Many flexible structures consist of a large number of components coupled end to end in the form of a chain. In this study, the authors consider a type of such structures formed by N one-dimensional coupled structures, with (N — 1) controllers at the coupled points. A method of solution for such a special class of optimal control problems is suggested by using an eigenfunction expansion and the maximum principle. This solution involves reducing the original problem to a system of ordinary differential equations. For comparative studies, the special problem is also solved by the variational approach, which leads to a system of integral equations as a necessary and sufficient condition of optimality The effectiveness of these approaches is demonstrated by means of a numerical solution for controlling the vibrations of two strings that are coupled at the connecting point.
Balas, M.J., 1978, "Active control of flexible systems," Journal of Optimization Theory and Applications25, 415-436.
2.
Balas, M.J., 1982, "Trends in large space structure control: Fondest hopes, wildest dreams," IEEE Transactions on Automatic Control27, 522-535.
3.
Bruch, J.C., Jr., Sadek, I.S., Adali, S., and Sloss, J.M., 1993, "Optimal open-loop/closed-loop control for one-dimensional structures with several state variables," Optimal Control Applications and Methods14, 169-179.
4.
Bruch, J.C., Jr., Sloss, J.M., Adali, S., and Sadek, I.S., 1990, "Orthotropic plates with shear rotation subject to optimal open-closed loop controls," Journal of Astronomical Sciences38(1), 105-119.
5.
Chan, W.L. and Zhu, G.B., 1991, "Pointwise stabilization for a chain of coupled vibrating strings," IMA Journal of Mathematical Control & Information7, 307-315.
6.
Chang, YE and Lee, T.T., 1986, "Applications of general orthogonal polynomials to the optimal control of time varying linear systems," International Journal of Control43(4), 1283-1304.
7.
Chen, G., 1979, "Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain," J. Math. Pures. Appl.58, 249-273.
8.
Chen, G., Coleman, M., and West, H.H., 1987, "Pointwise stabilization in the middle of the span for second order systems, nonuniform and uniform exponential decay of solutions ," SIAM Journal of Applied Mathematics47(4), 751-780.
9.
Chen, M.C., Delfour, M.C., Krall, A.M., and Payre G., 1987, "Modelling, stabilization and control of serially connected beams," SIAM Journal of Control Optimization25(3), 526-546.
10.
Conrad, E. and Pierre, M., 1990, "Stabilization of Euler-Bernoulli beams by nonlinear boundary feedback," Repports de Recherche, No. 1235, Institutut National de Recherche en Informatique et en Automatique, France.
11.
Ho, L.F., 1993, "Controllability and stability of coupled strings with control apphed at the coupled points," SIAM Journal of Control and Optimization31, 1416-1437.
12.
Krabs, W. and Leugering, G., 1994, "On boundary controllability of one-dimensional vibrating systems by W01,p -controls for p ∈ [2,∞] ," Mathematical Methods in the Applied Sciences17, 77-93.
13.
Lasiecka, 1989"Stabilization of wave and plate like equations with nonlinear dissipation on the boundary " Journal of Differential Equations79, 340-381.
14.
Liu, K.S., 1988, "Energy decay problems in the design of a point stabilizer for coupled string vibrating systems," SIAM Journal of Control and Optimization26(6), 1348-1356.
15.
Liu, K.S., Huang, F.L., and Chen, G., 1989, "Exponential stability analysis of a long chain of coupled vibrating strings with dissipative linkage," SIAM Journal of Applied Mathematics49, 1694-1707.
16.
Meirovitch, L. and Hagedorn, P., 1994, "A new approach to the modelling of distributed non-self-adjoint systems," Journal of Sound and Vibration178, 227-241.
17.
Meirovitch, L. and Silverberg, L.M. , 1983, "Globally optimal control of self-adjoint distributed systems," Optimal Control Applications and Methods4, 365-386.
18.
Sadek, I.S., 1988, "Optimal distributed control of continuous beam with damping ," Optimal Control Applications and Methods9, 79-85.
19.
Sadek, I.S., Adali, S., Sloss, J.M., and Bruch, J.C., Jr., 1987, "Optimal distributed control of continuous beam with damping," Journal of Sound and Vibration117(2), 207-218.
20.
Sadek, I.S., Jamiiru, L., and Al-Mohamad, H.A. , 1997, "Optimal boundary control of one-dimensional multi-span vibrating systems," Journal of Computational and Applied Mathematics94, 39-54.
21.
Sadek, I.S. and Melvin, V.H., 1996, "Oprimal control of serially connected structures using spatially distributed pomtwise controllers," IMA Journal of Mathematical Control & Information13, 335-358.
22.
Shih, D.H. and Wang, L.F., 1986, "Optimal control of deterministic systems described by intergrodifferential equations," International Journal of Control44(6), 1737-1745.
23.
Slemrod, M., 1976, "Stabilization and boundary control systems," Journal of Differential Equations22, 402-415.
24.
Sloss, J.M., Sadek, I.S., Bruch, J.C., Jr., and Adali, S., 1989, "Boundary feedback control of a vibrating beam subject to a displacement constraint," Optimal Control Applications and Methods, Short Communications10, 81-87.
25.
Sloss, J.M., Sadek, I.S., Bruch, J.C., Jr., and Adali, S., 1991, "Boundary feedback control of an electromechanical transducer ," European Journal of Mechanis A/Solids10(3), 295-307.
Sun, S.H., 1983, "Quasi-dynamical systems and the stabilization of elastic vibrations with controls whose range is restricted to a finite number of values ," Journal of Optimization Theory and Applications40, 237-253.
