On Applications of Maslov Optimization Theory

P. Del Moral ; M. Doisy

Matematičeskie zametki, Tome 69 (2001) no. 2, pp. 262-276.

Résumé

Maslov optimization theory has recently emerged as a new branch of functional analysis for studying deterministic control problems and Hamilton Jacobi equations. The main purpose of this work is to use an idempotent probability calculus to study the fixed points of nonexpansive transformations on nonnecessarily finite state spaces. We will see that these fixed points can be regarded as the $(\max,+)$-version of the invariant measure of Markov semi-groups. In the second part of this work we also present the $(\max,+)$-version of Dynkin's formula in the theory of stochastic processes and we apply this formula to study the stability properties of Bellman–Maslov processes.

Bibliographie

[1] Del Moral P., Doisy M., “Maslov Idempotent Probability Calculus. I; II”, Teoriya veroyatnostei i ee primenenie, 43:4 (1998), 735–751 ; 44:2 (1999), 384–400 | Zbl | Zbl

[2] Litvinov G., Maslov V. P., Correspondence principle for idempotent calculus and some computer applications, IHES-M-95-33, Institut des Hautes Études Scientifiques, Paris, 1995; Idempotency, ed. J. Gunawardena, Cambridge University Press, 1998, 420–443 | MR | Zbl

[3] Akian M., Quadrat J. P., Viot M., “Bellman Processes”, $11$-eme Conférence Internationale sur l'Analyse et l'Optimisation des Systèmes, École des Mines Sophia-Antipolis (France, 15–16–17 Juin 1994), Lecture Notes in Control and Information Sciences, 199, Springer-Verlag | Zbl

[4] Del Moral P., “Maslov Optimization Theory: Optimality versus Randomness”, Idempotency Analysis and its Applications, eds. V. N. Kolokoltsov, V. P. Maslov, Kluwer Academic Publishers, 1997

[5] Kolokoltsov V. N., Maslov V. P., “Idempotentnyi analiz kak apparat teorii upravleniya i optimalnogo sinteza”, Funktsionalnyi analiz i ego prilozheniya, 23:1 (1989), 1–14 ; “II”, Функциональный анализ и его приложения, 23:4, 53–62 ; English transl. in Functional Anal. Appl., 23 (1989) | MR | MR | Zbl

[6] Kolokoltsov V. N., Maslov V. P., “Obschii vid endomorfizmov v prostranstve nepreryvnykh funktsii so znacheniem v chislovom kommutativnom polukoltse (s operatsiei $\oplus =\max $)”, Dokl. AN SSSR, 295:2 (1987), 283–287

[7] Maslov V. P., Méthodes opératorielles, Mir, M., 1987

[8] Maslov V. P., “Quasilinear systems which are linear in some semimoduli”, Congrès international sur les problèmes hyperboliques (13–17 Janvier 1986, Saint Etienne, France)

[9] Maslov V. P., Samborskii S. N., Idempotent Analysis, Advances in Soviet Mathematics, 13, Amer. Math. Soc., Providence, 1992

[10] Trouvé A., Parallélisation massive du recuit simulé, Thèse de Doctorat, Université Paris XI, Janvier 1993

[11] P. I. Dudnikov, S. N. Samborskii, Endomorfizmy polumodulya nad polukoltsom s idempotentnoi operatsiei, Preprint No. 87-48, Institut matematiki AN USSR, Kiev, 1987

[12] Kolokoltsov V. N., “Polukoltsevye analogi lineinoi ekvivalentnosti prostranstv”, Vsesoyuznaya shkola “Optimalnoe upravlenie, geometriya i analiz” (Kemerovo), 1988, 30

[13] Meyn S. P., Tweedie R. L., Markov Chains and Stochastic Stability, Springer-Verlag, 1993

[14] Nussbaum R. D., “Convergence of iterates of a nonlinear operator arising in statistical mechanics”, Nonlinearity, 4, U.K, 1991, 1223–1240 | MR | Zbl

[15] Bacelli F., Cohen G., Olsder G. J., Quadrat J. P., Synchronization and Linearity, An Algebra for Discrete Event Systems, John Wiley and Sons Publ., 1992

[16] Bott R., Mayberry J. P., “Matrices and trees”, Economics Activity Analysis, John Wiley and Sons Publ., New York, 1954

[17] Freidlin M. I., Wentzell A. D., Random Perturbations of Dynamical Systems, Springer-Verlag, 1984

[18] Miclo L., “Recuit simulé sans potentiel sur un ensemble fini”, Séminaire de Probabilités, Lecture Notes in Mathematics, 1526, Springer-Verlag, 1992, 47–60

[19] Del Moral P., Miclo L., “On the convergence and the applications of the generalized simulated annealing”, SIAM J. Control Optim., 37:4 (1999), 1222–1250 | DOI | MR | Zbl