OPTIMAL CONTROL OF DISTRIBUTED PARAMETER SYSTEMS USING MULTILEVEL TECHNIQUES.
CALIFORNIA UNIV LOS ANGELES DEPT OF ENGINEERING
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The approximate solution of optimal control problems involving partial differential equations is studied by discretizing the space domain and considering the resultant set of ordinary differential equations. This approach is certainly not new. However, earlier efforts along this path have been hampered by the difficulties involved in solving the optimal control problem for the very large sets of interacting ordinary differential equations which arise as the discretization interval is decreased. This dissertation suggests the use of multilevel control techniques to overcome this computational difficulty. The major conclusion stemming from this research is that the multilevel approach does appear feasible in solving the optimal control problem for certain classes of distributed parameter systems namely, linear and nonlinear parabolic or elliptic equations. The convergence properties of the second-level controller are of paramount importance in accomplishing this task. Of the three types of second-level controllers discussed here feasible, nonfeasible, and Gauss-Seidel, the only one considered suitable in this application is the Gauss-Seidel controller. It is extremely simple and was found to have good convergence properties for this type of problem. In particular, the systems of semidiscrete equations may become very large and the number of subsystems likewise. By its very nature, the performance of the Gauss-Seidel controller does not seem to be degraded by increasing the number of subsystems as long as the discretization interval is not too small. The reason for the good convergence properties observed for this controller is largely the tridiagonal form of the A sub j matrix.
- Operations Research