The Lagrange-Newton method for nonlinear optimal control problems

We investigate local convergence of the Lagrange-Newton method for nonlinear optimal control problems subject to control constraints including the situation where the terminal state is fixed. Sufficient conditions for local quadratic convergence of the method based on stability results for the solutions of nonlinear control problems are discussed.     

Differential stability of solutions to convex,control constrained optimal control problems

A family of convex, control constrained optimal control problems that depend on a real parameter is considered. It is shown that under some regularity conditions on data the solutions of these problems, as well as the associated Lagrange multipliers are directionally differentiable with respect to parameter. The respective right-derivatives are given as the solution and the associated Lagrange multipliers for some quadratic optimal control problem. If a condition of strict complementarity type hold, then directional derivatives become continuous ones.     

Endpoint theory for set-valued nonlinear asymptotic contractions with respect to generalized pseudodistances in uniform spaces

In uniform spaces, inspired by ideas of Banach, Tarafdar and Yuan, we introduce the concepts of generalized pseudodistances and generalized gauge maps, for set-valued dynamic systems we define various nonlinear asymptotic contractions and contractions with respect to these pseudodistances and gauges, provide conditions on the iterates of these set-valued dynamic systems and present a method which is useful for establishing conditions guaranteeing the existence and uniqueness of endpoints (stationary points) of these set-valued dynamic systems and conditions that each generalized sequence of iterations (in particular, each dynamic process) converges and the limit of a generalized sequence of iterations is an endpoint. The definitions, the results and the method are new for set-valued dynamic systems in uniform, locally convex and metric spaces and even for single-valued maps. The paper includes a number of various examples which show a fundamental difference between our results and those existing in the literature.     

Two Cold Atoms in a Harmonic Trap

Two ultracold atoms moving in a trap interact weakly at a very short distance. This interaction can be modeled by a properly regularized contact potential. We solve the corresponding time-independent Schrödinger equation under the assumption of a parabolic, spherically symmetric trapping potential.     

Quadratic forms over formally real fields with eight square classes

Complete classification of formally real fields with 8 square classes with respect to the behaviour of quadratic forms is given. Two fields F and K are equivalent with respect to quadratic forms if the quadratic form schemes of the two fields are isomorphic or in other words, if the Witt rings W(F) and W(K) are isomorphic. It is shown here that for formally real fields with 8 square classes there are exactly seven possible quadratic form schemes and for each of the seven schemes a formally real field with 8 square classes and the given scheme is constructed.     

Particle representations of canonical commutation relations

The aim of the present paper is to find a necessary and sufficient condition for the existence of the operator of the total number of particles in a representation of canonical commutation relations. The result is formulated by means of generating functionals of representations. It is shown that an irreducible representation possesses a (generalized) number operator if and only if the representation obtained by averaging its generating functional over the group of phase transformations is a factor representation of type I.     

Analytical evaluation of lower critical hydrostatic pressure for imperfect cylindrical shells of moderate length

The paper applies generalized power series to determine effectively the lower critical hydrostatic pressure for imperfect cylindrical shells. The basic equations were assumed in the form derived by W.A. Nash. Two variants as regards the number of circumferential half-wavelengths are discussed: m constant during buckling, thus corresponding to upper critical pressure, and m furnishing minimal value of lower critical pressure. It is found that the dependence of lower critical pressure on an imperfection parameter V is linear, i.e. smaller than in the case of the upper critical pressure which decreases with $\text{V}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}$.