Dynamic mode decomposition of controlled H- and K-type transitions

In many situations, owing to the complexity of flow, access to detailed representation of the flow field is impossible. Therefore, the capability of representing the essential physics through fewer degrees of freedom (model reduction) is of great value. For example, control systems rely on models through differential equations. The use of partial differential equations governing the flow is intractable for this purpose. Therefore, one can reduce the governing equations by extracting modes that contribute most to the overall dynamics. In this approach, a high-dimensional flow field is typically projected onto a set of spatial modes. As an example, if U(x, y, z, t) is a flow variable, where x, y, and z are the spatial coordinates and t is time, the flow-field can be represented by