Introduction optogenetics is a recent and innovative technique which allows to induce or prevent electric shocks. The working paradigm of the fpbased control of stochastic models is the following. Numerical methods for simulation and optimization of. Boundary conditions given in vermes 1985 are weakened and replaced by boundary conditions in exittime optimal control problems as given in barles 1994. For this class of control problems, the value function can in general be characterized as the unique viscosity solution to the corresponding hamiltonjacobibellman.
Uniform assymptotics in the average continuous control of. A method of representing this controlled pdp as a discrete time decision process is presented, allowing the value. Piecewisedeterministic processes viscosity solutions markov processes. Optimal control of pure jump markov processes with noise. Approximation methods for piecewise deterministic markov. The trajectories of piecewise deterministic markov processes are solutions of an ordinary vectordifferential equation with possible random jumps between the. In order to assess the reliability of a complex industrial system by simulation, and in reasonable time, variance reduction methods such as importance sampling can be used. Continuoustime markov processes on general state spaces. Piecewisedeterministic markov processes pdmps have been introduced in the literature by m. Stochastic filtering and optimal control of pure jump markov. Phd in applied probability or related area, skills in continuoustime stochastic processes, markov processes, optimal stochastic control, numerical probability. Optimal impulsive control of piecewise deterministic markov processes dufour, f, horiguchi, m and piunovskiy, ab 2016 optimal impulsive control of piecewise deterministic markov processes. Optimal control of infinitedimensional piecewise deterministic markov processes and application to the control of neuronal dynamics via optogenetics. Optimal control of pure jump markov processes with noisefree.
For this class of control problems, the value function can in general be characterized as the unique viscosity solution to the corresponding hamiltonjacobibellman equation. We will prove under some continuity and compactness assumptions that an optimal stationary policy exists which is the solution of a deterministic control problem. Keywords optimal stopping piecewise deterministic markov processes quantization numerical method dynamic programming. Optimal control of infinitedimensional piecewise deterministic markov processes and application to the control of neuronal dynamics via optogenetics by vincent renault, michele thieullen and emmanuel trelat. This process is a coupling between a continuous time markov chain and a set of semilinear parabolic partial differential equations, both processes depending on the control. A piecewise deterministic markov process pdp is a continuous time markov process consisting of continuous, deterministic trajectories interrupted by random jumps. Pdf optimal control of piecewise deterministic markov.
Optimal impulsive control of piecewise deterministic. 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. These were only isolated rather recently but seen general enough to include as special cases practically all the nondi. Pdf impulse control of piecewise deterministic markov processes. This thesis describes a complete theory of optimal control of piecewise deterministic markov processes under weak assumptions. This process is a coupling between a continuous time markov chain and a set of semilinear parabolic partial differential equations, both processes depending. 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.
Optimal impulsive control of piecewise deterministic markov. This class of processes provides a rigorous framework for stochastic spatial models in which discrete random events are globally coupled with continuous space dependent variables solving partial differential equations, e. Keywords optimal stopping piecewise deterministic markov processes quantization. The pdp optimal full control problem with dynamic control plus impulse control is transformed. They form a family of markov processes involving deterministic. This paper concerns the optimal impulse control of piecewise deterministic markov processes pdps. Luiss roma, dipartimento di economia e finanza, via romania 32, 00197 roma, italy. Piecewise deterministic markov processes springerlink. In this paper the pl process is generalized in a minor but significant way to provide a class of processes we shall call piecewise deterministic pd markov processes. The trajectories may be controlled with the object of minimizing the expected costs associated with the process. Stochastic filtering partial observation control problem pure jump processes piecewisedeterministic markov processes markov decision processes this research was partially supported by three gnampaindam projects in 2015, 2016 and 2017 and by miurprin 2015 project deterministic and stochastic evolution equations. Average continuous control of piecewise deterministic. To prevent the use of relaxed controls, we will make several convexity assumptions. A turnpike improvement algorithm for piecewise deterministic.
Pdf a general class of nondiffusion stochastic models is introduced. Stochastic calculus for these processes is developed and a complete characterization of the extended generator is given. Multiconstrained finitehorizon piecewise deterministic. The aim of this paper is to investigate an optimal stopping problem under partial observation for piecewise deterministic markov processes pdmps both from the theoretical and numerical points of view. Qualitative properties of certain piecewise deterministic. Davis introduced the piecewise deterministic markov process pdmp class of stochastic hybrid models in an article in 1984. In the rst chapter we control a class of semimarkov processes on nite horizon. Deterministic and stochastic optimal control wendell.
Jan 19, 2018 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. Optimal control of piecewise deterministic markov processes. Stochastic filtering and optimal control of pure jump. The theory consists of a description of the processes, a nonsmooth stochastic maximum principle as a necessary optimality condition, a generalized bellmanhamiltonjacobi necessary and sufficient optimality condition involving the clarke generalized gradient, existence. For our applications this is no crucial restriction. Piecewisedeterministic processes and viscosity solutions. Piecewise deterministic markov processes, markov decision process, bellman equation, portfolio problems.
