Dichotomic Basis Research Articles

Manifold-Following Approximate Solution of Completely Hypersensitive Optimal Control Problems

The solution to a completely hypersensitive optimal control problem may shadow trajectories on the stable and unstable manifolds of an equilibrium point in the state---costate phase space. If such shadowing occurs, the solution of the Hamiltonian boundary value problem, that constitutes the first-order necessary conditions for optimality, can be approximated as a composite of an initial segment on the stable manifold leading to the equilibrium point and a final segment on the unstable manifold departing from the equilibrium point. Using a dichotomic basis, the Hamiltonian vector field can be decomposed into stable and unstable components, and the unspecified boundary conditions for the initial and terminal segments can be determined such that the initial and final conditions are, respectively, on the stable and unstable manifolds of the equilibrium point. In this paper, we propose and justify the use of finite-time Lyapunov vectors to construct an approximate dichotomic basis and develop a corresponding manifold-following solution approximation method. The method is illustrated on two examples and shown to be more accurate than a similar method that uses eigenvectors of the frozen-time linearized dynamics.

An immunohistochemically positive E-cadherin status is not always predictive for a good prognosis in human breast cancer

Background:In primary breast cancers dichotomic classification of E-cadherin expression, according to an arbitrary cutoff, may be inadequate and lead to loss of prognostic significance or contrasting prognostic indications. We aimed to assess the prognostic value of high and low E-cadherin levels in a consecutive case series (204 cases) of unilateral node-negative non-lobular breast cancer patients with a 8-year median follow-up and that did not receive any adjuvant therapy after surgery.Methods:Expression of E-cadherin was investigated by immunohistochemistry and assessed according to conventional score (0, 1+, 2+, 3+). Multiple correspondence analysis was used to visualise associations of both categorical and continuous variables. The impact of E-cadherin expression on patients outcome was evaluated in terms of event-free survival curves by the Kaplan–Meier method and proportional hazard Cox model.Results:Respect to intermediate E-cadherin expression values (2+), high (3+) or low (0 to 1+) E-cadherin expression levels had a negative prognostic impact. In fact, both patients with a low-to-nil (score 0 to 1+) expression level of E-cadherin and patients with a high E-cadherin expression level (score 3+) demonstrated an increased risk of failure (respectively, hazard ratio (HR)=1.71, confidence interval (CI)=0.72–4.06 and HR=4.22, CI=1.406–12.66) and an interesting association with young age.Conclusions:The findings support the evidence that high expression values of E-cadherin are not predictive for a good prognosis and may help to explain conflicting evidence on the prognostic impact of E-cadherin in breast cancer when assessed on dichotomic basis.

Scales, fields, and a problem of Hurewicz

Menger's basis property is a generalization of \sigma -comp\-actness and admits an elegant combinatorial interpretation. We introduce a general combinatorial method to construct non \sigma -compact sets of reals with Menger's property. Special instances of these constructions give known counterexamples to conjectures of Menger and Hurewicz. We obtain the first explicit solution to the Hurewicz 1927 problem, that was previously solved by Chaber and Pol on a dichotomic basis. The constructed sets generate nontrivial subfields of the real line with strong combinatorial properties, and most of our results can be stated in a Ramsey-theoretic manner. Since we believe that this paper is of interest to a diverse mathematical audience, we have made a special effort to make it self-contained and accessible.

Range Maximization for Emergency Landing After Engine Cutoff

A pilot who experiences engine cutoff when flying at an envelope point of high velocity and low altitude needs as much range and altitude as possible for safe landing. An algorithm is presented that converts the velocity excess into altitude and range in an optimal way. Such an algorithm can be implemented after engine cutoff as the first stage of an emergency landing. The optimization problem is identified as a two-timescale problem composed of a short-duration initial boundary layer, followed by a long segment dominated by slow behavior. The slow segment represents the classical quasi-steady-state glide, whereas the boundary layer determines the fast dynamic phase in which the excess of the dynamic pressure is converted into the steady-state dynamic pressure while reaching the highest altitude. The boundary-layer section is solved by the dichotomic basis method and applied to the F-16, and the ensuing solution is compared to the exact one. The match between the two is very good. with timescale separation based on three-timescale separation: The velocity and path angle were assumed to be faster than the altitude and range, and the path angle was assumed to be faster than the ve- locity. Such a separation produces a main boundary layer that keeps the altitude on hold while considering the dynamic fast change of the velocity and path angle. The main boundary layer has an internal sublayer that keeps the velocity on hold, changing the path angle first. This formulation defines boundary layers such that each one has only one active variable, thus, enabling an analytical solution of the boundary layers that produces an open-loop solution of the trajectory and a feedback law for the control. The ensuing solu- tions were compared to the exact ones, and the matching between the approximated solutions and the exact numerical solutions were studied. The open-loop solutions showed good agreement for those cases where the initial dynamic pressure was much higher than the stabilized one. The feedback solutions were reasonable except for induced oscillations that appeared when the boundary layer inter- cepted the steady-state segment. The oscillations result because the path-angle time constant is not faster than that of the velocity when approaching the stabilized section. In this paper, the same maximum range problem is treated, but as a two-time-constant problem instead of a three-time-constant problem. The path angle and the velocity time constants are con- sidered comparable, but faster than the altitude and range. Under this assumption, numerical solutions based on the dichotomic basis method 1 are derived. The dichotomic basis approach's objective is to develop an indirect solution method for nonlinear two-timescale optimal control problems, a solution method based on underlying geometric structure of the trajectories of the Hamiltonian system but not requiring a priori knowledge of this structure. Two capabilities are required: splitting the Hamiltonian system into slow and fast parts and splitting the fast part into contracting and expanding parts. The expanding part is suppressed in the initial boundary layer, thus, enabling an easier convergence of the solution.

