Category:Parabolic

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

A second order linear partial differential equation can be written as $\sum^n_{i,j=1} a_{ij} \frac{\partial^2u}{\partial x_i \partial x_j} +\, \text{lower-order terms} = 0$ .

If  $A=(a_{ij})_{ij}$  is positive or negative semidefinite with exact one eigenvalue zero, the  partial differential equation is called parabolic.

An example is the heat equation: $\frac{\partial u}{\partial t}-\Delta u = f$ , where $\Delta$ denotes the Laplace operator, $u$ is the unknown, and the function $f$ is given.