A unifying formulation of the Fokker–Planck–Kolmogorov equation for general stochastic hybrid systems

Bect, Julien

doi:10.1016/j.nahs.2009.07.008

Cited by 20 publications

(12 citation statements)

References 23 publications

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“…10. We consider a time horizon with T = 5 and time windows of size Δt = 0.5, so that t k = kΔt, k = 1, .…”

Section: Numerical Experimentsmentioning

confidence: 99%

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Optimal control of a class of piecewise deterministic processes

Annunziato¹,

Borzì²

2013

Eur. J. Appl. Math

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A new control strategy for a class of piecewise deterministic processes (PDP) is presented. In this class, PDP stochastic processes consist of ordinary differential equations that are subject to random switches corresponding to a discrete Markov process. The proposed strategy aims at controlling the probability density function (PDF) of the PDP. The optimal control formulation is based on the hyperbolic Fokker–Planck system that governs the time evolution of the PDF of the PDP and on tracking objectives of terminal configuration with a target PDF. The corresponding optimization problems are formulated as a sequence of open-loop hyperbolic optimality systems following a model predictive control framework. These systems are discretized by first-order schemes that guarantee positivity and conservativeness of the numerical PDF solution. The effectiveness of the proposed computational control framework is validated considering PDP with dichotomic noise

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“…10. We consider a time horizon with T = 5 and time windows of size Δt = 0.5, so that t k = kΔt, k = 1, .…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…The model for such distributions is the analogous to the FP equation for the stochastic Itō equation [10]. The model for such distributions is the analogous to the FP equation for the stochastic Itō equation [10].…”

Section: Introductionmentioning

confidence: 99%

Optimal control of a class of piecewise deterministic processes

Annunziato¹,

Borzì²

2013

Eur. J. Appl. Math

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“…We denote by L 1 (m × ν) the space of real functions integrable with respect to m × ν. As in [8], we assume the existence of a kernel Q * such that…”

Section: A Piecewise-deterministic Markov Processesmentioning

confidence: 99%

“…Rate functions have a fundamental role in the study of the probability of rare events in the context of large deviations theory. However, it is not common to solve the maximization posed in (8) explicitly. Our objective is to construct a Lyapunov function u by solving this maximization problem.…”

Section: Lyapunov Functions and Rate Functionsmentioning

confidence: 99%

Construction of Lyapunov functions for piecewise-deterministic Markov processes

Mesquita

Hespanha

2010

49th IEEE Conference on Decision and Control (CDC)

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Abstract-The purpose of this contribution is twofold: 1) to present for the first time a Lyapunov function that proves exponential ergodicity of a process studied by the authors in [1], where the problem of controlling the probability density of a swarm of robotic agents was solved; 2) to introduce alongside the method used to construct this Lyapunov function, which is of interest in its own since it may be applicable to a wide class of piecewise deterministic Markov processes. Our method searches for the Lyapunov function that maximizes a measure of the rate of convergence that appears in the theory of large deviations. Analytical solutions are often possible as shown by examples.

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“…An individual TCL unit is modeled as a Stochastic Hybrid System (SHS) with the Markov property, and the resulting population model is in the form of a Partial Differential Equation (PDE) or Partial Integro-Differential Equation (PIDE) system and boundary conditions. This PDE form corresponds to a generalized Fokker-Planck (Forward Kolmogorov) operator [1] associated with the TCL stochastic hybrid system.…”

Section: Introductionmentioning

confidence: 99%