SOME PROBLEMS IN THE THEORY OF APPROXIMATING SOLUTIONS OF DIFFERENTIAL-OPERATOR EQUATIONS IN HILBERT SPACE.
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JOHNS HOPKINS UNIV SILVER SPRING MD APPLIED PHYSICS LAB
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In the present paper the author studies certain aspects of the theory of approximating nonlinear differential-operator equations of the form dxdt f x, t phi c sub 1,..., c sub m, u sub 1 t,..., u sub n t,t, where fx,t and phi c sub 1, . . . ., c sub m, u sub 1 t, . . . ,u sub n tt are nonlinear operators satisfying certain conditions x,u sub 1, . . . , u sub n for given t are elements of a Hilbert space and c sub 1, c sub 2, ...., c sub m are numbers. An effective method is presented for choosing the functions u sub 1 t, u sub 2 t, . . . , u sub n t, called control functions, and the parameters c sub 1, . . . , c sub m, in order for the given function yt to have the least mean square deviation from the solution xt of equation. A relationship is pointed out between certain aspects of the stability of solutions of the equation and problems in the theory of approximating solutions.
- Theoretical Mathematics