Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-02T23:01:11.921Z Has data issue: false hasContentIssue false

Viability, invariance and reachability for controlled piecewise deterministic Markov processes associated to gene networks

Published online by Cambridge University Press:  13 April 2011

Dan Goreac*
Affiliation:
UniversitéParis-Est, Laboratoire d’Analyse et Mathématiques Appliquées, UMR 8050, Boulevard Descartes, Cité Descartes, 77450 Champs-sur-Marne, France. [email protected]
Get access

Abstract

We aim at characterizing viability, invariance and some reachability properties of controlled piecewise deterministic Markov processes (PDMPs). Using analytical methods from the theory of viscosity solutions, we establish criteria for viability and invariance in terms of the first order normal cone. We also investigate reachability of arbitrary open sets. The method is based on viscosity techniques and duality for some associated linearized problem. The theoretical results are applied to general On/Off systems, Cook’s model for haploinsufficiency, and a stochastic model for bacteriophage λ.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alvarez, O. and Tourin, A., Viscosity solutions of nonlinear integro-differential equations. Ann. Inst. Henri Poincaré, Anal. non linéaire 13 (1996) 293-317. Google Scholar
J.-P. Aubin, Viability Theory. Birkhäuser (1992).
Aubin, J.-P. and Da Prato, G., Stochastic viability and invariance. Ann. Sc. Norm. Pisa 27 (1990) 595694. Google Scholar
J.-P. Aubin and H. Frankowska, Set Valued Analysis. Birkhäuser (1990).
M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi- Bellman equations. Systems and Control : Foundations and Applications, Birkhäuser (1997).
M. Bardi and P. Goatin, Invariant sets for controlled degenerate diffusions : a viscosity solutions approach, in Stochastic analysis, control, optimization and applications, Systems Control Found. Appl., Birkhäuser, Boston, MA (1999) 191–208.
Bardi, M. and Jensen, R., A geometric characterization of viable sets for controlled degenerate diffusions. Set-Valued Anal. 10 (2002) 129141. Google Scholar
Barles, G. and Imbert, C., Second-order elliptic integro-differential equations : Viscosity solutions theory revisited. Ann. Inst. Henri Poincaré, Anal. non linéaire 25 (2008) 567–585. Google Scholar
Barles, G. and Jakobsen, E.R., On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations. ESAIM : M2AN 36 (2002) 3354. Google Scholar
Buckdahn, R., Peng, S., Quincampoix, M. and Rainer, C., Existence of stochastic control under state constraints. C. R. Acad. Sci. Paris Sér. I Math. 327 (1998) 1722. Google Scholar
Buckdahn, R., Goreac, D. and Quincampoix, M., Stochastic optimal control and linear programming approach. Appl. Math. Opt. 63 (2011) 257276. Google Scholar
Cook, D.L., Gerber, A.N. and Tapscott, S.J., Modelling stochastic gene expression : Implications for haploinsufficiency. Proc. Natl. Acad. Sci. USA 95 (1998) 1564115646. Google Scholar
A. Crudu, A. Debussche and O. Radulescu, Hybrid stochastic simplifications for multiscale gene networks. BMC Systems Biology 3 (2009).
M.H.A. Davis, Markov Models and Optimization, Monographs on Statistics and Applied probability 49. Chapman & Hall (1993).
Delbrück, M., Statistical fluctuations in autocatalytic reactions. J. Chem. Phys. 8 (1940) 120124. Google Scholar
Gautier, S. and Thibault, L., Viability for constrained stochastic differential equations. Differential Integral Equations 6 (1993) 13951414. Google Scholar
Hasty, J., Pradines, J., Dolnik, M. and Collins, J.J., Noise-based switches and amplifiers for gene expression. PNAS 97 (2000) 20752080. Google ScholarPubMed
Soner, H.M., Optimal control with state-space constraint. II. SIAM J. Control Optim. 24 (1986) 11101122. Google Scholar
Zhu, X. and Peng, S., The viability property of controlled jump diffusion processes. Acta Math. Sinica 24 (2008) 13511368.Google Scholar