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.
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