Optimal control of time evolution systems: Controllability investigations and numerical algorithms
Sorensen, Danny C.
Doctor of Philosophy
Optimal control problems for partial differential equations of evolution, mostly of parabolic type, are considered. The means of control are a nonhomogeneous boundary condition or forcing term. A hierarchical control problem is shown to be equivalent to two coupled linear least-squares problems and is solved using numerical methods involving the Singular Value Decomposition. This is an effective computational approach for linear, time-invariant equations if the dimension of the discrete problem is not too large. Sufficient conditions for convergence of the solution of the discrete control problem to the solution of the original problem are given in the case of a linear state equation and a quadratic objective function. The essential condition is a convergence condition on the derived, discrete adjoint equation. A spectral preconditioner is introduced, applicable to iterative solutions of control problems for a parabolic evolution equation involving a diagonalizable operator. Condition number estimates for the preconditioned problem are proven, and numerical experiments verify the effectiveness of the preconditioner. To overcome controllability difficulties for systems acted upon by pointwise sources, the strategy of allowing the location of the controller to move with time is investigated. In particular, the case when the controller is given a sinusoidally-shaped path is studied, analytically and numerically. The feasibility of the optimal-control approach for flow-control problems is demonstrated by numerical experiments for an unsteady two-dimensional channel flow modeled by the Navier-Stokes equations for a viscous, incompressible fluid. The Reynolds number is high enough for nonlinear effects to be important. The vorticity level in the domain is successfully damped by suction and blowing at a part of the boundary. A quasi-Newton algorithm solves the associated minimization problem. To obtain an accurate expression for the gradient of the objective function, the adjoint equation is derived in the fully discrete case. A memory-saving device allows memory requirements to be traded against some extra computations. This avoids the need of simultaneous storage of the solution at all time steps.
Mathematics; Computer science; Mechanical engineering