Discrete time McKean-Vlasov control problem: a dynamic programming approach

We consider the stochastic optimal control problem of nonlinear mean-field systems in discrete time. We reformulate the problem into a deterministic control problem with marginal distribution as controlled state variable, and prove that dynamic programming principle holds in its general form. We apply our method for solving explicitly the mean-variance portfolio selection and the multivariate linear-quadratic McKean-Vlasov control problem

Bellman equation and viscosity solutions for mean-field stochastic control problem

We consider the stochastic optimal control problem of McKean-Vlasov stochastic differential equation where the coefficients may depend upon the joint law of the state and control. By using feedback controls, we reformulate the problem into a deterministic control problem with only the marginal distribution of the process as controlled state variable, and prove that dynamic programming principle holds in its general form. Then, by relying on the notion of differentiability with respect to pro\-bability measures recently introduced by P.L. Lions in [32], and a special It{\^o} formula for flows of probability measures, we derive the (dynamic programming) Bellman equation for mean-field stochastic control problem, and prove a veri\-fication theorem in our McKean-Vlasov framework. We give explicit solutions to the Bellman equation for the linear quadratic mean-field control problem, with applications to the mean-variance portfolio selection and a systemic risk model. We also consider a notion of lifted visc-sity solutions for the Bellman equation, and show the viscosity property and uniqueness of the value function to the McKean-Vlasov control problem. Finally, we consider the case of McKean-Vlasov control problem with open-loop controls and discuss the associated dynamic programming equation that we compare with the case of closed-loop controls.Comment: to appear in ESAIM: COC

High curie temperature ternary piezoelectric ceramics

A preferred piezoelectric ceramic material is a BiFeO3—PbZrO3—PbTiO3 ternary solid solution wherein proportions of the constituent perovskite metal oxides are selected so that the material exhibits relatively high Curie temperatures above 380° C. and useful piezoelectric properties

Piezoelectric Ceramics with Compositions at the Morphotropic Phase Boundary in the BiFeO3–PbZrO3–PbTiO3 Ternary System

Since ceramics in the PbZrO3–PbTiO3 binary system display excellent piezoelectric properties and those in BiFeO3–PbTiO3 exhibit high Curie temperatures, morphotropic phase boundary (MPB) compositions in the BiFeO3–PbZrO3–PbTiO3 ternary solid solution system are investigated for the development of piezoelectric ceramics for high temperature applications. It is found that the MPB compositions in the ternary system deviate away from the mixture of two binary MPB compositions 0.70BiFeO3–0.30PbTiO3 and 0.52PbZrO3–0.48PbTiO3. With decreasing amount of BiFeO3 in the MPB compositions in the ternary system, the Curie temperature TC is observed to decrease while the piezoelectric coefficient d33 is found to increase. Accompanied with this trend are the decrease in the c/a ratio of the tetragonal phase, the increase in the dielectric constant and the decrease in the loss tangent of the ceramics at room temperature. It is further noticed that the compositions in the rhombohedral-rich side of MPB exhibit slightly better piezoelectric properties. An example of such compositions is 0.511BiFeO3–0.326PbZrO3–0.163PbTiO3, with TC of 431°C, d33 of 101 pC/N, and the electromechanical coupling factor kp of 0.50