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Some existence theorems for functional equations arising in dynamic programming. (English) Zbl 0533.90091
The authors prove some existence theorems for functional equations which arose in a certain type of continuous multistage decision processes which are described below. Let $S\subset X$ be the state space and $D\subset Y$ be the decision space. The state vectors are denoted by x, the decision vectors by y. Let T:$S\times D\to S$, g:$S\times D\to R$ and G:$S\times D\times R\to R$, R being the field of real numbers. The return function f:$S\to R$ of the continuous decision process is defined by the functional equation $$ (*)\quad f(x)=\sup\sb{y\in D}[g(x,y)+G(x,y,f(T(x,y)))]\quad(x\in S). $$ The authors prove the existence of a solution to the functional equation (*) under various conditions; in some cases they also prove uniqueness. Among the results are the following two typical ones: Theorem A. (i) Let g and G be bounded. (ii) Let $\vert G(x,y,z\sb 1)- G(x,y,z\sb 2)\vert \le \phi(\vert z\sb 1-z\sb 2\vert)$ for all $(x,y,z\sb 1)$, $(x,y,z\sb 2)$ in $S\times D\times R$, where $\phi$ :[0,$\infty)\to [0,\infty)$ is nondecreasing, continuous and $\phi(r)<r$ for every $r>0$. Then the functional equation (*) possesses a unique bounded solution on S. Theorem B. (i) Let $\vert g(x,y)\vert \le M\Vert x\Vert$ for (x,y)$\in S\times D$, where M is a positive constant. (ii) Let $\Vert T(x,y)\Vert \le \phi(\Vert x\Vert)$ for (x,y)$\in S\times D$, where $\phi$ :[0,$\infty)\to [0,\infty)$ is nondecreasing and $\sum \phi\sp n(r)$ converges for every $r>0$. Then the functional equation $$ f(x)=\sup\sb{y\in D}[g(x,y)+f(T(x,y))]\quad(x\in S) $$ possesses a unique solution $f\sb 0$ satisfying the condition (iii): (iii) If $x\in S$, $\{Y\sb n\}\sp{\infty}\!\sb{n=1}\subset D$ and $x\sb n=T(x\sb{n-1},Y\sb n)$, $(x\sb 0=x$, $n=1,2,3,...)$, then $f\sb 0(x\sb n)\to 0$ as $n\to \infty$.

MSC:
90C39Dynamic programming
49L20Dynamic programming method (infinite-dimensional problems)
90C40Markov and semi-Markov decision processes
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Full Text: DOI
References:
[1] Bellman, R.: Methods of non-linear analysis. (1973) · Zbl 0265.34002
[2] Bellman, R.; Kalaba, R.: Dynamic programming and modern control theory. (1965) · Zbl 0156.16902
[3] Browder, F. E.: Indag. math. MR 37. 30, No. No. 5743, 27-35 (1968)