Limiting discounted-cost control of partially observable stochastic systems (Q2753209)

This paper presents two main results on partially observable stochastic systems. The first one describes the partially observable stochastic system NEWLINE\[NEWLINEx_{t+1}= F(x_t,a_t, \xi_t);\;y_t=G(x_t, \eta_t),NEWLINE\]NEWLINE in which the state space \(X\) and the observation set \(Y\) are Borel spaces. The state and observation disturbances \(\xi_t\) and \(\eta_t\) take values in Borel spaces \(S\) and \(S'\). Control actions \(a_t\) are taken from a compact metric space \(A\). Conditions for the existence of \(\alpha\)-discounted optimal policies allowing the cost-per-stage being unbounded are given.NEWLINENEWLINENEWLINEThe second main result regards the additive-noise case NEWLINE\[NEWLINEx_{t+1}= F_t(x_t,a_t)+ \xi_t,\;t\in \mathbb{N};\;\mathbb{N}= \{0,1,2, \dots\}NEWLINE\]NEWLINE and observations \(\{y_t\}\) with NEWLINE\[NEWLINEy_t=G_t(x_t) +\eta_t,\;t \in \mathbb{N}.NEWLINE\]NEWLINE Assuming \(F_t\to F_\infty\) and \(G_t\to G_\infty\) are converging pointwise for \(t\to\infty\) for all \((x,a)\), and \(x\), respectively, conditions ensuring the existence of an optimal control policy for NEWLINE\[NEWLINEx_{t+1}= F_\infty (x_t,a_t)+ \xi_t,y_t= G_\infty(x_t)+ \eta_t,NEWLINE\]NEWLINE when the optimality criterion is the \(\alpha\)-discounted cost \((0<\alpha<1)\), are given.