Results 1  10
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16
Comparison of Riemann and Lebesgue sampling for first order stochastic systems
 In Proceedings of IEEE Conference on Decision and Control
, 2002
"... The normal approach to digital control is to sample periodically in time. Using an analog of integration theory we can call this Riemann sampling. Lebesgue sampling or event based sampling, is an alternative to Riemann sampling. It means that signals are sam pled only when measurements pass certain ..."
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Cited by 148 (0 self)
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The normal approach to digital control is to sample periodically in time. Using an analog of integration theory we can call this Riemann sampling. Lebesgue sampling or event based sampling, is an alternative to Riemann sampling. It means that signals are sam pled only when measurements pass certain limits. In this paper it is shown that Lebesgue sampling gives better performance for some simple systems. 1.
The substratum of impulse and hybrid control systems
 In HSCC’01, volume 2034 of LNCS
, 2001
"... The behavior of the run of an impulse differential inclusion, and, in particular, of a hybrid control system, is “summarized ” by the “ initialization map ” associating with each initial condition the set of new initialized conditions and more generally, by its “substratum”, that is a setvalued map ..."
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Cited by 17 (3 self)
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The behavior of the run of an impulse differential inclusion, and, in particular, of a hybrid control system, is “summarized ” by the “ initialization map ” associating with each initial condition the set of new initialized conditions and more generally, by its “substratum”, that is a setvalued map associating with a cadence and a state the next reinitialized state. These maps are characterized in several ways, and in particular, as “setvalued” solutions of a system of HamiltonJacobi partial differential inclusions, that play the same role than usual HamiltonJacobiBellman equations in optimal control.
Approximation of Viability Kernels and Capture Basin for Hybrid Systems
 Proc. of European Control Conference ECC’01
, 2001
"... This paper deals with hybrid dynamical systems with state constrains and target. We investigate the subset of initial positions from which there exists at least one run forever remaining in the constraint set the hybrid viability kernel or remaining in the constraint set until it reaches a given ..."
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Cited by 10 (1 self)
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This paper deals with hybrid dynamical systems with state constrains and target. We investigate the subset of initial positions from which there exists at least one run forever remaining in the constraint set the hybrid viability kernel or remaining in the constraint set until it reaches a given closed target in finite time the hybrid capture basin. We present an algorithm for approximating those sets and, under some regularity assumptions, we construct viable hybrid feedbacks providing viable runs. One example illustrates this study. It deals with an academic dynamical system revealing the complexity of the structure of the hybrid viability kernel and showing hybrid solutions. 1 Hybrid systems We consider a dynamical system describing the evolution of a state variable which, under some situations, may switch between a continuous evolution and an impulse evolution. Such system are called hybrid systems. Switches hold on a closed set. During some periods the state follows a continuous evolution until it reaches some position where a reset to a new position occurs. We will distinguish between slack solution for which the solution, when reaching may either be reset or keep going the continuous evolution, and strict solution for which the solution, when reaching, is necessarily reset. Such systems appear in control theory (see for instance [4, Bensoussan & Lions] and non linear systems (see
Cadenced runs of impulse and hybrid control systems
 International Journal Robust and Nonlinear Control
, 2001
"... Impulse differential inclusions, and in particular, hybrid control systems, are defined by a differential inclusion (or a control system) and a reset map. A run of an impulse differential inclusion is defined by a sequence of cadences, of reinitialized states and of motives describing the evolution ..."
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Cited by 9 (7 self)
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Impulse differential inclusions, and in particular, hybrid control systems, are defined by a differential inclusion (or a control system) and a reset map. A run of an impulse differential inclusion is defined by a sequence of cadences, of reinitialized states and of motives describing the evolution along a given cadence between two distinct consecutive impulse times, the value of a motive at the end of a cadence being reset as the next reinitialized state of the next cadence. A cadenced run is then defined by constant cadence, initial state and motive, where the value at the end of the cadence is reset at the same reinitialized state. It plays the role of a “discontinuous ” periodic solution of a differential inclusion. We prove that if the sequence of reinitialized states of a run converges to some state, then the run converges to a cadenced run starting from this state, and that, under convexity assumptions, that a cadenced run does exist.
