# One generalization of the dynamic programming problem

Aplikace matematiky (1970)

• Volume: 15, Issue: 2, page 79-96
• ISSN: 0862-7940

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## Abstract

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The main purpose of this article is to provide an exact theory of the dynamic programming on a sufficiently general basis. Let $M$ be a compact topological Hausdorff’s space, let ${\stackrel{˜}{T}}^{M}$ be the set of all continuous transformations of the space $M$ into itself. Suppose such a topology is introduced on ${\stackrel{˜}{T}}^{M}$ that ${\stackrel{˜}{T}}^{M}$ is Haousdorff’s space and that the transformation $\phi \left(x,y\right)=y\left(x\right)$ of the product $M\otimes {\stackrel{˜}{T}}^{M}$ into $M$ is continuous with respect to Tichonoff’s topology on $M\otimes {\stackrel{˜}{T}}^{M}$. Suppose ${\stackrel{˜}{T}}^{M}$ is a compact subspace of ${\stackrel{˜}{T}}^{M}$ and $ℳ=M\otimes {T}^{M}\otimes ...\otimes {T}^{M}\otimes ...$. We define the transformations $P,N$ and the set ${ℳ}^{\left({x}_{o}\right)}$ as followe: For each $X=\left({x}_{0},{y}_{0},{y}_{1},...\right)\in ℳ$ it is: $PX=\left({y}_{0}\left({x}_{0}\right),{y}_{1},...\right),\phantom{\rule{4pt}{0ex}}NX={x}_{0},\phantom{\rule{4pt}{0ex}}{ℳ}^{\left({x}_{0}\right)}=\left\{X;X\in ℳ,NX={x}_{0}\right\}$. Suppose $\Psi$ is a continuous function defined on $ℳ$ and $f\left({x}_{0}\right)=ma{x}_{x\in {ℳ}^{\left({x}_{0}\right)}}\phantom{\rule{4pt}{0ex}}\Psi \left(X\right)$. The dynamic programming problem can now be formulated as follows: For all $x\in M$ find the element (or elements) $\overline{X}\in {ℳ}^{\left(x\right)}$ for which $\Psi \left(\overline{X}\right)=f\left(x\right)$. Existence and uniqueness of the solution of this problem is proved and the method of succesive approximations is used to solve it in case when $\Psi \left(X\right)-\Psi \left(PX\right)=\Theta \left({x}_{0},{y}_{0}\right)$. Further the case $\Psi \left(X\right)={\sum }_{i=0}^{\infty }{\Theta }_{i}\left({x}_{i},{y}_{i}\right)$ is considered and one minor example is solved.

## How to cite

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Vlach, Milan, and Zimmermann, Karel. "One generalization of the dynamic programming problem." Aplikace matematiky 15.2 (1970): 79-96. <http://eudml.org/doc/14634>.

@article{Vlach1970,
abstract = {The main purpose of this article is to provide an exact theory of the dynamic programming on a sufficiently general basis. Let $M$ be a compact topological Hausdorff’s space, let $\tilde\{T\}^M$ be the set of all continuous transformations of the space $M$ into itself. Suppose such a topology is introduced on $\tilde\{T\}^M$ that $\tilde\{T\}^M$ is Haousdorff’s space and that the transformation $\phi (x,y)=y(x)$ of the product $M \otimes \tilde\{T\}^M$ into $M$ is continuous with respect to Tichonoff’s topology on $M \otimes \tilde\{T\}^M$. Suppose $\tilde\{T\}^M$ is a compact subspace of $\tilde\{T\}^M$ and $\mathcal \{M\} = M \otimes T^M \otimes \ldots \otimes T^M \otimes \ldots$. We define the transformations $P, N$ and the set $\mathcal \{M\}^\{(x_o)\}$ as followe: For each $X=(x_0, y_0, y_1, \ldots )\in \mathcal \{M\}$ it is: $PX=(y_0(x_0),y_1,\ldots ),\ NX=x_0,\ \mathcal \{M\}^\{(x_0)\}=\lbrace X; X \in \mathcal \{M\}, NX= x_0\rbrace$. Suppose $\Psi$ is a continuous function defined on $\mathcal \{M\}$ and $f(x_0)= max_\{x\in \mathcal \{M\}^\{(x_0)\}\}\ \Psi (X)$. The dynamic programming problem can now be formulated as follows: For all $x\in M$ find the element (or elements) $\bar\{X\}\in \mathcal \{M\}^\{(x)\}$ for which $\Psi (\bar\{X\})=f(x)$. Existence and uniqueness of the solution of this problem is proved and the method of succesive approximations is used to solve it in case when $\Psi (X)-\Psi (PX)=\Theta (x_0, y_0)$. Further the case $\Psi (X)=\sum ^\infty _\{i=0\}\Theta _i(x_i, y_i)$ is considered and one minor example is solved.},
author = {Vlach, Milan, Zimmermann, Karel},
journal = {Aplikace matematiky},
keywords = {operations research},
language = {eng},
number = {2},
pages = {79-96},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {One generalization of the dynamic programming problem},
url = {http://eudml.org/doc/14634},
volume = {15},
year = {1970},
}

