem, and the characterization of the analogues of “simplicial ” cones in terms of distributive lattices. 1
A policy iteration algorithm for zerosum stochastic games with mean payoff
, 2006
"... We give a policy iteration algorithm to solve zerosum stochastic games with finite state and action spaces and perfect information, when the value is defined in terms of the mean payoff per turn. This algorithm does not require any irreducibility assumption ..."
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Cited by 5 (1 self)
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We give a policy iteration algorithm to solve zerosum stochastic games with finite state and action spaces and perfect information, when the value is defined in terms of the mean payoff per turn. This algorithm does not require any irreducibility assumption
Imprecise markov chains and their limit behaviour
 Probability in the Engineering and Informational Sciences
"... ABSTRACT. When the initial and transition probabilities of a finite Markov chain in discrete time are not well known, we should perform a sensitivity analysis. This can be done by considering as basic uncertainty models the socalled credal sets that these probabilities are known or believed to belo ..."
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Cited by 4 (2 self)
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ABSTRACT. When the initial and transition probabilities of a finite Markov chain in discrete time are not well known, we should perform a sensitivity analysis. This can be done by considering as basic uncertainty models the socalled credal sets that these probabilities are known or believed to belong to, and by allowing the probabilities to vary over such sets. This leads to the definition of an imprecise Markov chain. We show that the time evolution of such a system can be studied very efficiently using socalled lower and upper expectations, which are equivalent mathematical representations of credal sets. We also study how the inferred credal set about the state at time n evolves as n → ∞: under quite unrestrictive conditions, it converges to a uniquely invariant credal set, regardless of the credal set given for the initial state. This leads to a nontrivial generalisation of the classical Perron–Frobenius Theorem to imprecise Markov chains. 1.
The Traffic Phases of Road Networks
"... We study the relation between the average traffic flow and the vehicle density on road networks that we call 2Dtraffic fundamental diagram. We show that this diagram presents mainly four phases. We analyze different cases. First, the case of a junction managed with a priority rule is presented, fou ..."
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Cited by 2 (1 self)
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We study the relation between the average traffic flow and the vehicle density on road networks that we call 2Dtraffic fundamental diagram. We show that this diagram presents mainly four phases. We analyze different cases. First, the case of a junction managed with a priority rule is presented, four traffic phases are identified and described, and a good analytic approximation of the fundamental diagram is obtained by computing a generalized eigenvalue of the dynamics of the system. Then, the model is extended to the case of two junctions, and finally to a regular city. The system still presents mainly four phases. The role of a critical circuit of nonpriority roads appears clearly in the two junctions case. In Section 4, we use traffic light controls to improve the traffic diagram. We present the improvements obtained by openloop, local feedback, and global feedback strategies. A comparison based on the response times to reach the stationary regime is also given. Finally, we show the importance of the design of the junction. It appears that if enough space is available in the junction, the traffic is almost not slowed down by the junction.
Observations on the Stability Properties of Cooperative Systems
, 2008
"... We extend two fundamental properties of positive linear timeinvariant (LTI) systems to homogeneous cooperative systems. Specifically, we demonstrate that such systems are Dstable, meaning that global asymptotic stability is preserved under diagonal scaling. We also show that a delayed homogeneous ..."
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Cited by 2 (0 self)
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We extend two fundamental properties of positive linear timeinvariant (LTI) systems to homogeneous cooperative systems. Specifically, we demonstrate that such systems are Dstable, meaning that global asymptotic stability is preserved under diagonal scaling. We also show that a delayed homogeneous cooperative system is globally asymptotically stable (GAS) for any nonnegative delay if and only if the system is GAS for zero delay.
Let f: R m1
, 905
"... In this paper we prove an analog of PerronFrobenius theorem for multilinear forms with nonnegative coefficients. ..."
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In this paper we prove an analog of PerronFrobenius theorem for multilinear forms with nonnegative coefficients.