Dynamic management of portfolios with transaction costs under tychastic uncertainty
 In: M. Breton and H. BenAmeur (Eds.). Numerical Methods in Finance
, 2005
"... We use in this paper the viability/capturability approach for studying the problem of dynamic valuation and management of a portfolio with transaction costs in the framework of tychastic control systems (or dynamical games against nature) instead of stochastic control systems. Indeed, the very defin ..."
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Cited by 7 (4 self)
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We use in this paper the viability/capturability approach for studying the problem of dynamic valuation and management of a portfolio with transaction costs in the framework of tychastic control systems (or dynamical games against nature) instead of stochastic control systems. Indeed, the very definition of the guaranteed valuation set can be formulated directly in terms of guaranteed viablecapture basin of a dynamical game. Hence, we shall “compute ” the guaranteed viablecapture basin and find a formula for the valuation function involving an underlying criterion, use the tangential properties of such basins for proving that the valuation function is a solution to HamiltonJacobiIsaacs partial differential equations. We then derive a dynamical feedback providing an adjustment law regulating the evolution of the portfolios obeying viability constraints until it achieves the given objective in finite time. We shall show that the Pujal & SaintPierre viability/capturability algorithm applied to this specific case provides both the valuation function and the associated portfolios. Acknowledgments The authors thank Giuseppe Da Prato, Francine Catté, Halim Doss, Hélène Frankowska, Georges Haddad, Nisard Touzi and Jerzy Zabczyk for many useful discussions and Michèle Breton and Georges Zaccour for inviting us in June 2004 to present these results in the Montréal’s GERAD (Groupe d’études et de recherche en analyse des décisions). Outline The first section is an introduction stating the problem and describing the main results presented. It is intended to readers who are not interested in the mathematical technicalities of the viability approach to financial dynamic valuation and management problems. The second section outlines the viability/capturability strategy and provides the minimal definitions and results of viability theory for deriving in the third and last section sketches the proofs of the main results 1Work supported in part by the European Community’s Human Potential Programme under contract HPRNCT
An Introduction to Viability Theory and Management of Renewable Resources
, 2002
"... The main purpose of viability theory is to explain the evolution of the state of a control system, governed by nondeterministic dynamics and subjected to viability constraints, to reveal the concealed feedbacks which allow the system to be regulated and provide selection mechanisms for implementing ..."
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Cited by 6 (0 self)
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The main purpose of viability theory is to explain the evolution of the state of a control system, governed by nondeterministic dynamics and subjected to viability constraints, to reveal the concealed feedbacks which allow the system to be regulated and provide selection mechanisms for implementing them. It assumes implicitly an “opportunistic” and “conservative ” behavior of the system: a behavior which enables the system to keep viable solutions as long as its potential for exploration (or its lack of determinism) — described by the availability of several evolutions — makes possible its regulation. We shall illustrate the main concepts and results of viability theory by revisiting the Verhulst type models in population dynamics, by providing the class of all Malthusian feedbacks (mapping states to growth rates) that guarantee the viability of the evolutions, and adapting these models to the management of renewable resources. Other examples of viability constraints are provided by architectures of networks imposing constraints described by connectionist tensors operating on coalitions of actors linked by the network. The question raises how to modify a given dynamical system governing the evolution of the signals, the connectionist tensors and the coalitions in such a way that the architecture remains viable. 1
BoundaryValue Problems for Systems of HamiltonJacobiBellman Inclusions with Constraints
 SIAM J. Control
"... We study in this paper boundaryvalue problems for systems of HamiltonJacobiBellman firstorder partial differential equations and variational inequalities, the solutions of which are constrained to obey viability constraints. They are motivated by some control problems (such as impulse control) ..."
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Cited by 6 (2 self)
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We study in this paper boundaryvalue problems for systems of HamiltonJacobiBellman firstorder partial differential equations and variational inequalities, the solutions of which are constrained to obey viability constraints. They are motivated by some control problems (such as impulse control) and financial mathematics. We shall prove the existence and uniqueness of such solutions in the class of closed setvalued maps, by giving a precise meaning to what a solution means in this case. We shall also provide explicit formulas to this problem. When we deal with HamiltonJacobiBellman equations, we obtain the existence and uniqueness of Frankowska contingent episolutions. We shall deduce these results from the fact that the graph of the solution is the viablecapture basin of the graph of the boundaryconditions under an auxiliary system, and then, from their properties and their characterizations proved in [12, Aubin].
