Optimal Control of Elastically Connected Mixed Complex Systems Ismail Kucuk and Ibrahim Sadek Department of Mathematics and Statistics, American University of Sharjah Sharjah, UAE 1 Abstract The problem of damping out the oscillations of an elastically connected rectangular plate-membrane system by point-wise actuators is considered. Mixed-type structures are widely used in many branches of modern civil, mechanical, and aerospace engineering. The basic control problem is to minimize the deflection and the velocity of displacements in a given period of time with the minimum possible expenditure of actuators. A quadratic performance is chosen as the cost functional which comprises the functionals of the defection, velocity and the distributed actuators of the two coupled structure. Necessary conditions of optimality are obtained from a variational approach and formulated in the form of independent integral equations which lead to explicit expressions for the point-wise actuators. The results are applied to a specific problem, and numerical results are presented to demonstrate the effectiveness of the proposed control mechanism. The numerical results are obtained using Maple. 2 Keywords Optimal Control, Plate, Membrane, Integral Equations, Elasticity 3 Introduction Mechanical structures are modeled as one- dimensional simple continuous systems such as a string and beam or two dimensional simple contin- uous systems such as membrane and plate. The simple models are used in complex continuous sys- tems such as an elastically connected double-solid system that consists of two elastics solids bounded continuously through a Winkler elastic layer [2]. The vibration analysis of complex continuous sys- tems is one of the tools used in civil and mechan- ical engineering for theoretical and practical ap- plications of the problems raising in engineering. Complex continuous systems of special types are studied in depth in [2, 5] for transverse vibration theory. Our interest in this study is motivated by the problems considered in [4, 5]. In these papers, the control of vibrations in an elastically connected double-beam system is investigated. An analytical method based on the maximum principle in [6] is applied to the double beam system in [4, 5]. The method consists of the basic control problem of minimizing a weighted sum of objective functional under given constraints, where the maximum prin- ciple is used to solve an initial-terminal-boundary- value problem in a system of partial differential equations [6]. In this paper, we analyze the free trans- verse vibrations of plate-membrane complex two- dimensional continuous system from the optimal control point of view. We suggest a new system where we include actuators as control parameters in order to prevent any unwanted resonance. The optimally controlled system is observed trough the performance index functional that consists of con- trol actuators. Necessary conditions of optimality leads us to coupled non-homogeneous Fredholm in- tegral equations with degenerate kernel. After de- coupling the integral equations, we obtain 4N×4N system of linear equations that results with the optimal actuators since the variational approach is considered. The advantage of using variational approach is that one needs to solve system of lin- ear equations rather than solving a partial differ- ential equation. The proposed approach is illus- trated by a numerical example in which the two discrete control actuators are applied to an elas- tically connected rectangular plate-membrane sys- tem. Numerical results indicate that the dynam- ical response of the mixed system can be reduced substantially in a given period of time. This can be achieved by selecting a suitable value of the mem- brane tension and applying an actuator in the do- main of the plate. 4 Formulation of the Problem Free transverse vibrations of an elastically con- nected rectangular plate-membrane (see Figure 1)