First-Order Necessary Conditions in Optimal Control

Journal of Optimization Theory and Applications - In an earlier analysis of strong variation algorithms for optimal control problems with endpoint inequality constraints, Mayne and Polak provided...     

Existence of Neighboring Feasible Trajectories: Applications to Dynamic Programming for State-Constrained Optimal Control Problems

In this paper, the value function for an optimal control problem with endpoint and state constraints is characterized as the unique lower semicontinuous generalized solution of the Hamilton-Jacobi equation. This is achieved under a constraint qualification (CQ) concerning the interaction of the state and dynamic constraints. The novelty of the results reported here is partly the nature of (CQ) and partly the proof techniques employed, which are based on new estimates of the distance of the set of state trajectories satisfying a state constraint from a given trajectory which violates the constraint.     

Duality theorems for convex problems with time delay

In this paper, we examine a class of convex problems of Bolza type, involving a time delay in the state. It encompasses a variety of time-delay problems arising in the calculus of variations and optimal control. A duality analysis is carried out which, among other things, leads to a characterization of minimizers in terms of the Euler-Lagrange inclusion. The results obtained improve in significant respects on what is achievable by techniques previously employed, based on elimination of the time delay by introduction of an infinite-dimensional state space or on the method of steps.     

COSMIC(90): An improved molecular mechanics treatment of hydrocarbons and conjugated systems

Summary Four modifications to the COSMIC molecular mechanics force field are described, which greatly increase both its versatility and the accuracy of calculated conformational energies. The Hill non-bonded van der Waals potential function has been replaced by a two-parameter Morse curve and a new H-H potential, similar to that in MM3, incorporated. Hydrocarbon energies in particular are much improved.A simple iterative Hückel pi-electron molecular orbital calculation allows modelling of conjugated systems. Calculated bond lengths and rotational barriers for a series of conjugated hydrocarbons and nitrogen heterocycles are shown to be as accurate as those determined by the MM2 SCF method.Explicit hydrogen-bonding potentials for H-bond acceptor-donor atom pairs have been included to give better hydrogen bond energies and lengths. The van der Waals radii of protonic hydrogens are reduced to 0.5 Å and the energy well depth is increased to 1.0 kcal mol^-1.Two new general atom types, N⁺ _sp ² and O^- _sp ³ , have been introduced which allow a wide variety of charged conjugated systems to be studied. A minimum of parameterisation is required, as the new types are easily included in the Hückel scheme which automatically adjusts bond and torsional parameters according to the defined bond-order relationships.     

Relaxation of optimal control problems to equivalent convex programs

Relaxed controls induce convexity of the velocity sets in optimal control problems, permitting a general existence theory. Here we obtain complete convexity, of the set of control-trajectory pairs, by relaxing the problem constraints to admit certain measures on the product of the control and trajectory spaces. It is proved that these measures are just unit mixtures of control-trajectory pairs and that admitting them does not alter the minimum value of the control problems. This can be used to derive necessary and sufficient conditions for optimality of dynamic programming type.     

Extended electron distributions applied to the molecular mechanics of some intermolecular interactions

Summary Extended electron distributions (XEDs) have been added to the molecular mechanics Coulombic term and applied to a selection of intermolecular interactions. The results from this approach have been compared with the commonly used atom-centred charges and more rigorous AM1-derived natural atom orbital point densities. The use of XEDs generally improves the simulation of experimental and ab initio results over the other two charge allocations and corrects geometries in those cases for which the others yield wrong results.     

Self-consistent calculation of impurity scattering in inversion layers

The local density-functional method has been applied to the calculation of the influence of an ionized impurity on the electrons in an inversion layer on the surface of silicon. Compared to linear screening theories this theory shows improved agreement with experiments on the mobility. The bound state associated with the self-consistent potential has a very small binding energy.