TY - JOUR
AU - Vlach, Milan
AU - Zimmermann, Karel
TI - One generalization of the dynamic programming problem
JO - Aplikace matematiky
PY - 1970
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 15
IS - 2
SP - 79
EP - 96
AB - The main purpose of this article is to provide an exact theory of the dynamic programming on a sufficiently general basis. Let $M$ be a compact topological Hausdorff’s space, let $\tilde{T}^M$ be the set of all continuous transformations of the space $M$ into itself. Suppose such a topology is introduced on $\tilde{T}^M$ that $\tilde{T}^M$ is Haousdorff’s space and that the transformation $\phi (x,y)=y(x)$ of the product $M \otimes \tilde{T}^M$ into $M$ is continuous with respect to Tichonoff’s topology on $M \otimes \tilde{T}^M$. Suppose $\tilde{T}^M$ is a compact subspace of $\tilde{T}^M$ and $\mathcal {M} = M \otimes T^M \otimes \ldots \otimes T^M \otimes \ldots$. We define the transformations $P, N$ and the set $\mathcal {M}^{(x_o)}$ as followe: For each $X=(x_0, y_0, y_1, \ldots )\in \mathcal {M}$ it is: $PX=(y_0(x_0),y_1,\ldots ),\ NX=x_0,\ \mathcal {M}^{(x_0)}=\lbrace X; X \in \mathcal {M}, NX= x_0\rbrace$. Suppose $\Psi$ is a continuous function defined on $\mathcal {M}$ and $f(x_0)= max_{x\in \mathcal {M}^{(x_0)}}\ \Psi (X)$. The dynamic programming problem can now be formulated as follows: For all $x\in M$ find the element (or elements) $\bar{X}\in \mathcal {M}^{(x)}$ for which $\Psi (\bar{X})=f(x)$. Existence and uniqueness of the solution of this problem is proved and the method of succesive approximations is used to solve it in case when $\Psi (X)-\Psi (PX)=\Theta (x_0, y_0)$. Further the case $\Psi (X)=\sum ^\infty _{i=0}\Theta _i(x_i, y_i)$ is considered and one minor example is solved.
LA - eng
KW - operations research
UR - http://eudml.org/doc/14634
ER -

## References

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1. Шрейбep H. А., Задача динамического планирования и автоматы, Проблемы кибернетики V, Москва, 1961. (1961)
2. Alexandrov P. S., Úvod do obecné teorie množin a funkcí, Praha 1956. (1956)
3. Бурбаки, Общая топология, Москва, 1961. (1961) Zbl1160.68305
4. Bellmann R., Dynamic Programming, Princeton University Press, Princeton, New Jersey, 1957. (1957) Zbl0081.36902MR0090477
5. Zimmermann K., O jednom zobecnění úlohy dynamického programování, kandidátská disertační práce, Praha MFF UK, 1968. (1968)

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