Impulse control of piecewise deterministic markov processes pdf

By stability here we mean the existence of an invariant probability measure for the pdmp. Our approach is in contrast with the pomdp and quasivariational formulations in the aforementioned literature that make use of more analytic tools. Impulsive control of piecewise deterministic processes. The continuous component evolves according to a smooth vector. Uniform assymptotics in the average continuous control of. The construction of such strategies is explicit and only necessitates the previous knowledge of the cost of the no impulse. Optimal impulsive control of piecewise deterministic markov processes. Dufour abstractin this paper we consider the long run average continuous control problem of piecewisedeterministic markov processes pdps for short. This paper considers existence and uniqueness results for viscosity solutions of integrodifferential equations associated with the impulse control problem for piecewise deterministic processes on bounded domains and on r n. On gradualimpulse control of continuoustime markov decision processes with multiplicative cost. Convergence results are obtained and more importantly we are able to give a convergence.

Optimal strategies for impulse control of piecewise deterministic. This paper deals with the general discounted impulse control problem of a piecewise deterministic markov process. Piecewise deterministic markov process archive ouverte hal. Today it is used to model a variety of complex systems in the fields of engineering, economics, management sciences, biology, internet traffic, networks and many more. Controlled stochastic process encyclopedia of mathematics. This chapter contains the basic theory for piecewise deterministic markov processes, whether homogeneous or not, based exclusively on the theory of marked point processes from the previous chapters and presented through the device of viewing a pdmp as a process adapted to the filtration generated by an rcm.

Average continuous control of piecewise deterministic. Stability of piecewisedeterministic markov processes. The construction of such strategies is explicit and only necessitates the previous knowledge of the cost of the no impulse strategy. In this paper we consider the long run average continuous control problem of piecewisedeterministic markov processes pdps for short. Impulse control of piecewise deterministic markov processes created date. The main result is the markovian form of the value function of this problem. In this paper, we study the infinitehorizon expected discounted continuoustime optimal control problem for piecewise deterministic markov. Impulse control of piecewisedeterministic processes researchgate. We prove a uniform abelian result for controlled systems with piecewise deterministic. Average continuous control of piecewise deterministic markov processes o. Numerical method for optimal stopping of piecewise deterministic markov processes, annals of applied probability, 2010, vol.

Persistent random walks, variable length markov chains and. Read optimal strategies for impulse control of piecewise deterministic markov processes, automatica on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Finally, we obtain explicit value functions for optimal stopping of straddle options, when the state process is a re. The stochastic process is a controlled jump markov process. This paper concerns the optimal impulse control of piecewise deterministic markov processes pdps. An impulse control strategy consists in a sequence of single interventions introducing a jump. The piecewise deterministic markov processes denoted pdmps were. Numerical method for control of piecewise deterministic markov processes the aim of this chapter is to present the computational method developed in des 10 for the control of a piecewise deterministic markov process pdmp x t t0 and to explain how similar ideas can. In the first part of the paper we study the impulse control of p. Continuous average control of piecewise deterministic. Citeseerx document details isaac councill, lee giles, pradeep teregowda. The mathematics of stochastic manufacturing systems, edited by george yin and qing zhang, vol. This paper presents a numerical method to calculate the value function for a general discounted impulse control problem for piecewise deterministic markov processes.

For a class of piecewise deterministic differential games in finite or infinite horizons we formulate conditions for the value functions in the. Optimal control of piecewise deterministic markov processes with application to. An impulse control problem of a production model with interruptions to follow stochastic demand an impulse control problem of a production model with interruptions to follow stochastic demand salles, j. Numerical method for control of piecewisedeterministic. Viscosity solutions associated with impulse control. The modelization is deeply inspired by the theory of renewal for markov processes by p. Impulse control of piecewisedeterministic processes springerlink.

The class of models is wide enough to include as special cases virtually all the nondiffusion models of applied probability. Abstract this paper deals with the general discounted impulse control problem of a piecewise deterministic markov process. Numerical methods for optimal control of piecewise deterministic markov processes. Davis introduced the piecewisedeterministic markov process pdmp class of stochastic hybrid models in an article in 1984. The pdp optimal full control problem with dynamic control plus impulse. In a recent paper we presented a numerical technique for solving the optimal stopping problem for a piecewise deterministic process p. This work we will focus on impulse control problems when the controller punctually. The notion of piecewise deterministic process is very intuitive and simple to. Hence this leads to a control problem in continuoustime which can be tackled for example via the. Impulse control of piecewisedeterministic processes. Qualitative properties of certain piecewise deterministic. We will henceforth call these piecewise deterministic processes or pdps. Continuous average control of piecewise deterministic markov processes oswaldo luiz do valle costa, francois dufour the intent of this book is to present recent results in the control theory for the long run average continuous control problem of piecewise deterministic markov processes pdmps. In this paper we apply these results to the impulse control problem.

Impulse control total cost discounted cost randomized strategy piecewise deterministic markov process. The jump rates may depend on the whole position of the process. We consider an infinitehorizon discounted optimal control problem for piecewise deterministic markov processes, where a piecewise openloop control acts continuously on the jump dynamics and on the deterministic flow. Using the tools of the markov decision processes, we justify the dynamic programming approach to the optimal impulse control of deterministic dynamical systems. Optimal control of piecewise deterministic markov processes. Piecewise deterministic markov processes springerlink. Pdmps impulse control numerical scheme numerical results perspectives discretization scheme recurrenceonrandomvariables v nz n.

Piecewise deterministic markov processes and dynamic. Our approach is based on a quantization technique for the underlying markov chain defined by the post jump location and interarrival time. This paper is concerned with the markovian feedback strategies of piecewise deterministic differential games and their applications to business and management decisionmaking problems that involve multiple agents and continuous and impulse controls. The pdp optimal full control problem with dynamic control plus impulse control is transformed to an equivalent dynamic control problem. Impulse control of piecewise deterministic markov processes.

Piecewise deterministic markov processes for continuous. Ye, dynamic programming and maximum principle for control of piecewise deterministic markov processes, ams lectures in applied mathematics. In this paper, we study a form of stability for a general family of nondiffusion markov processes known in the literature as piecewise deterministic markov process pdmp. Piecewise deterministic markov processes have been introduced by m. On value functions for impulsive control of piecewise deterministic processes. Numerical methods for simulation and optimization of. Impulse control of piecewisedeterministic processes, math. Ye university of essex and dalhousie university, and university of victoria this paper concerns the optimal impulse control of piecewise deterministic markov processes pdps.

Numerical method for impulse control of piecewise deterministic. The control markov property of a controlled jump markov process means that from a known present, the past enters in the righthand side of 1a1b only through the strategy. Optimal strategies for impulse control of piecewise. Numerical methods for optimal control of piecewise. The pdp optimal full control problem with dynamic control plus impulse control is transformed. We investigate a new family of epsilonoptimal strategies. Pdf impulse control of piecewise deterministic markov. Optimal control of piecewise deterministic markov processes with. Then under smooth additional conditions, we prove the existence of an optimal control. The existence of an optimal full control and a generalized bellmanhamilton. Davis introduced the piecewise deterministic markov process pdmp class of stochastic hybrid models in an article in 1984. If ip working conference on stochstic systems and optimization, warsaw, 1218 september 1988. In probability theory, a piecewise deterministic markov process pdmp is a process whose behaviour is governed by random jumps at points in time, but whose evolution is deterministically governed by an ordinary differential equation between those times.