By Maksimov V. I.

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12 Let y ∈ RN . Then l(y) = {Ay | A ∈ A(N )}. 1 is a ﬁnite dimensional form of the Borwein–Preiss variational principle [58]. 2 was suggested by [137]. 5). Geometrically, Gordan’s alternative [129] is clearly a consequence of the separation theorem: it says either 0 is contained in the convex hull of a0 , . . , aM or it can be strictly separated from this convex hull. 3 shows that with an appropriate auxiliary function variational method can be used in the place of a separation theorem – a fundamental result in analysis.

4. 6). 3 The Uniqueness of Viscosity Solutions We now use the nonlocal approximate sum rule to prove a uniqueness theorem for the viscosity solution of the following Hamilton–Jacobi equation u + H(x, u ) = 0. 15) This equation is closely related to the optimal value function of certain optimal control problems. 16) where f and g are Lipschitz functions, c is a measurable function modeling the control and C is a compact set modeling the admissible range of the control function. We assume that for any given x, an optimal control for the above problem exists.

X This shows that S(f + h; a/3) ⊂ S(f + g; a). Thus, h ∈ Ui . To see that each Ui is dense in Y , suppose that g ∈ Y and ε > 0; it suﬃces to produce h ∈ Y such that h Y < ε and for some a > 0 diam S(f + g + 34 2 Variational Principles h; a) < 1/i. By hypothesis (iv), Y contains a bump function φ. Without loss of generality we may assume that φ Y < ε. By hypothesis (ii) we can assume that φ(0) = 0, and therefore that φ(0) > 0. Moreover, by hypothesis (iii) we can assume that supp(φ) ⊂ B(0, 1/2i).

### A Boundary Control Problem for a Nonlinear Parabolic Equation by Maksimov V. I.

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