By Ma J. M., Qiu R. F.

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**Extra resources for 3-Manifold Containing Separating Incompressible Surfaces of All Positive Genera**

**Example text**

This should be the case when a wastewater system is going to be built. In the next section we consider a simplified version of this problem. Further results concerning pointwise control problems can be found in Saguez[1974], Lions[1979] and Simon[1982]. 1 Statement of the problem and theoretical results Optimal placement of an outfall can be formulated as an optimal control problem where the control is the position of the outlet, the cost function is the distance to the wastewater treatment plant and the state equations are those modelling pollutant concentration.

E. conditions to be satisfied by the state. In many practical situations this corresponds to the fact that it has to belong to a certain subset of a functional space. Optimal control theory deals with the following problems: 1. Existence of solution 2. Optimality conditions 3. Numerical methods, including discretization and iterative algorithms for solving the discretized problem. In this presentation we consider the case where the state equation is a partial differential equation of parabolic type.

The values of VIand Q'I', in accordance with (7), are given in this table. The Coriolis parameter is denoted 1('" ± 10-4 S-1 at mid latitudes, =0 at the Equator), while ez' p and P represent the vertical unit vector, the pressure and the density, respectively. 8 m2 s-I). The variable c represents any scalar quantity, for instance, temperature, salinity, concentration of a pollutant... Q'I' 'If continuity equation 0 -Iez xu - p0-IVp horiz. momentum eq. u vert. momentum eq. 0 _ p-l ap/dz _ g scalar quantity budget c = 0 if passive tracer, "# 0 otherwise 3.

### 3-Manifold Containing Separating Incompressible Surfaces of All Positive Genera by Ma J. M., Qiu R. F.

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