Controllability properties for the one-dimensional Heat equation under multiplicative or nonnegative additive controls with local mobile support

We discuss several new results on nonnegative approximate controllability for the one-dimensional Heat equation governed by either multiplicative or nonnegative additive control, acting within a proper subset of the space domain at every moment of time. Our methods allow us to link these two types of controls to some extend. The main results include approximate controllability properties both for the static and mobile control supports

A Coin Vibrational Motor Swimming at Low Reynolds Number

Low-cost coin vibrational motors, used in haptic feedback, exhibit rotational internal motion inside a rigid case. Because the motor case motion exhibits rotational symmetry, when placed into a fluid such as glycerin, the motor does not swim even though its oscillatory motions induce steady streaming in the fluid. However, a piece of rubber foam stuck to the curved case and giving the motor neutral buoyancy also breaks the rotational symmetry allowing it to swim. We measured a 1 cm diameter coin vibrational motor swimming in glycerin at a speed of a body length in 3 seconds or at 3 mm/s. The swim speed puts the vibrational motor in a low Reynolds number regime similar to bacterial motility, but because of the oscillations of the motor it is not analogous to biological organisms. Rather the swimming vibrational motor may inspire small inexpensive robotic swimmers that are robust as they contain no external moving parts. A time dependent Stokes equation planar sheet model suggests that the swim speed depends on a steady streaming velocity V stream ~ Re 1/2s U 0 where U 0 is the velocity of surface oscillations, and streaming Reynolds number Re s = U 20/(ων) for motor angular frequency ω and fluid kinematic viscosity ν

Addendum to “The well-posedness of a swimming model in the 3-D incompressible fluid governed by the nonstationary Stokes equation”

In this addendum we address some unintentional omission in the description of the swimming model in our recent paper (Khapalov, 2013

Source localization and sensor placement in environmental monitoring

Source localization and sensor placement in environmental monitoringIn this paper we discuss two closely related problems arising in environmental monitoring. The first is the source localization problem linked to the questionHow can one find an unknown "contamination source"?The second is an associated sensor placement problem:Where should we place sensors that are capable of providing the necessary "adequate data" for that?Our approach is based on some concepts and ideas developed in mathematical control theory of partial differential equations