Necessary Conditions for Free End-Time, Measurably Time Dependent Optimal Control Problems with State Constraints

Recently, necessary conditions have been derived for fixed-time optimal control problems with state constraints, formulated in terms of a differential inclusion, under very weak hypotheses on the data. These allow the multifunction describing admissible velocities to be unbounded and possibly nonconvex valued. This paper extends the earlier necessary conditions, to allow for free end-times. A notable feature of the new free end-time necessary conditions is that they cover problems with measurably time dependent data. For such problems, standard analytical techniques for deriving free-time necessary conditions, which depend on a transformation of the time variable, no longer work. Instead, we use variational methods based on the calculus of 'essential values".     

Many-body effects in the first excited subband of the n-inversion layer of Si

We have calculated the self-energies of electrons in the lowest and first excited sub-bands of Si inversion layers. The self-consistent wavefunctions calculated in the Hartree approximation were used, and dynamic screening was approximated by the Lundqvist-Overhauser model. The correlation energy of an electron in the excited band is quite large: about ?10 meV at an inversion layer density of 10¹¹ cm^?2 to about ?16 meV at 3 × 10¹² cm^?2. The calculated separation between subbands is in very good agreement with available experimental measurements. An exciton is predicted with a binding energy of 0.9 meV at N_inv = 10¹² cm^?2 calculated in the static approximation.     

A maximum principle for optimal control problems with mixed constraints

Necessary conditions in the form of maximum principles are derivedfor optimal control problems with mixed control and state constraints.Traditionally, necessary condtions for problems with mixed constraintshave been proved under hypothesis which include the requirementthat the Jacobian of the mixed constraint functional, with respectto the control variable, have full rank. We show that it canbe replaced by a weaker ‘interiority’ hypothesis.This refinement broadens the scope of the optimality conditions,to cover some optimal control problems involving differentialalgebraic constraints, with index greater than unity.     

Strategic approaches to drug design. I. An integrated software framework for molecular modelling

Summary An integrated molecular graphics and computational chemistry framework is described which has been designed primarily to handle small molecules of up to 300 atoms. The system provides a means of integrating software from any source into a single framework. It is split into two functional subsystems. The first subsystem, called COSMIC. runs on low-cost, serial-linked colour graphics terminals and allows the user to prepare and examine structural data and to submit them for extensive computational chemistry. Links also allow access to databases, other modelling systems and user-written modules. Much of the output from COSMIC cannot be examined with low level graphics. A second subsystem, called ASTRAL, has been developed for the high-resolution Evans & Sutherland PS300 colour graphics terminal and is designed to manipulate complex display structures. The COSMIC minimisers, geometry investigators, molecular orbital displays, electrostatic isopotential generators and various interfaces and utilities are described.     

Second Order Sufficient Conditions for Optimal Control Problems with Non-unique Minimizers: An Abstract Framework

Standard second order sufficient conditions in optimal control theory provide not only the information that an extremum is a weak local minimizer, but also tell us that the extremum is locally unique. It follows that such conditions will never cover problems in which the extremum is continuously embedded in a family of constant cost extrema. Such problems arise in periodic control, when the cost is invariant under time translations, in shape optimization, where the cost is invariant under Euclidean transformations (translations and rotations of the extremal shape), and other areas where the domain of the optimization problem does not really comprise elements in a linear space, but rather an equivalence class of such elements. We supply a set of sufficient conditions for minimizers that are not locally unique, tailored to problems of this nature. The sufficient conditions are in the spirit of earlier conditions for ‘non-isolated’ minima, in the context of general infinite dimensional nonlinear programming problems provided by Bonnans, Ioffe and Shapiro, and require coercivity of the second variation in directions orthogonal to the constant cost set. The emphasis in this paper is on the derivation of directly verifiable sufficient conditions for a narrower class of infinite dimensional optimization problems of special interest. The role of the conditions in providing easy-to-use tests of local optimality of a non-isolated minimum, obtained by numerical methods, is illustrated by an example in optimal control.     

Quantitative determination of intermolecular interactions with fluorinated aromatic rings

The chemical double mutant cycle approach has been used to investigate substituent effects on intermolecular interactions between aromatic rings and pentafluorophenyl pi-systems. The complexes have been characterised using 1H and 19F NMR titrations, X-ray crystal structures of model compounds and molecular mechanics calculations. In the molecular zipper system used for these experiments, H-bonds and the geometries of the interacting surfaces favour the approach of the edge of the aromatic ring with the face of the pentafluorophenyl pi-system. The interactions are generally repulsive and this repulsion increases with more electron-withdrawing substituents up to a limit of +2.2 kJ mol(-1), when the complex distorts to minimise the unfavourable interaction. Strongly electron-donating groups cause a change in the geometry of the aromatic interaction and attractive stacking interactions are found (-1.6 kJ mol(-1) for NMe2). These results are generally consistent with an electrostatic model: the polarisation of the pentafluorophenyl ring leads to a partial positive charge located at the centre and this leads to repulsive interactions with the positive charges on the protons on the edge of the aromatic ring; when the aromatic ring has a high pi-electron density there is a large electrostatic driving force in favour of the stacked geometry which places this pi-electron density over the centre of the positive charge on the pentafluorophenyl group.