Dynamical qualitative analysis of evolutionary systems
 In Hybrid Systems: Computation and Control, LNCS 2289
, 2002
"... Kuipers ’ QSIM algorithm for tracking the monotonicity properties of solutions to differential equations has been revisited by Dordan by placing it in a rigorous mathematical framework. The Dordan QSIM algorithm provides the transition laws from one qualitative cell to the others. We take up this id ..."
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Cited by 5 (1 self)
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Kuipers ’ QSIM algorithm for tracking the monotonicity properties of solutions to differential equations has been revisited by Dordan by placing it in a rigorous mathematical framework. The Dordan QSIM algorithm provides the transition laws from one qualitative cell to the others. We take up this idea and revisit it at the light of recent advances in the field of “hybrid systems ” and, more generally, “impulse differential equations and inclusions”. Let us consider a family of “qualitative cells Q(a) ” indexed by a parameter a ∈ A: We introduce a dynamical system on the discrete set of qualitative states prescribing an order of visit of the qualitative cells and an evolutionary system govening the “continuous ” evolution of a system, such as a control system. The question arises to study and characterize the set of any pairs of qualitative and quantitative initial states from which start at least one order of visit of the qualitative cells and an continuous evolution visiting the qualitative cells in the prescribed order. This paper is devoted to the issues regarding this question using tools of setvalued analysis and viability theory.
PathDependent Impulse and Hybrid Systems
 in Hybrid Systems: Computation and Control, 119132, Di Benedetto & SangiovanniVincentelli Eds, Proceedings of the HSCC 2001 Conference, LNCS 2034, SpringerVerlag
, 2001
"... Pathdependent impulse differential inclusions, and in particular, pathdependent hybrid control systems, are defined by a pathdependent differential inclusion (or pathdependent control system, or differential inclusion and control systems with memory) and a pathdependent reset map. In this paper, ..."
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Cited by 4 (4 self)
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Pathdependent impulse differential inclusions, and in particular, pathdependent hybrid control systems, are defined by a pathdependent differential inclusion (or pathdependent control system, or differential inclusion and control systems with memory) and a pathdependent reset map. In this paper, we characterize the viability property of a closed subset of paths under an impulse pathdependent differential inclusion using the Viability Theorems for pathdependent differential inclusions. Actually, one of the characterizations of the Characterization Theorem is valid for any general impulse evolutionary system that we shall defined in this paper.
First Order Impulsive Differential Inclusions with Periodic Conditions
"... In this paper, we present an impulsive version of Filippov’s Theorem for the firstorder nonresonance impulsive differential inclusion y ′(t) − λy(t) ∈ F(t,y(t)), a.e. t ∈ J\{t1,...,tm},) − y(t− k) = Ik(y(tk)), k = 1,...,m, y(0) = y(b), y(t + k where J = [0,b] and F: J × R n → P(R n) is a setv ..."
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Cited by 2 (1 self)
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In this paper, we present an impulsive version of Filippov’s Theorem for the firstorder nonresonance impulsive differential inclusion y ′(t) − λy(t) ∈ F(t,y(t)), a.e. t ∈ J\{t1,...,tm},) − y(t− k) = Ik(y(tk)), k = 1,...,m, y(0) = y(b), y(t + k where J = [0,b] and F: J × R n → P(R n) is a setvalued map. The functions Ik characterize the jump of the solutions at impulse points tk (k = 1,...,m.). Then the relaxed problem is considered and a FilippovWasewski result is obtained. We also consider periodic solutions of the first order impulsive differential inclusion y(t + k y ′(t) ∈ ϕ(t,y(t)), a.e. t ∈ J\{t1,...,tm},) − y(t− k) = Ik(y(tk)), k = 1,...,m, y(0) = y(b), where ϕ: J × R n → P(R n) is a multivalued map. The study of the above problems use an approach based on the topological degree combined with a Poincaré operator.