Contents Preface ix Part 1. Controllability of linear control systems 1 Chapter 1. Finite-dimensional linear control systems 3 1.1. Definition of controllability 3 1.2. An integral criterion for controllability 4 1.3. Kalman's type conditions for controllability 9 1.4. The Hilbert Uniqueness Method 19 Chapter 2. Linear partial differential equations 23 2.1. Transport equation 24 2.2. Korteweg-de Vries equation 38 2.3. Abstract linear control systems 51 2.4. Wave equation 67 2.5. Heat equation 76 2.6. A one-dimensional Schrodinger equation 95 2.7. Singular optimal control: A linear 1-D parabolic-hyperbolic example 99 2.8. Bibliographical complements 118 Part 2. Controllability of nonlinear control systems 121 Chapter 3. Controllability of nonlinear systems in finite dimension 125 3.1. The linear test 126 3.2. Iterated Lie brackets and the Lie algebra rank condition 129 3.3. Controllability of drift less control affine systems 134 3.4. Bad and good iterated Lie brackets 141 3.5. Global results 150 3.6. Bibliographical complements 156 Chapter 4. Linearized control systems and fixed-point methods 159 4.1. The Linear test: The regular case 159 4.2. The linear test: The case of loss of derivatives 165 4.3. Global controllability for perturbations of linear controllable systems 177 Chapter 5. Iterated Lie brackets 181 Chapter 6. Return method 187 6.1. Description of the method 187 6.2. Controllability of the Euler and Navier-Stokes equations 192
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