Incomplete sensitivities in design and control of fluidic channels

Control of distributed systems has various industrial applications, as it is often desired to keep complex multi-disciplinary systems in some given state. Definition or parameterization of control space is the first main issue we face when formulating a control problem. Usually, one wishes to keep the parameterization space dimension as small as possible to limit the complexity of the problem. In addition, for any control approach to be effective, it should be realizable during the time the system is still controllable. Computational cost is therefore another critical issue. Our aim in this paper is to discuss alternative remedies for these two problems. We discuss the behavior of an electrokinetic microchannel system where the control variables include both the geometry of the microchannels and the temporal control of potentials. In a real system, the geometric control is achieved by the realization of etched microchannel structures using microlithography techniques. Flow control is accomplished by applying electric potentials along microchannels. We discuss the behavior of our design and control platform for two complementary classes of problems: the situation where the number of controls is small (a potential field) and where the number of controls is large (the geometry of a microchannel turn). We use our sub-optimal control technique, using accurate gradient evaluation, for the first class of problems. For second class, we show that the sub-optimal control is also efficient using incomplete evaluation of the gradient, but only for a limited class of cost functions. Our motivation here comes from the fact that, for a control algorithm based on gradient methods to be efficient, the design should have the same complexity as the direct problem. We therefore need a cheap and easy gradient evaluation somehow avoiding the adjoint equation solution. Since the problem involves electrostactics, electromigration, and fluid motion, we couple several differential state equations in the simulation. In that context, the gradientbased minimization algorithm is reformulated as a dynamic system, which is considered as an extra state equation for the parameterization. This formulation makes it easier to understand the coupling between different components of the simulation. We look for the solutions to our optimization problem as stationary solutions of a second order dynamic system. In addition, for the system to have global search features, we use the natural instability of second order hyperbolic systems (Attouch & Cominetti 1996).