# Linearization Techniques for Controlled Piecewise Deterministic Markov Processes; Application to Zubov's Method

## Abstract

We aim at characterizing domains of attraction for controlled piecewise deterministic processes using an occupational measure formulation and Zubov's approach. Firstly, we provide linear programming (primal and dual) formulations of discounted, infinite horizon control problems for PDMPs. These formulations involve an infinite-dimensional set of probability measures and are obtained using viscosity solutions theory. Secondly, these tools allow to construct stabilizing measures and to avoid the assumption of stability under concatenation for controls. The domain of controllability is then characterized as some level set of a convenient solution of the associated Hamilton-Jacobi integral-differential equation. The theoretical results are applied to PDMPs associated to stochastic gene networks. Explicit computations are given for Cook's model for gene expression.

- Authors:

- Universite Paris-Est, LAMA, UMR8050 (France)
- Univ. Perpignan Via Domitia, Laboratoire de Matheematiques et PhySique (France)

- Publication Date:

- OSTI Identifier:
- 22092059

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Applied Mathematics and Optimization; Journal Volume: 66; Journal Issue: 2; Other Information: Copyright (c) 2012 Springer Science+Business Media, LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; CONTROL THEORY; HAMILTON-JACOBI EQUATIONS; INTEGRO-DIFFERENTIAL EQUATIONS; LINEAR PROGRAMMING; MARKOV PROCESS; MATHEMATICAL SOLUTIONS; PROBABILITY

### Citation Formats

```
Goreac, Dan, E-mail: Dan.Goreac@univ-mlv.fr, and Serea, Oana-Silvia.
```*Linearization Techniques for Controlled Piecewise Deterministic Markov Processes; Application to Zubov's Method*. United States: N. p., 2012.
Web. doi:10.1007/S00245-012-9169-X.

```
Goreac, Dan, E-mail: Dan.Goreac@univ-mlv.fr, & Serea, Oana-Silvia.
```*Linearization Techniques for Controlled Piecewise Deterministic Markov Processes; Application to Zubov's Method*. United States. doi:10.1007/S00245-012-9169-X.

```
Goreac, Dan, E-mail: Dan.Goreac@univ-mlv.fr, and Serea, Oana-Silvia. Mon .
"Linearization Techniques for Controlled Piecewise Deterministic Markov Processes; Application to Zubov's Method". United States. doi:10.1007/S00245-012-9169-X.
```

```
@article{osti_22092059,
```

title = {Linearization Techniques for Controlled Piecewise Deterministic Markov Processes; Application to Zubov's Method},

author = {Goreac, Dan, E-mail: Dan.Goreac@univ-mlv.fr and Serea, Oana-Silvia},

abstractNote = {We aim at characterizing domains of attraction for controlled piecewise deterministic processes using an occupational measure formulation and Zubov's approach. Firstly, we provide linear programming (primal and dual) formulations of discounted, infinite horizon control problems for PDMPs. These formulations involve an infinite-dimensional set of probability measures and are obtained using viscosity solutions theory. Secondly, these tools allow to construct stabilizing measures and to avoid the assumption of stability under concatenation for controls. The domain of controllability is then characterized as some level set of a convenient solution of the associated Hamilton-Jacobi integral-differential equation. The theoretical results are applied to PDMPs associated to stochastic gene networks. Explicit computations are given for Cook's model for gene expression.},

doi = {10.1007/S00245-012-9169-X},

journal = {Applied Mathematics and Optimization},

number = 2,

volume = 66,

place = {United States},

year = {Mon Oct 15 00:00:00 EDT 2012},

month = {Mon Oct 15 00:00:00 EDT 2012}

}