Discrete Structures Hidden in Nonlinear Partial Differential Equations: For Control Methods that Have High Compatibility with Computers

Abstract

It is well known that “nonlinear partial differential equations,” which are typified by mathematical models for transforming objects such as strings, beams, and films, Burgers equations in fluid mechanics, and Schrödinger equations in quantum mechanics, can represent a lot of phenomena in the world of nature. However, it is quite difficult to control such systems because of nonlinearity and infinite dimensionality; hence, they are actively studied by various approaches. There are some approaches, called “discrete mechanics” and “ultradiscretization,” based on discrete structures of nonlinear partial differential equations, and it corresponds to the extraction of the “discrete bony framework” from the systems. Moreover, we can expect control synthesis methods that are highly compatible with computers treating only digital data. This paper shall provide some fundamental and applied research studies on this topic with our group's results.