Time-iteration methods for controllability

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.