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Sektion 1
Dienstag, 19.09.2000, 17.30–17.50 Uhr, POT 151

Optimal Control of Systems Governed by Infinite Order Operators

Somaya El-Zahaby, Al-Azhar University, Cairo

This is a review article for previous results which were obtained by I.M. Gali, H. A. El-Saify and S. A. El-Zahaby. The results obtained there are for the case of operators with an infinite number of variables which are elliptic, parabolic, hyperbolic or well-posed in the sense of Petrowsky.

Subsequently, J. L. Lions suggested a problem related to this result but in different direction by taking the case of operators of infinite order with finite dimension in the form

        oo 
Bu  =  sum   (- 1)|a|a D2au,  B  (-  L(W  oo  {a ,2},W - oo  {a ,2}).
                  a                0   a       0    a
      |a|=0

This operator has a self - adjoint closure. Here W 0 oo {aa, 2} is the set of all functions of W  oo {a a, 2} which vanish on the boundary G of Rn and

                                 sum  oo 
W  oo {aa, 2}= {u(x)   (-  C oo (Rn) :    aa||Dau  ||22 <  oo }
                                |a|=0

is the Sobolev space of infinite order of periodic functions defined on all G of Rn .

A necessary and sufficient condition for optimality in distributed control governed by Dirichlet and Neumann problem, for elliptic equations of infinite order is given. The optimality condition is expressed in terms of a set of inequalities.

The solvability of the mixed problem for nonlinear infinite order hyperbolic equations is shown. A necessary and sufficient condition for the control to be optimal is expressed by a set of inequalities. We established a necessary and sufficient condition for the existence of the infinite tensor product of operators acting in the infinite tensor product of Hilbert spaces and present the set of inequalities defining an optimal control of a system governed by infinite tensor product of elliptic operators.

Also a distributed control problem for hyperbolic operators

-d2     oo 
dt2 + k ox =1Ak
is considered, and necessary and sufficient conditions for the control to be optimal are obtained.

Gali and El Zahaby found the existence of optimal control of system governed by variational inequalities in the case of infinite number of variables. The problem of the system governed by very strongly nonlinear variational inequalities for infinite order elliptic operator is also considered.