This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

Control

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Control Theory studies dynamical systems with input designed to influence their behavior to achieve a desired goal (a specific state, stability, etc.). Its applications are abundant and include control of a missile, spacecraft, aircraft, chemical reaction, electric or computer network, drug intake, protein folding, and computer graphics.

The objective is to develop methods for computing topological characteristics of a system that indicate that it possesses certain properties. In particular these properties include the presence of equilibria and controllability. Other characteristics will describe the solvability and complexity of algorithms for the motion planning, feedback, and the inverse kinematic problems. Currently, the main tool is geometry while topology has seen virtually no use. The essence of the topological approach is in simultaneously capturing the behavior of the system and as well as that of any perturbed system. This is necessary as a way of dealing with uncertainty emerging from errors in computations, measurements, or the environment. This goal is achieved by allowing the geometric objects and transformations that describe the system to be continuously deformed. The result is that the output has built-in robustness.

Normally robustness is treated in terms of perturbations that are assumed "small". However unless actual estimates are available, uncertainty remains an issue as we don't know how "small" are the perturbations of the real system. Therefore in order to take into account the "worst possible scenario" we consider arbitrarily large, but still continuous, perturbations of the model. Thus the use of topology provides answers with a new, for control theory, degree of robustness. The significance of this lies in a better ability to design systems capable of working in highly uncertain environments. The examples of such environments are any applications of robots outside the factory especially the battlefield.