On the limited memory BFGS method for large scale optimization
Mathematical Programming: Series A and B
Nash equilibria for the multiobjective control of linear partial differential equations
Journal of Optimization Theory and Applications
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This work deals with some numerical experiments regarding the distributed control ofsemilinear parabolic equations of the type yt - y@g@g + f (y) = u@g@w,in (0, 1) x (0, T), with Neumann and initial auxiliary conditions, where w is an open subset of (0, 1), f is a C1 nondecreasing real function, u is the output control and T 0 is (arbitrarily) fixed. Given a target state YT we study the associated approximate controllability problem (given e 0, find u E L2(0, T), such that y(T; u) -YT@?L^2(0,1) @? @? by passing to the limit (when k - ~) in the penalized optimal control problem (find uk as the minimum of Jk(u) = 1/2 @?u@?"L"("0","T")^2 + (k/2) y(T; u) -YT@?"L^22L2(o,T) L2(0,1)). In the superlinear case (e.g., f (y) = @?y@?n-1"y, n 1) the existence of two obstruction functions Y+/-~ shows that the approximate controllability is only possible if Y-~ (x, T)