  We theoretically studied the optimal control, frequency lock-in, and phase
lock-in phenomena due to the spatially localized periodic forcing in the flow
past the inclined plate. Although frequency lock-in is evident in many fluid
phenomena, especially fluid-structure interactions, not many researchers have
investigated it using a theoretical approach based on flow details. We obtained
detailed information on the lock-in phenomena to external periodic forcing
using phase reduction theory, a mathematical method for extracting the dynamics
near the limit cycle. Furthermore, the optimal forces applied to the velocity
field were determined under the condition of the minimum forcing energy and
maximum lock-in range. The study of uniform periodic forces applied within
spatially confined regions led us to conclude that the effective lock-in
position, which includes both the upstream and downstream areas of the plate,
depends on the principal frequency of the force. The frequency lock-in range of
these forces was analyzed and compared with theoretical predictions.
