Shape optimization problems which are one of our main topics, are related to control problems governed by PDE. An understanding of boundary control problems will help us to get the proper insight into shape optimization problems.

We consider the small disturbance equation, for zero Mach number,
in two dimensions. The domain is a rectangle
whose bottom boundary has a control variable to be optimized.
It is required to achieve a certain pressure distribution on that boundary.
Denote by the bottom boundary and by the rest of the
boundary . The potential satisfies the equation

(30) |

where is the design variable. This problem is related to a shape design problem in which the bottom boundary is described by the function , and the boundary condition for on this boundary is . We will come to this relation later on.

We consider the cost functional

(31) |

where , and we would like to construct a formula for the gradient of this functional. We consider a perturbation of the design variable by and the corresponding change in by , which satisfies

(32) |

The variation in the functional is

(33) |

where integration by parts have been used in the last equality. As in the abstract constrained optimization problem we discussed in section (2.2), we see that the change in the functional depends on the the sensitivity derivatives , and we would like to eliminate this dependence, in order to get an efficient computation of the gradient. We do it by adding a term to which is the differential analog of the term in the algebraic case. Then a proper choice for will result in the desired form for the variation in the functional.

Let be an arbitrary function defined in the same domain as .
From equation (32) for we have

(34) |

where the second equality follows from integration by parts. This is the analog of equation (10) that we have in the algebraic case. Adding the right hand side of (34) (multiplied by ) to we get

(35) |

Since we can break the integral into and then combine the terms, to obtain

(36) |

In order to eliminate the dependence of on we make the following choice for

(37) |

Therefore,

(38) |

where integration by parts was used in the last equality. The expression for the changes in the functional is given as as a function of the changes in the design variable, as well the adjoint variable which satisfies the adjoint equation (37), (or costate equation in control terminology). We would like now to pick a direction of change that will result in reduction of the cost functional. We distinguish two cases.

**I. Finite Dimensional Control.** In this case we assume that the
design variable has a representation

(39) |

and a similar expression for , where the functions are prescribed. This implies

(40) |

and hence

(41) |

and the choice

(42) |

will result in a reduction of the cost functional by

(43) |

At a minimum the following conditions hold,

(44) |

Note that the gradient given in (44) is in terms of .

**II. Infinite Dimensional Control.** In this case we regard the variable
as a function defined on the boundary . A proper choice that will
result in a reduction of the functional is given in terms of ,

(45) |

and the corresponding reduction in the functional is given this time by

(46) |

An algorithm for solving this control problem consists of repeated application of the following three steps, until convergence (of the gradient to zero),

`ALGORITHM`

`(1) Solve the state equation (30) for `

`(2) Solve the adjoint equation (37) for `

`(3) Update by
, where is found by line search.
is given by (42) or (45).`