Tunneling Estimates and Approximate Controllability for Hypoelliptic Equations.

Cover
Title page
Chapter 1. Introduction and main results
1.1. Introduction
1.2. Main results
1.3. Comparison to other works
1.4. Sketch of the proofs and plan of the paper
1.5. Some remarks and further comments
Chapter 2. The quantitative Holmgren-John theorem of [LL19]
2.1. A typical quantitative unique continuation result of [LL19]
2.2. Definitions and tools for propagating the information
2.3. Semiglobal estimates along foliation by hypersurfaces
Chapter 3. The hypoelliptic wave equation, proof of Theorem 1.15
3.1. Step 1: Geometric setting and non-characteristic hypersurfaces
3.2. Step 2: Propagation of smallness
3.3. Step 3: Energy estimates
Chapter 4. The hypoelliptic heat equation
4.1. Approximate controllability with polynomial cost in large time: Proof of Theorem 1.22
4.2. Approximate controllability in Gevrey-type spaces: Proof of Theorem 1.20
4.3. Approximate controllability in natural spaces with exponential cost: Proof of Theorem 1.18
4.4. Technical lemmata used for the heat equation
Chapter 5. A partially analytic example: Grushin type operators
5.1. The geometric context
5.2. A proof of Estimate (1.26)
5.3. An observation term in ² in quantitative unique continuation estimates
Appendix A. On the optimality: Proof of Proposition 1.14
Appendix B. Subelliptic estimates
B.1. ^{ } subelliptic estimates on compact manifolds
B.2. Subelliptic estimates for manifolds with boundaries
Appendix C. Sub-Riemannian norm of normal vectors
Bibliography
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