Time-iteration methods for controllability

TL;DR

This paper reviews three time-iteration control strategies for PDEs and ODEs, highlighting classical and recent methods for null and local controllability, with applications to linear and nonlinear systems.

Contribution

It provides a comprehensive survey of classical and modern time-iteration methods in control theory, including new black-box formulations for nonlinear PDE controllability.

Findings

Lebeau-Robbiano method effectively proves null controllability for linear PDEs.

Liu-Takahashi-Tucsnak method offers a practical approach to local controllability of nonlinear PDEs.

Tangent vectors method establishes local exact controllability for nonlinear ODEs.

Abstract

These notes are based on a short course delivered at the Summer School EUR MINT 2025 "Control, Inverse Problems and Spectral Theory", held in June 2025 in Toulouse, France. The course presents three important strategies in control theory, formulated as time-iteration methods, where each time step brings the state of the system closer to the desired target. For linear PDEs, we survey the classical Lebeau-Robbiano method and its more recent developments. This method combines spectral inequalities and dissipation estimates to prove null controllability of a dissipative linear system. For nonlinear PDEs, we reinterpret the Liu-Takahashi-Tucsnak method, which establishes local controllability of a nonlinear system by analyzing the control cost of its linearization. We provide an easily applicable black-box formulation of their method. Finally, for nonlinear ODEs, we present the tangent vectors method, which establishes local exact controllability starting from approximately reachable directions.