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- W124483976 abstract "In this chapter, we study the control problem of the linear wave equation $$frac{partial^2 u}{partial t^2} = c^2 nabla^2 u.$$ This equation can serve as a mathematical model for many physical problems, such as the vibration of a membrane. In the membrane problem, u = u(x, y, t) represents the vertical displacement of the membrane from its equilibrium and the positive constant c (m/s) is a wave speed. In this chapter, we use the following notation. Let Ω be a bounded open set in ℝ n and x 0 ∈ ℝ n . Set (see Figure 6.1) 6.1 $$Gamma = partialOmega,$$ 6.2 $$mathbf{m}(mathbf{x}) = mathbf{x} - mathbf{x}^0 = (x_1 - x_{1}^{0}, ldots, x_{n} - x_{n}^{0}),$$ 6.3 $$Gamma (mathbf{x}^{0}) = {mathbf{x} in Gamma : mathbf{m}(mathbf{x}) cdot mathbf{n}(mathbf{x}) > 0},$$ 6.4 $$Gamma_{ast}(mathbf{x}^0) = Gamma - Gamma (mathbf{x}^0) = {mathbf{x} in Gamma : mathbf{m}(mathbf{x}) cdot mathbf{n}(mathbf{x}) leq 0},$$ where n denotes the unit normal pointing towards the outside of Ω." @default.
- W124483976 created "2016-06-24" @default.
- W124483976 creator A5066152182 @default.
- W124483976 date "2009-11-22" @default.
- W124483976 modified "2023-10-05" @default.
- W124483976 title "Higher-dimensional Wave Equation" @default.
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- W124483976 doi "https://doi.org/10.1007/978-3-642-04613-1_6" @default.
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