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Any single permanent or electro magnet will always attract a magnetic fluid. For this reason it is difficult to precisely position and manipulate ferrofluid at a distance from magnets. We develop and experimentally demonstrate optimal (minimum electrical power) 2-dimensional manipulation of a single droplet of ferrofluid by feedback control of 4 external electromagnets. The control algorithm we have developed takes into account, and is explicitly designed for, the nonlinear (fast decay in space, quadratic in magnet strength) nature of how the magnets actuate the ferrofluid, and it also corrects for electro-magnet charging time delays. With this control, we show that dynamic actuation of electro-magnets held outside a domain can be used to position a droplet of ferrofluid to any desired location and steer it along any desired path within that domain -an example of precision control of a ferrofluid by magnets acting at a distance. KeywordsMagnetic particles; magnetic carriers; nano-particles; ferrofluid; magnetic drug delivery; manipulation at a distance; control; feedback; electromagnets; dielectrophoresis IntroductionWe consider an initial ferrofluid control problem: the precise manipulation of a single drop of ferrofluid by four external electromagnets. Precision control is achieved by feedback: we sense the location of the droplet by a camera and imaging software and then correctly actuate the electromagnets at each time to move it from where it is to closer to where it should be (Figure 1). Repeating this magnetic correction at each time quickly forces the droplet to the desired stationary or moving target and allows us to precisely control its position over time.Control design, the mathematical development of the algorithm that determines how to turn on the magnets to create the needed position correction at each time, is challenging. It is recognized that each magnet can only pull the fluid towards it; any single magnet cannot

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