Category:Hyperbolic: Difference between revisions

Revision as of 14:29, 24 February 2016

This category contains all control problems which are governed by a hyperbolic partial differential equation.

A second order linear partial differential equation can be written as $\sum_{i, j = 1}^{n} a_{i j} \frac{\partial^{2} u}{\partial x_{i} \partial x_{j}} + lower - orderterms = 0$ . If $A = (a_{i j})_{i j}$ is indefinite such that $n - 1$ eigenvalues have the same sign and the remaining eigenvalue the other sign, the partial differential equation is called hyperbolic. An example is the wave equation: $\frac{\partial^{2} u}{\partial t^{2}} - Δ u = f$ , where $Δ$ denotes the Laplace operator, $u$ is the unknown, and the function $f$ given.

Pages in category "Hyperbolic"

This category contains only the following page.

C

Control of Transmission Lines

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 This category contains all control problems which are governed by a hyperbolic partial differential equation.
+ <p>
+ A second order linear partial differential equation can be written as
+ <math>\sum^n_{i,j=1} a_{ij} \frac{\partial^2u}{\partial x_i \partial x_j} +\quad \textrm{ lower-order terms} = 0</math>.
+ If <math>A=(a_{ij})_{ij}</math> is indefinite such that <math>n-1</math> eigenvalues have the same sign and the remaining eigenvalue the other sign, the partial differential equation is called hyperbolic.
+ An example is the wave equation: <math>\frac{\partial^2 u}{\partial t^2}-\Delta u = f</math>,
+ where <math>\Delta</math> denotes the Laplace operator, <math>u</math> is the unknown, and the function <math>f</math> given.
+</p>
 <!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->