Asian Journal of Control, Vol. 13, No. 4, pp. 480 491, July 2011 Published online 26 March 2010 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.181 TIME OPTIMAL CONTROL OF A DIELECTROPHORETIC SYSTEM Matthew P. Melnyk and Dong Eui Chang ABSTRACT This paper examines a time optimal control problem for a dielec- trophoretic system. The system consists of a neutrally buoyant and neutrally charged particle in a chamber filled with a fluid flowing with a low Reynolds number. At the bottom of this chamber is a series of parallel electrodes with a controlled time-varying voltage. The voltage on the electrodes creates a time- varying nonuniform electric field inducing a dipole moment in the particle. This induced dipole moment interacts with the electric field to generate a force on the particle. There are two state variables x and y , where x is the position of the particle and y is the induced dipole moment in the particle. The system has two parameters  and c which depend on the electric characteristics of the particle and the ambient fluid. The parameter c is always positive by the laws of physics, but  can have either sign. We solve the time optimal control problem for this system when  ≥ 0 and y (0) is arbitrarily prescribed. Key Words: Time optimal control, dielectrophoresis, nanotechnology, bio- technology. I. INTRODUCTION Dielectrophoresis refers to the translational mo- tion of particles induced by polarization effects in non-uniform electric fields [1–3]. The force on the particles by these effects is called a dielectrophoretic force. The dielectrophoretic force has the advantage that it can move even neutrally charged particles, and it has been actively used in biotechnology with the advent of micro/nano-technology [1]. Since the first introduction of dielectrophoretic systems to the control community in [4], there have been slowly increasing interests in control of dielectrophoretic systems [5, 6]. Manuscript received April 27, 2009; revised October 1, 2009; accepted November 19, 2009. M. P. Melynk is with the Department of Economics, Uni- versity of British Columbia, Vancouver, BC, V6T 1Z1, Canada (e-mail: matwithahat@hotmail.com). D. E. Chang (corresponding author) is with the Department of Applied Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada (e-mail: dechang@math.uwaterloo.ca). This work was partially supported by the NSERC. We would like to thank Harsha Simha and Jean L´ evine for their invaluable comments. In this paper, we study the time optimal control problem for the following dielectrophoretic system: ˙ x = yu + u 2 , (1) ˙ y =-cy + u (2) with the state (x , y ) ∈ R 2 , the control u ∈ R and the constraints: x (0) = x 0 = given, y (0) = y 0 = given, (3) x (T ) = x f = given, y (T ) = free, (4) |u |≤ 1 (5) where the constant parameters  and c satisfy  ≥ 0, c>0. (6) This system describes the vertical motion of a neu- trally buoyant and neutrally charged particle in a cham- ber with a series of parallel electrodes at the bottom where the ambient fluid in the chamber flows with a low Reynolds number; see Fig. 1. After a nonlinear coordinate transformation, the variable x measures the 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society