By Wahlen E.

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71) = 1,2 are updated by a gradient with projection method. 1 does not apply. However similar results to the previous ones can be obtained. When discharges take place at points rather than in subsets of non zero measure, it is reasonable to try to optimize the system with respect to the position of these points too. 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].

J) (i+ Ifl,j+ 1(2) (i - 1(2. J) (i,j) ' - - - - - . - . . j-l/2) ~"_-----i""'1 ~x Figure 1. Location of the unknowns (1], u, v) on the A, B, C, and D grids; i and} are integer indices associated with the space discretization. y, where & and l'1y denote the space increments, Stability analysis. e" every unknown r is assumed to be of the form r(t,x,y) = R(t) exp[l (kxx + k;)], with I = (_1)112. Wavenumbers kx and ky obey the inequalities set out below. A 3x3 amplification matrix is obtained, the eigenvalues, A, of which are given by (A-1)(A 2 -2bA+1) = O.

However similar results to the previous ones can be obtained. When discharges take place at points rather than in subsets of non zero measure, it is reasonable to try to optimize the system with respect to the position of these points too. 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.

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A blow-up result for the periodic Camassa-Holm equation by Wahlen E.


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