| Pattern-forming systems are used to model many diverse phenomena from biology, chemistry, and physics. These systems of differential equations have the property that as a bifurcation (or control) parameter passes through a critical value, a stable spatially uniform equilibrium state gives way to a stable pattern state, which may have spatial variation, time variation, or both. There is a large body of experimental and mathematical work on pattern-forming systems. However, these ideas have not yet been adequately exploited in engineering, particularly in the control of smart systems; i.e., feedback systems having large numbers of actuators and sensors. With dramatic recent improvements in micro-actuator and micro-sensor technology, there is a need for control schemes better than the conventional approach of reading out all of the sensor information to a computer, performing all the necessary computations in a centralized fashion, and then sending out commands to each individual actuator. Potential applications for large arrays of micro-actuators include adaptive optics (in particular, micromirror arrays), suppressing turbulence and vortices in fluid boundary-layers, micro-positioning small parts, and manipulating small quantities of chemical reactants.; The main theoretical result presented is a Lyapunov functional for the cubic nonlinearity activator-inhibitor model pattern-forming system. Analogous Lyapunov functionals then follow for certain generalizations of the basic cubic nonlinearity model. One such generalization is a complex activator-inhibitor equation which, under suitable hypotheses, models the amplitude and phase evolution in the continuum limit of a network of coupled van der Pol oscillators, coupled to a network of resonant circuits, with an external oscillating input. Potential applications for such coupled van der Pol oscillator networks include quasi-optical power combining and phased-array antennas.; In addition to the Lyapunov functional, a Lyapunov function for the truncated modal dynamics is derived, and the Lyapunov functional is also used to analyze the stability of certain equilibria. Basic existence, uniqueness, regularity, and dissipativity properties of solutions are also verified, engineering realizations of the dynamics are discussed, and finally, some of the potential applications are explored. |
