Simultaneously determining the optimal control and state in an vibrating circular surface system by a discretization method


Department of Mathematics, Shiraz University of Technology, Shiraz, Iran


Presenting a new method for simultaneous determination of the control and trajectory functions in an optimal control problem governed by a vibrating circular surface system, is the main purpose of this paper. First, by identifying the form of the trajectory function, doing discretization and using the properties of measure, the problem is linearized. Then, by doing some approximation steps, the nearly optimal pair of trajectory and control with the optimal value of the objective function are obtained via solving a finite linear programming problem. A numerical example is also presented.


 Bazaraa, M.S., Jarvis, J.J. and Sherali, H.D. (1990). Linearprogramming and network flows, John Wiley and Sons (2Ed). Carlsson, J. (2008). Symplectic Reconstruction of data for heat and wave equations,
Report arxiv: 0809.3621v1. Conway, J. B. (1990). A course in functional analysis, University of Tennessee, Springer. تعیین همزمان کنترل و وضعیت بهینه در سیستم... 41 Fakharzadeh, A. (2003). Finding the optimal domain of a nonlinear wave optimal control system by measure, J. Appl. Math and Computing, 13,183- 194. Fakharzadeh, A. and Rubio, J.E. (2009). Best domain for an elliptic problem in cartesian coordinate by means of shape-measure, AJOP Asian J. of Control, 11, 536-547. Farahi, M.H. (1996). The boundary control of the wave equation. PhD thesis, Dept. of Applied Mathematical Studies, Leeds University. Gerdts, M., Greif, G. and Pesch, H.J. (2006). Numerical optimal control of the wave equation: optimal boundary control of a string to rest in finite time, Proceedings 5th Mathmod Vienna. Goberna, M.A. and Lopez, M.A. (1998). Linear semi-infinite optimization, John Wiley and Sons, Chichester. Mikhailov, V. P. (1978). Partial Differential Equations, Moscow, Mir Pub. Nazemi A.R., Farahi M.H. and Zamirian, M. (2008). Filtration problem in inhomogenous dam by using embedding method, J. of Applied Mathematics and Computting, 28, 313-332. Nowakowski, A. (2008). Sufficient optimality conditions for dirichlet boundary control of wave equation, Proceedings of the 47th IEEE Conference on Decision and Control
Cancun, Mexico. Simmons, G.F. (1972). Differential equations with applications and historical notes, McGraw-Hill Inc. Rubio, J.E. (1986). Control and optimization the linear treatment of nonlinear problems, Manchester University Press, Manchester. Rosenbloom, P.C. (1952). Qudques classes de problems exteremaux, Bulletin de SocieteMathematique de France, 80, 183-216. Rudin, W. (1976). Principles of mathematical analysis, McGraw-Hill, New york. Waziri, V.O .and Reju, S.A. (2006). Control operator for
the twodimensional energized wave equation ,Leonardo Journal of Sciences, 9, 33- 44. Wylie, C.R. and Barret, L.C. (1982). Advanced engineering mathematics (5th Ed), McGraw-Hill. Young, L.C. (1969).
Lectures on the calculus of variations and optimal control theory, Philadelphian: W. B. Saunders Company