Second order optimality conditions and their role in PDE control

If f : Rn R is twice continuously differentiable, f’(u) = 0 and f’’(u) is positive definite, then u is a local minimizer of f. This paper surveys the extension of this well known second order suffcient optimality condition to the case f : U R, where U is an infinite-dimensional linear normed space. The reader will be guided from the case of finite-dimensions via a brief discussion of the calculus of variations and the optimal control of ordinary differential equations to the control of nonlinear partial differential equations, where U is a function space. In particular, the following questions will be addressed: Is the extension to infinite dimensions straightforward or will unexpected difficulties occur? How second order sufficient optimality conditions must be modified, if simple inequality constraints are imposed on u? Why do we need second order conditions and how can they be applied? If they are important, are we able to check if they are fulfilled order sufficient optimality condition to the case f : U R, where U is an infinite-dimensional linear normed space. The reader will be guided from the case of finite-dimensions via a brief discussion of the calculus of variations and the optimal control of ordinary differential equations to the control of nonlinear partial differential equations, where U is a function space. In particular, the following questions will be addressed: Is the extension to infinite dimensions straightforward or will unexpected difficulties occur? How second order sufficient optimality conditions must be modified, if simple inequality constraints are imposed on u? Why do we need second order conditions and how can they be applied? If they are important, are we able to check if they are fulfilled? It turns out that infinite dimensions cause new difficulties that do not occur in finite dimensions. We will be faced with the surprising fact that the space, where f’’(u) exists can be useless to ensure positive definiteness of the quadratic form v f’’(u)v2. In this context, the famous two-norm discrepancy, its consequences, and techniques for overcoming this difficulty are explained. To keep the presentation simple, the theory is developed for problems in function spaces with simple box constraints of the form a = u = ß. The theory of second order conditions in the control of partial differential equations is presented exemplarily for the nonlinear heat equation. Different types of critical cones are introduced, where the positivity of f’’(u) must be required. Their form depends on whether a so-called Tikhonov regularization term is part of the functional f or not. In this context, the paper contains also new results that lead to quadratic growth conditions in the strong sense. As a first application of second-order sufficient conditions, the stability of optimal solutions with respect to perturbations of the data of the control problem is discussed. Second, their use in analyzing the discretization of control problems by finite elements is studied. A survey on further related topics, open questions, and relevant literature concludes the paper.The first author was partially supported by the Spanish Ministerio de Economía y Competitividad under project MTM2011-22711, the second author by DFG in the framework of the Collaborative Research Center SFB 910, project B6

Thermostabilisation of human serum butyrylcholinesterase for detection of its inhibitors in water and biological fluids

The ability of gelatine-trehalose to convert the normally fragile, dry human serum BChE into a thermostable enzyme and its use in the detection of cholinesterase inhibitors in water and biological fluids is described. Gelatine or trehalose alone is unable to protect the dry enzyme against exposure to high temperature, while a combination of gelatine and trehalose were able to protect the enzyme activity against prolonged exposure to temperature as high as +50°C. A method for rapid, simple and inexpensive means of screening for cholinesterase inhibitors such as carbamates and organophosphates in water, vegetables and human blood has been developed.<br>A capacidade da gelatina-trehalose em converter a frágil BChE do soro humano em uma enzima termoestável e seu uso na descoberta de inibidores de colinesterase em água e fluidos biológicos é apresentado. A Gelatina ou trehalose são incapazes de proteger a enzima seca BchE do soro humano contra exposição a elevadas temperaturas, enquanto que uma combinação de gelatina e trehalose são capazes de proteger a atividade de enzima contra exposição prolongada a temperaturas elevadas e da ordem de 50° C. Um método barato, simples e rápido de screening para inibidores de colinesterase tal como carbamatos e organofosfatos em água, verduras e sangue humano foi desenvolvido