Controllability of McKean–Vlasov Stochastic Integrodifferential Evolution Equation in Hilbert Spaces

Abstract

Sufficient conditions for controllability of McKean-Vlasov stochastic integrodifferential evolution equation in Hilbert spaces are obtained by using Banach fixed point analysis approach. This paper is motivated primarily by the work of Ahmed and Ding (Stochastic Processes Appl. 1995; 60:65-85), which is extended to a more general class of McKean-Vlasov stochastic integrodifferential equation in Hilbert space for establishing controllability result.

Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equations

“…In recent years, related topics and problems have attracted more and more researchers' attentions, see, for examples, Veretennikov [32], Huang-Malhamé-Caines [17], , Buckdahn-Djehiche-Li-Peng [8], Buckdahn-Li-Peng [9], Borkar-Kumar [6], Crisan-Xiong [12], Kotelenez-Kurtz [20], Kloeden-Lorenz [19], and so on. More interestingly, control problems of McKean-Vlasov equation or MF-FSDEs were investigated by Ahmed-Ding [2], Ahmed [3], Buckdahn-Djehiche-Li [7], Park-Balasubramaniam-Kang [28], Andersson-Djehiche [4], Meyer-Brandis-Oksendal-Zhou [25], and so on. This paper can be regarded as an addition to the study of optimal control for MF-FSDEs.…”

A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon

“…For the relevant works of recent years, see, for examples, Veretennikov [34], Huang-Malhamé-Caines [22], Buckdahn-Djehiche-Li-Peng [12], Buckdahn-Li-Peng [13], Borkar-Kumar [9], Crisan-Xiong [16], Kotelenez-Kurtz [25], and so on. Control problems of McKean-Vlasov equation or MF-FSDEs were investigated by Ahmed-Ding [3], Ahmed [4], Buckdahn-Djehiche-Li [11]), Park-Balasubramaniam-Kang [30], Andersson-Djehiche [6], Meyer-Brandis-Oksendal-Zhou [29], and so on. In Yong [35], a linearquadratic (LQ, for short) problem was introduced and investigated for MF-FSDEs in finite horizons.…”

Exact controllability of linear mean‐field stochastic systems and observability inequality for mean‐field backward stochastic differential equations

“…Moreover, it has the smallest norm among all feasible controls; that is, it is also the optimal control of Problem (NOC) at the point (x 0 , x T ). The smallest norm is given by (22). Furthermore, let ð xðÁÞ, yðÁÞ, zðÁÞÞ denote the solution to MF-FBSDE (19) with the parameter (x 0 , x T ).…”

“…However, we find there exists little research on the controllability of McKean-Vlasov SDEs except the work of Goreac [21] concerning the approximate controllability. We also mention that Park et al [22] studied the controllability of McKean-Vlasov stochastic integrodifferential evolution equation in Hilbert spaces. The exact controllability seems to be a brand new problem, incredulously ignored in the theory of McKean-Vlasov SDEs.…”