One generalization of the dynamic programming problem
Applications of Mathematics, Tome 15 (1970) no. 2, pp. 79-96
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library
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 $\Cal M = M \otimes T^M \otimes \ldots \otimes T^M \otimes \ldots$. We define the transformations $P, N$ and the set $\Cal M^{(x_o)}$ as followe: For each $X=(x_0, y_0, y_1, \ldots)\in \Cal M$ it is: $PX=(y_0(x_0),y_1,\ldots),\ NX=x_0,\ \Cal M^{(x_0)}=\{X; X \in \Cal M, NX= x_0\}$. Suppose $\Psi$ is a continuous function defined on $\Cal M$ and $f(x_0)= max_{x\in \Cal M^{(x_0)}}\ \Psi(X)$. The dynamic programming problem can now be formulated as follows: For all $x\in M$ find the element (or elements) $\bar{X}\in \Cal M^{(x)}$ 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.
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 $\Cal M = M \otimes T^M \otimes \ldots \otimes T^M \otimes \ldots$. We define the transformations $P, N$ and the set $\Cal M^{(x_o)}$ as followe: For each $X=(x_0, y_0, y_1, \ldots)\in \Cal M$ it is: $PX=(y_0(x_0),y_1,\ldots),\ NX=x_0,\ \Cal M^{(x_0)}=\{X; X \in \Cal M, NX= x_0\}$. Suppose $\Psi$ is a continuous function defined on $\Cal M$ and $f(x_0)= max_{x\in \Cal M^{(x_0)}}\ \Psi(X)$. The dynamic programming problem can now be formulated as follows: For all $x\in M$ find the element (or elements) $\bar{X}\in \Cal M^{(x)}$ 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.
@article{10_21136_AM_1970_103272,
author = {Vlach, Milan and Zimmermann, Karel},
title = {One generalization of the dynamic programming problem},
journal = {Applications of Mathematics},
pages = {79--96},
year = {1970},
volume = {15},
number = {2},
doi = {10.21136/AM.1970.103272},
mrnumber = {0270736},
zbl = {0193.19402},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.21136/AM.1970.103272/}
}
TY - JOUR AU - Vlach, Milan AU - Zimmermann, Karel TI - One generalization of the dynamic programming problem JO - Applications of Mathematics PY - 1970 SP - 79 EP - 96 VL - 15 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.21136/AM.1970.103272/ DO - 10.21136/AM.1970.103272 LA - en ID - 10_21136_AM_1970_103272 ER -
Vlach, Milan; Zimmermann, Karel. One generalization of the dynamic programming problem. Applications of Mathematics, Tome 15 (1970) no. 2, pp. 79-96. doi: 10.21136/AM.1970.103272
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