We present a computational method for the optimal control of linear distributed systems. Its derivation is based on the functional calculus of self-adjoint operators, and on the Dunford-Schwartz representation formula. It has been devised to be implementable on very fine grained computing processors with semi-decentralized coordination. Finally, it is illustrated by an example related to vibration stabilization of a micro-cantilever array.

Abstract. In this paper we deal with the derivation of state-realizations of linear operators that are solutions to certain operator linear differential equations in one-dimensional bounded domains. We develop two approaches in the framework of diffusive representations: one with complex diffusive symbols; the other with real diffusive symbols. Then, we illustrate the theories and develop numerical methods for a Lyapunov equation arising from optimal control theory of the heat equation. A practical purpose of this approach is real-time computation on a semi-decentralized architecture with low granularity.

We apply the method of semi-decentralized approximation, introduced in Lenczner and Yakoubi [2009] and Yakoubi [2010], to the linear quadratic regulation of a one-dimensional array of cantilevers with regularly spaced actuators and sensors. It is based on two mathematical concepts, namely on functions of operators, and on the Cauchy integral formula. We evaluate its performances and the errors of approximation. We also propose its implementation in terms of an analog processor, namely a periodic network of resistors. The presented application is based on a twoscale model representing an array of cantilevers. We shortly explain its genesis before to state it in details, and to show validation results.

International audienceIn Yakoubi [2010] and Lenczner et al. [2010] we developed a theoretical framework of diffusive realization for state-realizations of some linear operators. Those are solutions to certain operator linear differential equations inone-dimensional bounded domains. We also illustrated the theory and developed a numerical method for a Lyapunov equation arising from optimal control theory of the heat equation. However, the principles of our numerical methods were only sketched, and now we provide more details. Then, we do not only provide validation results of the method, but we also report our experience in its implementation on a Field Programmable Gate Arrays (FPGA), for the purpose of promoting embedded real-time computation