A maximum principle approach to optimal control for one-dimensional hyperbolic systems with several state variables. (English) Zbl 1134.49013
The maximum principle developed by J. M. Sloss et al. [Optimal control of structural dynamic systems in one space dimension using a maximum principle, J. Vibr. Control 11, 245–261 (2005)] is used to determine the optimal control functions for a class of one-dimensional distributed parameter structures. The distributed parameter structures are governed by systems of fourth order hyperbolic equations with constant coefficients. A quadratic performance index is formulated as the cost functional of the problem and can be used to represent the energy of the structure and the force spent in the control process. The developed maximum principle establishes a theoretical foundation for the solution of the optimal control problem and relates the optimal control vector to an adjoint variable vector. The method of solution is outlined which involves reducing the original problem to a system of ordinary differential equations. The solution of the general problem is given and a structural control problem is solved to illustrate the solution procedure. The effectiveness of the proposed control solution is shown by comparing the behavior of controlled and uncontrolled systems.
Reviewer: Jagdish Prakash (Mumbai)
MSC:
| 49K15 | Optimality conditions for problems involving ordinary differential equations |
| 74H45 | Vibrations in dynamical problems in solid mechanics |
| 74M05 | Control, switches and devices (“smart materials”) in solid mechanics |
Keywords:
maximum principle for optimal control; Flextural and torsional vibrations; Structural control; System of hyperbolic equationsReferences:
| [1] | Onoda, J.; Haftka, R. T., An approach to structure/control simultaneous optimization for large flexible spacecraft, AIAA J., 25, 1133-1138 (1987) |
| [2] | Grandhi, R. V., Structural and control optimization of space structures, Comput. Struct., 31, 139-150 (1989) |
| [3] | Miller, R. K.; Masri, S. F.; Dehghanyar, T. J.; Caughey, T. K., Active vibration control of large civil structures, ASCE J. Eng. Mech., 114, 1542-1570 (1988) |
| [4] | Yang, J. N.; Soong, T. T., Recent advances in active control of civil engineering structures, Probabilistic Eng. Mech., 3, 179-188 (1988) |
| [5] | Wang, P. K.C., Feedback control of vibrations in an extendible wing, Dynamic Stability of Systems, 3, 109-133 (1988) · Zbl 0662.93056 |
| [6] | Chung, L. L.; Reinhorn, A. M.; Soong, T. T., Experiments on active control of seismic structures, ASCE J. Eng. Mech., 114, 241-256 (1988) |
| [7] | Pan, X.; Hansen, C. H., Effect of end conditions on the active control of beam vibration, J. Sound Vib., 168, 429-448 (1993) · Zbl 0925.73636 |
| [8] | Bruch, J. C.; Sloss, J. M.; Adali, S. A.; Sadek, I. S., Orthotropic plates with shear rotation subject to optimal open-closed loop controls, J. Astronaut. Sci., 38, 105-119 (1990) |
| [9] | Basile, N.; Mininni, M., An extension of the maximum principle for a class of optimal control problems in infinite-dimensional spaces, SIAM J. Control Optim., 28, 1113-1135 (1990) · Zbl 0717.49017 |
| [10] | Fattorini, H. O., Optimal control problems for distributed parameter systems in banach spaces, Appl. Math. Optim., 28, 225-257 (1993) · Zbl 0797.49017 |
| [11] | Bruch, J. C.; Adali, S.; Sloss, J. M.; Sadek, I. S., Maximum principle for the optimal control of a hyperbolic equation in one space dimension, Part 2: application, J. Optim. Theory Appl., 87, 2, 287-300 (1995) · Zbl 0867.49018 |
| [12] | Sloss, J. M.; Sadek, I. S.; Bruch, J. C.; Adali, S., Optimal control of structural dynamic systems in one space dimension using a maximum principle, Journal Vib. Control, 11, 245-261 (2005) · Zbl 1182.74177 |
| [13] | Bruch, J. C.; Sadek, I. S.; Adali, S.; Sloss, J. M., Optimal open-loop/closed-loop control for one-dimensional structures with several state variables, Optimal Control Appl. Meth., 14, 169-179 (1993) · Zbl 0797.49031 |
| [14] | Wang, G.; Chen, S., Maximum principle for optimal control of some parabolic systems with two point boundary conditions, Numer. Funct. Anal. Optim., 20, 1, 163-174 (1999) · Zbl 0924.49017 |
| [15] | Sadek, I. S.; Sloss, J. M.; Adali, S.; Bruch, J. C., Maximum principle for optimal control using spatially distributed pointwise controllers, J. Vib. Control, 4, 4, 445-462 (1998) · Zbl 0949.49502 |
| [16] | Sadek, I. S.; Jamiiru, L.; Al-Mohamad, H. A., Optimal boundary control of one-dimensional multi-span vibrating systems, J. Comput. Appl. Math., 94, 1, 39-54 (1998) · Zbl 0937.74044 |
| [17] | Margaliot, M.; Langholz, G., Hyperbolic optimal control and fuzzy control, IEEE Trans. Systems Man and Cybern. Part A: Systems and Humans, 29, 1, 1-10 (1999) |
| [18] | Kowalewski, A., Optimal control of a distributed hyperbolic system with multiple time-varying lags, Int. J. Control, 71, 3, 419-435 (1998) · Zbl 0945.49018 |
| [20] | Kong, D.-X., Maximum principle in nonlinear hyperbolic systems and its applications, Nonlinear Anal., 32, 7, 871-880 (1998) · Zbl 0962.35031 |
| [21] | Holden, H.; Risebro, N. H.; Tveito, A., Maximum principles for a class of conservation laws, SIAM J. Appl. Math., 55, 3, 651-661 (1995) · Zbl 0843.35056 |
| [22] | Belbas, S. A., Dynamic programming approach to the optimal control of systems governed by Goursat-Darboux equations, Int. J. Control, 51, 6, 1279-1294 (1990) · Zbl 0701.49031 |
| [23] | Boltyansky, V. G.; Poznyak, A. S., Robust maximum principle in minimax control, Int. J. Control, 72, 4, 305-314 (1999) · Zbl 0943.49017 |
| [24] | Magrab, E. B., Vibrations of Elastic Structural Members (1979), Sijthoff & Noordhoff: Sijthoff & Noordhoff Alphen aan den Rijn, The Netherlands · Zbl 0417.73056 |
| [25] | Bellman, R.; Cooke, K. L., Differential-Difference Equations (1963), Academic Press: Academic Press New York · Zbl 0118.08201 |
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