Viability Theorem for Deterministic Mean Field Type Control Systems
Tóm tắt
A mean field type control system is a dynamical system in the Wasserstein space describing an evolution of a large population of agents with mean-field interaction under a control of a unique decision maker. We develop the viability theorem for the mean field type control system. To this end we introduce a set of tangent elements to the given set of probabilities. Each tangent element is a distribution on the tangent bundle of the phase space. The viability theorem for mean field type control systems is formulated in the classical way: the given set of probabilities on phase space is viable if and only if the set of tangent distributions intersects with the set of distributions feasible by virtue of dynamics.
Tài liệu tham khảo
Ahmed, N., Ding, X.: Controlled McKean-Vlasov equation. Commun. Appl. Anal. 5, 183–206 (2001)
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows: in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics. ETH Zurich. Birkhäuser, Basel (2005)
Andersson, D., Djehiche, B.: A maximum principle for SDEs of mean-field type. Appl Math Optim 63(3), 341–356 (2011)
As Soulaimani, S.: Viability with probabilistic knowledge of initial condition, application to optimal control. Set-Valued Anal. 16(7), 1037–1060 (2008)
Aubin, J.P.: Viability Theory. Birkhäuser, Basel (2009)
Aubin, J.P., Bayen, A.M., Saint-Pierre, P.: Viability Theory. New Directions. Springer, New York (2011)
Aubin, J.P., Cellina, A.: Differential Inclusions. Set-valued Maps and Viability Theory. Springer, New York (1984)
Bayraktar, E., Cosso, A., Pham, H.: Randomized dynamic programming principle and Feynman-Kac representation for optimal control of McKean-Vlasov dynamics. Trans. Amer. Math. Soc. 370, 2115–2160 (2018)
Bensoussan, A., Frehse, J., Yam, P.: Mean Field Games and Mean Field Type Control Theory. Springer, New York (2013)
Bensoussan, A., Frehse, J., Yam, S.: The master equation in mean field theory. Journal de Mathématiques Pures et Appliquées 103, 1441–1474 (2015)
Buckdahn, R., Djehiche, B., Li, J.: A general stochastic maximum principle for SDEs of mean-field type. Appl. Math. Optim. 64(2), 197–216 (2011)
Carmona, R., Delarue, F.: Forward-backward stochastic differential equations and controlled McKean–Vlasov dynamics. Preprint at arXiv:1303.5835 (2013)
Carmona, R., Delarue, F.: The master equation for large population equilibriums. Stoch. Anal. Appl. 100, 77–128 (2014). Springer
Carmona, R., Delarue, F., Lachapelle, A.: Control of McKean-Vlasov dynamics versus mean field games. Math. Financ. Econ. 7(2), 131–166 (2013)
Cavagnari, G., Marigonda, A., Nguyen, K., Priuli, F.: Generalized control systems in the space of probability measures. Set-Valued and Variational Analysis. (2017). https://doi.org/10.1007/s11228-017-0414-y
Gigli, N.: On the inverse implication of brenier-McCann theorems and the structure of \((\mathcal {P}_{2}({M}), W_{2})\). Methods and Applications of Analysis 18, 127–158 (2009)
Huang, M., Malhamé, R., Caines, P.: Nash equilibria for large population linear stochastic systems with weakly coupled agents. In: Boukas, E.K., Malhamé, R. (eds.) Analysis, Control and Optimization of Complex Dynamic Systems, pp. 215–252. Springer (2005)
Khaled, B., Meriem, M., Brahim, M.: Existence of optimal controls for systems governed by mean-field stochastic differential equations. Afr. Stat. 9(1), 627–645 (2014)
Lacker, D.: Limit theory for controlled McKean-Vlasov dynamics. SIAM J. Control Optim. 55, 1641–1672 (2017)
Lasry, J.M., Lions, P.L.: Jeux à champ moyen. I. Le cas stationnaire (French) [Mean field games. I. The stationary case]. C. R. Math. Acad. Sci. Paris 343, 619–625 (2006)
Lasry, J.M., Lions, P.L.: Jeux à champ moyen. II. Horizon fini et contrôle optimal (French) [Mean field games. II. Finite horizon and optimal control]. C. R. Math. Acad. Sci. Paris 343, 679–684 (2006)
Laurière, M., Pironneau, O.: Dynamic programming for mean-field type control. C. R. Math. Acad. Sci. Paris 352(9), 707–713 (2014)
Lott, J.: On tangent cones in wasserstein space. Proc. Amer. Math. Soc. 145, 3127–3136 (2017)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basic Theory. Springer, New York (2006)
Pham, H., Wei, X.: Dynamic programming for optimal control of stochastic McKean-Vlasov dynamics. SIAM J. Control Optim. 55, 1069–1101 (2017)
Pham, H., Wei, X.: Bellman equation and viscosity solutions for mean-field stochastic control problem. ESAIM Control Optim. Calc. Var. Accepted (2018)
Pogodaev, N.: Optimal control of continuity equations. NoDEA Nonlinear Differential Equations Appl. 23, 24 (2016). Art21
Subbotin, A.I.: Generalized solutions of first-order PDEs. The dynamical perspective. Birkhäuser, Basel (1995)
Warga, J.: Optimal Control of Differential and Functional Equations. Academic Press, New York (1972)