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By Andjelic T. (ed.)

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64)  ∂ψ  ∗ − + A ψ = 0 in Q, ψ(T ) = k(yT − y(T )), ψ = 0 on . 64) (two points with respect to the time variable, the two points being t = 0 and t = T ). 5). 63). Given ε > 0, there exists a control w such that y(T ; w) − yT ≤ ε. 2) and it is not constructive. 67) w2 dx dt + ε2 . 63). w2 dx dt ≤ 1 2 β . ). 64) if we have a way to choose k. 69) (where the solution has been denoted by f ∗ , instead of f ). 62) reads as follows: (k −1 I + )f = yT . 70) by f we obtain ( f , f ) + k −1 f 2 = (yT , f ).

152b) where ψ n ≈ ψ(n t) (ψ(n t) : x → ψ(x, n t)). 153a) φ 0 = 0; then, assuming that φ n−1 is known, we solve the following Dirichlet problem for n = 1, . . , N , φ n − φ n−1 + Aφ n = ψ n χO in t Finally, we approximate t by , φ n = 0 on . 153b) defined by t g = φN . 153b) have a unique solution; we have, furthermore, the following. 37 Operator L2 ( ). is symmetric and positive semidefinite from L2 ( ) into Proof. Consider a pair {g, g} ˜ ∈ L2 ( )×L2 ( ). We have then (with obvious notation) t ( g, g) ˜ L2 ( ) = φ N ψ˜ N +1 dx.

49). 55) with ψ obtained from f via the solution of − We have then ∂ψ + A∗ ψ = 0 in Q, ∂t ψ(T ) = f , ψ = 0 on y(0) = 0, y = 0 on . 56) f = y(T ), where, ∂y + Ay = ψχO×(0,T ) in Q, ∂t . 53). 56) to compute the corresponding value of ψ. 54). 8. Before that, several remarks are in order. 56). Therefore, ψ is smooth (indeed, the smoother the coefficients of operator A, the smoother will be ψ). In other words, the control u is a smooth function of x and t. This observation excludes the possibility of finding an optimal control of the “bangbang” type.

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Set theory. Foundations of mathematics: Proceedings Belgrade, 1977 by Andjelic T. (ed.)


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