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Second order necessary conditions of optimality for constrained optimization problems in finite dimensional spaces with vanished cost function gradient at optimal point
Conditions of optimality for constrained optimization problem
Higher order boundary conditions involve derivatives of the fields beyond the first and were recently shown to be more effective than traditional first order conditions in modeling dielectric coatings and layers. In this report an application of a third order generalized boundary condition to scattering by a filled rectangular groove is presented. Deficiencies of such higher-order boundary conditions are addressed and a correction is proposed for the present case. As part of the process of examining and correcting the accuracy of the proposed generalized boundary conditions, an exact solution is developed and a comparison is provided with a solution based on the standard impedance boundary condition.
Second order endpoint condition considered in solution of Bolza problem in calculus of variations
Second order necessary conditions of optimality with straightforward application to nonlinear programming of optimal control problems
Second order variational endpoint condition numerical application used for low thrust minimum fuel orbital transfer problems
Sufficient conditions for a weak relative minimum are derived for a form of the Bolza problem of variational calculus. The derivation ties together first-order and second-order conditions in a unified consistent manner, addresses controllability considerations in some detail, and clarifies some small inconsistencies in earlier work. The resulting second-order conditions of optimality involve the integration of fewer backward-sweep matrix elements than the standard conditions in the literature for problems with an unspecified final time. As a result, the backward-sweep matrices have the same general structure and dimensionality whether the final time is specified or free, with the terminal values of two of three sweep matrices being more complicated in the latter case.
There are two tasks described in this report. First, an extension of a two dimensional formulation is presented for a three dimensional body of revolution. With the introduction of a Fourier expansion of the vector electric and magnetic fields, a coupled two dimensional system is generated and solved via the finite element method. An exact boundary condition is employed to terminate the mesh and the fast fourier transformation is used to evaluate the boundary integrals for low O(n) memory demand when an iterative solution algorithm is used. Second, the diffraction by a material discontinuity in a thick dielectric/ferrite layer is considered by modeling the layer as a distributed current sheet obeying generalized sheet transition conditions (GSTC's).
There are two tasks described in this report. First, an extension of a two dimensional formulation is presented for a three dimensional body of revolution. A Fourier series expansion of the vector electric and magnetic fields is employed to reduce the dimensionality of the system, and an exact boundary condition is employed to terminate the mesh. The mesh termination boundary is chosen such that it leads to convolutional boundary operators for low O(n) memory demand. Second, rigorous uniform geometrical theory of diffraction (UTD) diffraction coefficients are presented for a coated convex cylinder simulated with generalized impedance boundary conditions. Ray solutions are obtained which remain valid in the transition region and reduce uniformly those in the deep lit and shadow regions. A uniform asymptotic solution is also presented for observations in the close vicinity of the cylinder.
Derivation of the equations of condition of order eight for a general system of second-order differential equations approximated by the basic Runge-Kutta-Nystrom algorithm. For this general case, the number of equations of condition is considerably larger than for the special case where the first derivative is not present. Specifically, it is shown that, for orders two through eight, the number of equations for each order is 1, 1, 1, 2, 3, 5, and 9 for the special case and is 1, 1, 2, 5, 13, 34, and 95 for the general case.
Sufficient conditions for a weak relative minimum for a form of the Bolza problem of variational calculus are derived. Testing of the second-order conditions requires the backward integration of fewer matrix elements than in the case of most previously published sets of conditions. The derivation is felt to be more complete and straightforward than previous derivations. The variational problem considered is relatively simple, with just a scalar constraint to implicitly or explicitly determine the final time, in order to avoid the complexities associated with controllability considerations. Sufficient conditions for a local minimum for more general optimal control problems can be approached by building upon the derivation and results presented here.
The discharge-flow resonance fluorescence technique has been used to obtain absolute rate data for the O + HO2 reaction from 229 to 372 K at a total pressure of 1 torr. Pseudo-first-order conditions were used with HO2 concentrations in large excess over initial atomic oxygen in order to minimize interference from secondary reactions. The results are independent of the method used to generate HO2 and atomic oxygen. At 299 K, the result is (6.1 + or - 0.4) x 10 to the -11th cu cm/molecule s. The temperature dependence expressed in Arrhenius form is (3.1 + or - 0.3) x 10 to the -11th exp/(+200 + or - 28)/T/. The error limits given are twice the standard deviation; overall experimental error is estimated to be + or - 25%.
In this paper we develop the theory of high-order radiation boundary conditions for wave propagation problems. In particular, we study the convergence of sequences of time-local approximate conditions to the exact boundary condition, and subsequently estimate the error in the solutions obtained using these approximations. We show that for finite times the Pade approximants proposed by Engquist and Majda lead to exponential convergence if the solution is smooth, but that good long-time error estimates cannot hold for spatially local conditions. Applications in fluid dynamics are also discussed.
Sufficient conditions for a weak relative minimum of a form of the Bolza problem of variational calculus are derived. The variational problem considered includes arbitrary numbers of constraints on the initial and terminal points, but assumes no control constraints or interior point constraints. Testing of the second-order conditions requires the backward integration of fewer matrix elements than in the case of previously published sets of conditions. The derivation, though lengthy, is felt to be simpler in concept than previous derivations. The application of these conditions is demonstrated in the context of a simple geometric problem.
Boundary and interface conditions for high order finite difference methods applied to the constant coefficient Euler and Navier-Stokes equations are derived. The boundary conditions lead to strict and strong stability. The interface conditions are stable and conservative even if the finite difference operators and mesh sizes vary from domain to domain. Numerical experiments show that the new conditions also lead to good results for the corresponding nonlinear problems.
Absolute rate parameters for the reaction of atomic hydrogen with hydrazine H + N2H4 yields H2 + N2H3 have been determined in a direct manner using flash photolysis of dilute mixtures of hydrazine in helium and time dependent observation of H via resonance fluorescence. By measuring the H-atom decay under pseudo-first-order conditions, the bimolecular rate constant K sub 1 was obtained over the temperature range 228-400 K. The data were fitted with good linearity to the Arrhenius expression K sub 1 = (9.87 plus or minus 1.17) x 10 to the -12th exp(-2380 plus or minus 100/RT) cu cm/molecule/s. The data were shown to be free of any contributions from secondary reactions involving H as a reactant or product.
Experiments were conducted in a conventional fast flow apparatus to study the rate constant and the activation energy for the reaction O(3P) + ClONO2. The air-afterglow technique was used for the detection of oxygen atoms under pseudo first order conditions. A value of (3.4 plus or minus 0.6) x 10 to the -12th exp/(-840 plus or minus 60)T/cu cm/molecule/s was obtained for the rate constant. The observed rate is sufficiently slow so that oxygen atom attack is of minor importance in the stratosphere compared to photolysis.
The absolute rate constant of the reaction between chlorine (2P) atoms and hydrogen peroxide was determined from 298 to 424 K, using the discharge flow resonance fluorescence technique. Pseudo-first-order conditions were used with hydrogen peroxide in large excess. A fast flow-sampling procedure limited hydrogen peroxide decomposition to less than 5% over the temperature range studied. At 298 K, the rate constant is (4.1 plus or minus 0.2) x 10 to the minus 13th cu cm/molecule-sec.