Three algorithms for efficient solution of optimal control problems for high-dimensional systems are presented. Each bypasses the intermediate (and, unnecessary) step of open-loop model reduction. Each also bypasses the solution of the full Riccati equation corresponding to the LQR problem, which is numerically intractable for large n. Motivation for this effort comes from the field of model-based flow control, where open-loop model reduction often fails to capture the dynamics of interest (governed by the Navier–Stokes equation). Our minimum control energy method is a simplified expression for the well-known minimum-energy stabilizing control feedback that depends only on the left eigenvectors corresponding to the unstable eigenvalues of the system matrix A. Our Adjoint of the Direct-Adjoint method is based on the repeated iterative computation of the adjoint of a forward problem, itself defined to be the direct-adjoint vector pair associated with the LQR problem. Our oppositely-shifted subspace iteration (OSSI, the main new result of the present paper) method is based on our new subspace iteration method for computing the Schur vectors corresponding, notably, to the m≪n\documentclass[12pt]{minimal}
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\begin{document}$$m\ll n$$\end{document} central eigenvalues (near the imaginary axis) of the Hamiltonian matrix related to the Riccati equation of interest. Prototype OSSI implementations are tested on a low-order control problem to illustrate its behavior.