Riccati Dichotomic Basis Method for Solving Hypersensitive Optimal Control Problems

The solution of a completely hyper-sensitive Hamiltonian boundary-value problem (HBVP) arising in optimal control can be accurately approximated by concatenating an initial boundary-layer segment, an equilibrium segment, and a terminal boundary-layer segment. The approximations in the boundary-layers are essentially similar; only the direction of time is different. Consequently, it is sufficient to focus on the initial boundary-layer. In this paper it is shown that by using a dichotomic basis, the Hamiltonian vector field can be decomposed into its stable and unstable components. Furthermore, it is shown that a dichotomic basis can be constructed from by solving a Riccati differential equation. Using this result, a successive approximation procedure arises from which from which the approximate solution in the initial boundary-layer can be computed. The successive approximation procedure is illustrated on a problem in supersonic aircraft trajectory optimization and its range of applicability is discussed.

Fuzzy-Logic-Based Closed-Loop Optimal Law for Homing Missiles Guidance

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Application of a Dichotomic Basis Method to Performance Optimization of Supersonic Aircraft

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Eigenvector approximate dichotomic basis method for solving hyper-sensitive optimal control problems

Eigenvector approximate dichotomic basis method for solving hyper-sensitive optimal control problems

Dichotomic basis approach to solving hyper-sensitive optimal control problems

As a step toward developing a general method for determining the underlying geometric structure of two time-scale optimally controlled nonlinear systems, we define a degenerate class of two time-scale optimal control problems, called completely hypersensitive problems, and propose an indirect solution method for this class of problems. The method uses a dichotomic basis to split the Hamiltonian vector field into its stable and unstable components. An accurate approximation to the optimal solution is constructed by matching the initial and terminal boundary-layer segments with the equilibrium solution. A variation of the method for the case of an approximate dichotomic basis is also developed and is applied to a nonlinear spring-mass problem. The challenging problem of determining a dichotomic basis or a sufficiently accurate approximation to one is discussed only briefly, but some potential solutions are identified.

Eigenvector approximate dichotomic basis method for solving hyper- sensitive optimal control problems

The dichotomic basis method is further developed for solving completely hyper-sensitive Hamiltonian boundary value problems arising in optimal control. For this class of problems, the solution can be accurately approximated by concatenating an initial boundary-layer segment, an equilibrium segment, and a terminal boundary-layer segment. Constructing the solution in this composite manner alleviates the sensitivity. The method uses a dichotomic basis to decompose the Hamiltonian vector field into its stable and unstable components, thus allowing the missing initial conditions needed to specify the initial and terminal boundary-layer segments to be determined from partial equilibrium conditions. The dichotomic basis reveals the phase-space manifold structure in the neighbourhood of the optimal solution. The challenge is to determine a sufficiently accurate approximation to a dichotomic basis. In this paper we use an approximate dichotomic basis derived from local eigenvectors. An iterative scheme is proposed to handle the approximate nature of the basis. The method is illustrated on an example problem and its general applicability is assessed. Copyright © 1999 John Wiley & Sons, Ltd.

Eigenvector approximate dichotomic basis method for solving hyper‐ sensitive optimal control problems

Eigenvector approximate dichotomic basis method for solving hyper‐ sensitive optimal control problems

Eigenvector approximate dichotomic basis method for solving hyper‐ sensitive optimal control problems

Eigenvector approximate dichotomic basis method for solving hyper‐ sensitive optimal control problems