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This is the fourth English edition of the book Convexity and Optimization in Banach Spaces. With respect to the previous edition published by Kluwer in 1986, this book contains new results pertaining to new concepts of subdifferential for convex functions and new duality results in convex programming. The last chapter of the book, concerned with convex control problems, was rewritten for this edition and completed with new results concerning boundary control systems, the dynamic programming equations in optimal control theory, periodic optimal control problems. Also, the bibliographical list and bibliographical comments were updated. The contents, as well as the structure of the book, were modified in order to include a few fundamental results and progress in the theory of infinite-dimensional convex analysis which were obtained in the last 25 years.

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Abstract.We show here that a variant of the maximum principle holds for the time-optimal trajectories of control systems governed by certain variational inequalities of parabolic type. The approach is constructive and relies on the approximation of the time-optimal problem by a family of optimal control problems with infinite time horizon.

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Abstract. This paper is devoted to the study of noise effects on blow-up solutions to stochastic nonlinear Schrödinger equations. It is a continuation of our recent work [2], where the (local) well-posedness is established in H 1 , also in the non-conservative critical case. Here we prove that in the non-conservative focusing mass-(super)critical case, by adding a large multiplicative Gaussian noise, with high probability one can prevent the blow-up on any given bounded time interval [0, T ], 0 < T < ∞. Moreover, in the case of spatially independent noise, the explosion even can be prevented with high probability on the whole time interval [0, ∞). The noise effects obtained here are completely different from those in the conservative case studied in [5].

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