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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 13 (1 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.
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 6 (5 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.
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.
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 3 (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
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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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 1 (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.
Structure of Solutions Sets and a Continuous Version of Filippov’s Theorem for First Order Impulsive Differential Inclusions with Periodic Conditions
"... In this paper, the authors consider the firstorder nonresonance impulsive differential inclusion with periodic conditions y ′(t) − λy(t) ∈ F(t,y(t)), a.e. t ∈ J\{t1,...,tm},) − y(t− k) = Ik(y(tk)), k = 1,2,...,m, y(0) = y(b), y(t + k where J = [0,b] and F: J × R n → P(R n) is a setvalued map. ..."
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In this paper, the authors consider the firstorder nonresonance impulsive differential inclusion with periodic conditions y ′(t) − λy(t) ∈ F(t,y(t)), a.e. t ∈ J\{t1,...,tm},) − y(t− k) = Ik(y(tk)), k = 1,2,...,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 = 1,2,...,m). The topological structure of solution sets as well as some of their geometric properties (contractibility and Rδsets) are studied. A continuous version of Filippov’s theorem is also proved.
FILIPPOV’S THEOREM FOR IMPULSIVE DIFFERENTIAL INCLUSIONS WITH FRACTIONAL ORDER
"... In this paper, we present an impulsive version of Filippov’s Theorem for fractional differential inclusions of the form: Dα ∗ y(t) ∈ F(t,y(t)), a.e. t ∈ J\{t1,...,tm}, α ∈ (1,2],) − y(t− k) = Ik(y(tk)), k = 1,...,m, k) = Ik(y ′(t − k)), k = 1,...,m, y(0) = a, y ′(0) = c, y(t + k y ′(t + k) − ..."
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In this paper, we present an impulsive version of Filippov’s Theorem for fractional differential inclusions of the form: Dα ∗ y(t) ∈ F(t,y(t)), a.e. t ∈ J\{t1,...,tm}, α ∈ (1,2],) − y(t− k) = Ik(y(tk)), k = 1,...,m, k) = Ik(y ′(t − k)), k = 1,...,m, y(0) = a, y ′(0) = c, y(t + k y ′(t + k) − y ′ (t − where J = [0,b], D α ∗ denotes the Caputo fractional derivative and F is a setvalued map. The functions Ik, Ik characterize the jump of the solutions at impulse points tk (k = 1,...,m). Key words and phrases: Fractional differential inclusions, fractional derivative, fractional integral.