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

The quotient space approach is considered as an effective and pioneering tool to model the transient dynamics of dimension‐varying systems recently. In this paper, we focus on the controllability of dimension‐varying systems that are modeled by the quotient space approach. First, we investigate the relation of controllability matrix between so‐called equivalent lifting systems. The result of the investigation prompts us to propose a new concept called equivalent class controllability for dimension‐varying systems on quotient space. Moreover, the coordinate transformation of states on quotient space is studied. Based on it, the controllable normal form of dimension‐varying systems on quotient space, which follows the criterion for the equivalent class controllability, is presented.

Section: Decomposition Of the Linear System On Quotient Spacementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Equivalence‐based controllability of dimension‐varying systems on quotient space

Fan

Zhang

2023

“…Mean field control can be seen as one-agent version of mean field games. Ye and Yu [28] are concerned with the exact controllability of linear mean field stochastic systems with time-variant random coefficients. Djehiche et al [29] studied a kind of mean field type control problems with risk-sensitive performance functionals.…”

Section: Introductionmentioning

confidence: 99%

Risk‐sensitive large‐population linear‐quadratic‐Gaussian Games with major and minor agents

2023

This paper studies large‐population dynamic games involving a linear‐quadratic‐Gaussian (LQG) system with an exponential cost functional. The parameter in the cost functional can describe an investor's risk attitude. In the game, there are a major agent and a population of minor agents where is very large. The agents in the games are coupled via both their individual stochastic dynamics and their individual cost functions. The mean field methodology yields a set of decentralized controls, which are shown to be an ‐Nash equilibrium for a finite population system where . Numerical results are established to illustrate the impact of the population's collective behaviors and the consistency of the mean field estimation.

“…It is well-known that the stochastic stability of system (1) has attracted a lot of attention of researchers in the field of stochastic control, for example, earlier studies [1][2][3][4][5][6][7][8][9][10][11][12] and references therein. Being different from asymptotic stability, which describes the asymptotic behavior of the solution trajectories as time goes to infinity, finite-time stability means that the solution trajectories converge to an equilibrium state in finite-time.…”

Section: Introductionmentioning

confidence: 99%

A new theorem on finite‐time stability of stochastic homogeneous systems and its application

Guo

Yin

et al. 2023

This paper provides a novel Lyapunov theorem on finite‐time stability of stochastic homogeneous systems. Different from the existing results, the differential operator in this paper is not required to be negative definite and could be negative semidefinite. The obtained condition is somehow similar to the LaSalle's condition. Moreover, the existing theorem of finite‐time stability for stochastic homogeneous systems is a special case of our result. As an application of the obtained theorem, stochastic finite‐time stabilization is considered for stochastic homogeneous affine control systems. Some examples and simulations are provided to show the effectiveness of the results.