This text discusses singularities in boundary value problems and exact controllability of hyperbolic systems, focusing on the wave equation. The main problem is to find a control function v on the lateral boundary Σ such that the solution u of the wave equation satisfies u(T) = u'(T) = 0, i.e., the system is brought to rest at time T. The goal is to find the smallest possible T and minimize the support of v. The Hilbert Uniqueness Method (HUM), introduced by J.L. Lions in 1988, provides a systematic approach to this problem. It treats the problem as an optimization problem, minimizing the integral of v² over Σ. The method relies on the existence of a subset Σ₀ of Σ such that a certain inequality holds for all pairs (φ₀, φ₁) in H₀¹(Ω) × L²(Ω). This inequality ensures exact boundary controllability for any initial data (u₀, u₁) in L²(Ω) × H⁻¹(Ω). The minimal smoothness assumptions on the boundary Γ are discussed. J.L. Lions uses a multiplier method with integration by parts to prove the result, which works when Γ is C². However, C. Bardos, G. Lebeau, and J. Rauch extend this to Γ being C∞, allowing them to find optimal T₀ and minimal support Σ₀ for exact controllability. It is necessary that ∂φ/∂ν ∈ L²(Σ₀) for the inequality to make sense. The integral ∫ m.v |∂φ/∂ν|² ds dt, derived from integration by parts, plays a key role in the calculations. The method ensures that the system can be controlled exactly to rest at time T, with the minimal possible T and support of the control function.This text discusses singularities in boundary value problems and exact controllability of hyperbolic systems, focusing on the wave equation. The main problem is to find a control function v on the lateral boundary Σ such that the solution u of the wave equation satisfies u(T) = u'(T) = 0, i.e., the system is brought to rest at time T. The goal is to find the smallest possible T and minimize the support of v. The Hilbert Uniqueness Method (HUM), introduced by J.L. Lions in 1988, provides a systematic approach to this problem. It treats the problem as an optimization problem, minimizing the integral of v² over Σ. The method relies on the existence of a subset Σ₀ of Σ such that a certain inequality holds for all pairs (φ₀, φ₁) in H₀¹(Ω) × L²(Ω). This inequality ensures exact boundary controllability for any initial data (u₀, u₁) in L²(Ω) × H⁻¹(Ω). The minimal smoothness assumptions on the boundary Γ are discussed. J.L. Lions uses a multiplier method with integration by parts to prove the result, which works when Γ is C². However, C. Bardos, G. Lebeau, and J. Rauch extend this to Γ being C∞, allowing them to find optimal T₀ and minimal support Σ₀ for exact controllability. It is necessary that ∂φ/∂ν ∈ L²(Σ₀) for the inequality to make sense. The integral ∫ m.v |∂φ/∂ν|² ds dt, derived from integration by parts, plays a key role in the calculations. The method ensures that the system can be controlled exactly to rest at time T, with the minimal possible T and support of the control function.