Ergodic control in \(L^\infty\)
The authors prove six theorems concerning asymptotic properties, either as \(t\to \infty\) or as \(\delta \to 0_+\) of the value functions \[ u_\delta (t,x):= \inf_{\alpha \in {\mathcal A}} \operatorname {ess}\sup_{s\in [0,t]} \Biggl[\int_0^se^{-\delta r}g(x(r),\alpha (r)) dr + e^{-\delta s}h(x(s),\alpha (s))\Biggr], \;\delta \geq 0, \;t\in (0,\infty], \] where \({\mathcal A}=L^\infty ([0,t];A), \;A\subset R^m\), is the set of all measurable (bounded) controls and \(x(.)\) is the unique solution of the problem \[ x'(t)=f(x(t),\alpha (t)) \text{ a.e. in }(0,\infty), \quad x(0)=x_0\in R^n. \] In the first part of the paper one considers the case with ``no running cost'' (i.e., \(g(x,a)\equiv 0\)) and proves that the functions \(\sup_{\delta >0} u_\delta (\infty ,.)\), \(\sup_{t\geq 0}u_0(t,.)\) are ``extremal'' viscosity semi-solutions of an associated Hamilton-Jacobi equation \(H(x,u,Du)=0\). In the second part one proves a ``relaxation'' result for the function \( \sup_{t\geq 0}u_0(t,.)\) in the presence of the running cost, \(g(.,.)\), and in the last section the authors study the so-called ``ergodic problem''.
scientific article; zbMATH DE number 1523575