Dynamical low-rank approximations for the optimal control of matrix ODEs

Abstract

We consider the open-loop optimal control of a dynamical system whose state is described by a time-dependent matrix $X(t)\in\mathbb{R}^{m\times n}$. Gradient-based optimization methods may incur substantial computational and memory costs, particularly for large-scale systems and fine temporal discretizations. To alleviate this computational burden, we develop a mathematical framework to use Dynamical Low Rank Approximations (DLRA) in optimal control problems. We study two distinct paradigms: the first one, called Optimize-then-Project (OtP), first derives the optimality system and then approximates both the state and adjoint equations using DLRA. The second one, called Project-then-Optimize (PtO), first replaces the state equation with its DLR approximation, and then derives the optimality conditions for the resulting optimization problem. The two approaches differ fundamentally concerning the role of the adjoint variable. In OtP, the adjoint variable evolves on the manifold of fixed rank matrices, whereas in PtO, the adjoint variable belongs to the tangent space to the fixed-rank manifold at the evolving state variable. This latter formulation leads to a covariant ODE describing the dynamics of the adjoint variable as the underlying tangent space evolves in time. For both approaches we establish rigorous error estimates for the approximated gradients, identifying how their accuracy depends on the low-rank approximations of initial and terminal conditions, the components of the state and adjoint vector fields normal to the relevant tangent spaces, the stability properties of the dynamics, and the manifold curvature. PtO further requires tailored time discretization strategies for the adjoint covariant ODE, which we show to be robust against small singular values of the state variable. To illustrate the framework, we apply these methods to the optimal control of a random parabolic PDE.