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Time-iteration methods for controllability

Frédéric Marbach

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.

Time-iteration methods for controllability

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.
Paper Structure (79 sections, 67 theorems, 238 equations)

This paper contains 79 sections, 67 theorems, 238 equations.

Table of Contents

  1. A prequel on the controllability of ODEs
  2. The rank condition for linear systems
  3. About the cost of fast controls
  4. About the cost of large systems
  5. Small-time local controllability of control-affine systems
  6. The linear test for control-affine systems
  7. Nonlinear motions and obstructions
  8. From linear ODEs to linear dissipative PDEs
  9. Input-output formalism
  10. A general time iteration
  11. The role of spectral inequalities
  12. The commutative case
  13. The non-commutative case
  14. Historical perspective
  15. The 1995 paper of Lebeau and Robbiano.
  16. ...and 64 more sections

Key Result

Lemma 1.1

For any $T > 0$, any initial data $y_0 \in \mathbb{R}^n$ and any control $u \in L^p((0,T); \mathbb{R}^m)$, there exists a unique mild solution $y \in C^0([0,T];\mathbb{R}^n)$ to eq:lti_system with $y(0) = y_0$ given by:

Theorems & Definitions (165)

  • Lemma 1.1
  • Definition 1.2: Controllability
  • Theorem 1.3: Rank condition
  • proof
  • Example 1.4: Integrator chain
  • Remark 1.5
  • Example 1.6: Diagonal system
  • Proposition 1.7
  • proof
  • Example 1.8: Integrator chain
  • ...and 155 more