We go here briefly on the main ideas we need for the analysis of
optimization problems.
Let us begin by a simple problem, the linearized small disturbance potential
equation of
fluid dynamics. This equation will demonstrate the different issues we
are concerned with, and will set the foundation for the treatment of general
problems, including systems of equations such as the Euler and the Navier-Stokes equations.
**Example I: A Scalar Equation.**
Consider the equation

(1) |

We want to analyze the solution in terms of its
boundary values on .
We consider boundary data in terms of Fourier components

(2) |

(3) |

(4) |

(5) |

For the supersonic case, , there are two bounded solutions for , and only one bounded solution for . The two bounded solutions can be viewed as an incident wave and a reflected wave, whose sum satisfies the boundary condition,

(6) |

A general solution for the problem can be written as a sum (integral) of these Fourier components.

**Example II: A System of PDE.** We next show the use of Fourier analysis for studying systems of partial differential equation. Consider the system

(7) |

where and . Here one considers vector Fourier components, that is

(8) |

The problem is to determine but also the relation of and . This is done by substituting this form into the equations leading to an algebraic equation for ,

(9) |

This is a linear set of equations for the unknown and it has a non zero solution if and only if the determinant of the system is zero. This gives an equation for in terms of ,

(10) |

For the subsonic case and we have only one solution , which correspond to a bounded solution in . To find the relation of and we substitute this value of into the system, and solve for yielding,

(11) |

For the supersonic case, , we have two solutions and corresponding to two bounded solution in , and the coefficients are given by

(12) |

The solution for given by (11) or (12) can be multiplied by an arbitrary constant. The actual value of that constant depends on the boundary condition which is required for the problem. This complete the solution of the problem in .

**Remark:**
In general, there may be several solutions for and to each of them
there would correspond a vector solution such as
(11) or (12) in the example above. The general solution is then a
linear combination of these vectors with some weights to be determined from the
boundary conditions.
If the problem is well posed, then the number of boundary condition equals exactly
to the number of bounded solutions that we are seeking and the analysis
can be completed.

Some of the important question regarding general systems of equations can be answered to some extent using this tool. These include well-posedness of the problem, the choice of boundary conditions, their effect on the solutions and more.