The main goal of the paper is to characterize the optimality equation of the problem in terms of integrodifferential equations for the continuoustime problem as well as. Numerical method for optimal stopping of piecewise. The relevance of the extended generator concept in applied problems is discussed and some recent results on optimal control of piecewise. Piecewise deterministic controlled systems 20, 40 occupy an intermediate niche and are useful for encoding nondi usive stochastic perturbations. The continuous component evolves according to a smooth vector. In the second part of the book we give an introduction to stochastic optimal control for markov diffusion processes. Piecewise deterministic markov processes pdmp have been introduced by davis 8, 10. A dynamic programming algorithm for the optimal control of. We consider an infinite horizon 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. This process is a coupling between a continuous time markov chain and a set of semilinear parabolic. In this paper we define an infinitedimensional controlled piecewise deterministic markov process pdmp and we study an optimal control problem with finite time horizon and unbounded cost. In this paper we consider a control problem for a partially observable piecewise deterministic markov process of the following type.
Continuous control of piecewise deterministic markov. This algorithm exploits the structure of markov decision processes with continuous state and action spaces that can be associated with piecewise deterministic control systems. The aim of this paper is to propose a new family of. Piecewise deterministic markov processes and dynamic. The goal is to minimize one type of expected finitehorizon cost over historydependent policies while keeping some other types of expected finitehorizon costs lower than some tolerable bounds. 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. 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. Davis introduced the piecewisedeterministic markov process pdmp class of stochastic hybrid models in an article in 1984. Primary 90c40, secondary 60j25, 91g10 1 introduction in this paper we deal with optimization problems where the state process is a piecewise deterministic markov process pdmp. The class of models is wide enough to include as special cases virtually all the nondiffusion models of applied probability.
This algorithm exploits the structure of markov decision processes with continuous state and action spaces that can be associated with piecewise deterministic control. We consider an infinite horizon discounted optimal control problem for piecewise deterministic markov processes, where a piecewise. Numerical methods for optimal control of piecewise. Optimal control of piecewise deterministic markov processes with. In probability theory, a piecewisedeterministic 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. Pdmps have been introduced by davis as a general class of stochastic models. For stochastic optimal control in discrete time, see bertsekas and shreve 1996.
Optimal control of piecewise deterministic markov processes with finite time horizon article pdf available january 2010 with 1 reads how we measure reads. The main goal of this chapter is to derive sufficient conditions for the existence of an optimal control strategy for the long run average continuous control problem of piecewise deterministic markov processes pdmps taking values in a general borel space and with compact action space depending on the state variable. Stochastic filtering partial observation control problem pure jump processes piecewise deterministic markov processes markov decision processes this research was partially supported by three gnampaindam projects in 2015, 2016 and 2017 and by miurprin 2015 project deterministic and stochastic evolution equations. The aim of this paper is to investigate an optimal stopping problem under partial observation for piecewisedeterministic markov processes pdmps both from the theoretical and numerical points of view. Optimal control of piecewise deterministic markov process.
The aim of this paper is to propose a computational method for optimal stopping of a piecewise deterministic markov process by using a quantization technique for an underlying discretetime markov chain related to the continuoustime process and pathadapted time discretization grids. The literature on optimal control topics in connection to these processes is extremely wide 9, 20, 1, 11, etc. Pdf impulse control of piecewise deterministic markov. In this context, one is interested in computing certain quantities of interest such as the probability of ruin of an insurance company, or the. Piecewise deterministic markov processes, introduced in 21, evolve through. The jump rates may depend on the whole position of the process. Numerical methods for simulation and optimization of piecewise deterministic markov processes. Piecewise deterministic markov processes and dynamic reliability. Controlledpiecewisedeterministicmarkovprocesses introduction davis80s generalclassofnondi. Optimal strategies for impulse control of piecewise. Pdf piecewise deterministic markov processes pdps are. 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. This paper studies a multiconstrained problem for piecewise deterministic markov decision processes pdmdps with unbounded cost and transition rates. Backward stochastic differential equations, optimal control problems, piecewise.
This article concerns the optimal control of piecewise deterministic processes in the viscosity solutions context. This paper proposes a numerical technique, called turnpike improvement, for the approximation of the solution of a class of piecewise deterministic control problems typically associated with manufacturing flow control models. In this paper the pl process is generalized in a minor but significant way to provide a class of processes we shall call piecewisedeterministic pd markov processes. We consider the infinite horizon expected discounted impulse control problem where the controller instantaneously moves. Piecewise deterministic markov processes, optimal control, semilinear parabolic equations, dynamic programming, markov decision processes, optogenetics. We present limit theorems for a sequence of piecewise deterministic markov processes pdmps taking values in a separable hilbert space. Yet, despite this, there is very little in the way of literature devoted to the development of. Piecewisedeterministic controlled systems 20, 40 occupy an intermediate niche and are useful for encoding nondi usive stochastic perturbations. Unconstrained and constrained optimal control of piecewise. Pdf necessary and sufficient optimality conditions for control of. Davis introduced the piecewisedeterministic markov process pdmp class of stochastic hybrid models in. Probabilistic representation of hjb equations for optimal. Optimal control of partially observable piecewise deterministic markov processes nicole bauerle and dirk langey abstract. The numerical method is applicable whenever a turnpike property holds for some associated infinite horizon deterministic control problem.
First, one reasonably assumes that the initial pdf of the state variable is known at the initial time, and the state variable x. Numerical methods for optimal control of piecewise deterministic markov processes. Lectures on stochastic control and nonlinear filtering. Average continuous control of piecewise deterministic markov. Control, optimisation and calculus of variations 24. The modeling is a key step in order to study the properties of the involved physical process.
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