Stochastic Control of Dynamical Systems

Abstract

While Chapter 8 deals with Markov decision processes, this chapter is concerned with stochastic dynamical systems with the state x^ε (t) ∈ ℝⁿ and the control satisfying

$$ \frac{{d{x^{\varepsilon }}\left( t \right)}}{{dt}} = f\left( {{x^{\varepsilon }}\left( t \right),u\left( t \right),{\alpha ^{\varepsilon }}\left( t \right)} \right),{x^{\varepsilon }}\left( 0 \right) = x,t \geqslant 0 $$

(9.1)

where ε > 0 is a small parameter, and α ^ε(t), t ≥, is a Markov chain defined on a probability space (ΩF, P) taking values in

$$M = \left\{ {{s_{{11, \ldots ,}}}{s_{{1{m_{{11}}}, \ldots ,}}}{s_{{l1}}}, \ldots ,{s_{{l{m_{1}}}}}} \right\}$$