Notes on Dynamic Systems and the Value Function

 

Dynamic systems can be described by differential and difference equations. Without a loss of generality, consider a dynamic system represented by a difference equation: $x_{k+1}=f\left(x_{k}\right)$. The state of the system is represented by $x$ and the function $f$ is the mapping from one state to the next. One way to characterize a dynamic system is with an additive cost: $J$. An additive cost summarizes the cost of operation for the system from some initial state, $x_{0}$. To ensure that the sums are finite an exponential weighting factor, $alpha$, is introduced. This factor has a value between between 0 and 1.  Under some circumstances, $alpha$ can be equal to one. One special case is where the system is guaranteed to eventually enter a zero cost state. However, in general, it will need to be less than one. This additive cost is a function of each initial condition:

$J\left(x_{0}\right)=\sum_{k=0}^{\infty}\left(\alpha^{k}\cdot c\left(x_{k}\right)\right)$.

(1)

The value of the additive cost can be solved using the dynamic programming equations:

$V\left(x\right)=c\left(x\right)+\alpha^{k}\cdot V\left(f\left(x\right)\right)$.

(2)

The function $V$ is referred to as the value function.

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