The Mean Field Type Control Problems

  • Alain Bensoussan
  • Jens Frehse
  • Phillip Yam
Chapter
First Online:
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

We need to assume that the
$$\displaystyle\begin{array}{lll}\,\, m \rightarrow f(x,m,v),\;g(x,m,v),\;h(x,m) \\ \text{are} \text{ differentiable in}\;m \in {L}^{2}({\mathbb{R}}^{n}){}\end{array}$$
(4.1)
and we use the notation \(\dfrac{\partial f} {\partial m}(x,m,v)(\xi )\) to represent the derivative, so that
$$\displaystyle{\frac{d} {d\theta }f(x,m +\theta \tilde{ m},v)_\vert {}_{\theta =0} =\int _{{\mathbb{R}}^{n}} \dfrac{\partial f} {\partial m}(x,m,v)(\xi )\tilde{m}(\xi )\,d\xi.}$$
Here, x, v are simply parameters. Coming back to the definition ( 2.9)–( 2.11), consider a feedback v(x) and the corresponding trajectory defined by ( 2.9). The probability distribution m v(. )(t) of x v(. )(t) is a solution of the FP equation
$$\displaystyle\begin{array}{rcl} \frac{\partial m_{v(.)}} {\partial t} + {A}^{{\ast}}m_{ v(.)} + \text{div }(g(x,m_{v(.)},v(x))m_{v(.)})& =& 0, \\ m_{v(.)}(x,0)& =& m_{0}(x){}\end{array}$$
(4.2)
and the objective functional J(v(. ), m v(. )) can be expressed as follows
$$\displaystyle\begin{array}{rcl} J(v(.),m_{v(.)}(.))& =& \int _{0}^{T}\int _{{ \mathbb{R}}^{n}}f(x,m_{v(.)}(x),v(x))m_{v(.)}(x)dxdt\; \\ & & \quad +\int _{{\mathbb{R}}^{n}}h(x,m_{v(.)}(x,T))m_{v(.)}(x,T)dx.{}\end{array}$$
(4.3)
Consider an optimal feedback \(\hat{v}(x)\) and the corresponding probability density \(m_{\hat{v}(.)}(x,t) = m(x,t)\).

Keywords

Nash Equilibrium Feedback Control Optimal Feedback Field Type Convexity Assumption 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Copyright information

© Alain Bensoussan, Jens Frehse, Phillip Yam 2013

Authors and Affiliations

  • Alain Bensoussan
    • 1
    • 2
  • Jens Frehse
    • 3
  • Phillip Yam
    • 4
  1. 1.Naveen Jindal School of ManagementUniversity of Texas at DallasRichardsonUSA
  2. 2.Department of Systems Engineering and Engineering ManagementCity University of Hong KongKowloonHong Kong SAR
  3. 3.Institut für Angewandte MathematikUniversitat BonnBonnGermany
  4. 4.Department of StatisticsThe Chinese University of Hong KongShatinHong Kong SAR

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