Introduction:
The optimization method based on dynamic programming is a powerful technique used to solve control problems. It involves viewing the control problem as a multistage decision problem, where the control input is considered as a time sequence of decisions. Additionally, a sampled data system gives rise to a sequence of transformations of the original state vector. All of these aspects are taken into account in the optimization method based on dynamic programming.

Control Problem as a Multistage Decision Problem:
In dynamic programming, the control problem is viewed as a multistage decision problem. This means that the control input is not determined in a single step but rather over multiple stages or time periods. Each stage involves making a decision based on the current state and the desired objective. The decisions made at each stage affect the future states and the overall performance. By considering the problem in this way, dynamic programming allows for the optimization of the control input over a sequence of decisions.

Control Input as a Time Sequence of Decisions:
Dynamic programming treats the control input as a time sequence of decisions. The decisions made at each time period are based on the current state and the desired objective. The control input is not determined all at once but rather evolves over time. By considering the control input as a time sequence of decisions, dynamic programming allows for the optimization of the control input over the entire time horizon.

Sampled Data System and Transformations:
A sampled data system refers to a system where the continuous-time signals are converted into discrete-time signals through sampling. In the context of dynamic programming, a sampled data system gives rise to a sequence of transformations of the original state vector. These transformations occur at each time period and are based on the current state and the control input. The objective of dynamic programming is to find the optimal sequence of transformations that maximize or minimize a certain performance criterion.

Conclusion:
The optimization method based on dynamic programming takes into account various aspects of control problems. It views the control problem as a multistage decision problem, considers the control input as a time sequence of decisions, and recognizes that a sampled data system gives rise to a sequence of transformations of the original state vector. By considering these aspects, dynamic programming provides a powerful method for optimizing control systems.