Maximum survival capability of an aircraft in a severe windshear

This paper is concerned with guidance strategies and piloting techniques which ensure near-optimum performance and maximum survival capability in a severe windshear. The take-off problem is considered with reference to flight in a vertical plane. In addition to the horizontal shear, the presence of a downdraft is assumed.First, six particular guidance schemes are considered, namely: constant alpha guidance; maximum alpha guidance; constant velocity guidance; constant absolute path inclination guidance; constant rate of climb guidance; and constant pitch guidance. Among these, it is concluded that the best one is the constant pitch guidance.Next, in an effort to improve over the constant pitch guidance, three additional trajectories are considered: the optimal trajectory, which minimizes the maximum deviation of the absolute path inclination from a reference value, while employing global information on the wind flow field; the gamma guidance trajectory, which is based on the absolute path inclination and which approximates the behavior of the optimal trajectory, while employing local information on the windshear and the downdraft; and the simplified gamma guidance trajectory, which is the limiting case of the gamma guidance trajectory in a severe windshear and which does not require precise information on the windshear and the downdraft.The essence of the simplified gamma guidance trajectory is that it yields a quick transition to horizontal flight. Comparative numerical experiments show that the survival capability of the simplified gamma guidance trajectory is superior to that of the constant pitch trajectory and is close to that of the optimal trajectory.Next, with reference to the simplified gamma guidance trajectory, the effect of the feedback gain coefficient is studied. It is shown that larger values of the gain coefficient improve the survival capability in a severe windshear; however, excessive values of the gain coefficient are undesirable, because they result in larger altitude oscillations and lower average altitude.Finally, with reference to the simplified gamma guidance trajectory, the effect of time delays is studied, more specifically, the time delay ₁ in reacting to windshear onset and the time delay ₂ in reacting to windshear termination. While time delay ₂ has little effect on survival capability, time delay ₁ appears to be critical in the following sense: smaller values of ₁ correspond to better survival capability in a severe windshear, while larger values of ₁ are associated with a worsening of the survival capability in a severe windshear.This research was supported by NASA-Langley Research Center, Grant No. NAG-1-516, and by Boeing Commercial Airplane Company.     

Discontinuous self-excited systems: An asymptotic analysis

In this paper we derive a uniformly valid asymptotic approximation of the periodic solution of a self-excited system given by the differential equation and β₁,β₂, are positive constants. By uniformly valid asymptotic approximation we mean that no secular terms are present. Our procedure makes use of a nonlinear change of independent variable that transforms the problem from one in which the discontinuities are ? dependent to one in which the discontinuities are ? independent. We obtain an asymptotic approximation up to order ? of the periodic solution and an asymptotic approximation up to order ?² of the period. Some comparisons between our asymptotic results and numerically derived results are given. Application of our technique to other examples of self-excited systems is discussed. The equation is investigated in detail.     

Inverse Spectral Transform for the Nonlinear Evolution Equation Generating the Davey-Stewartson and Ishimori Equations

A 2 + 1-dimensional nonlinear differential equation integrable by the inverse-spectral-transform method with the quartet operator representation is proposed. This GL(2, C)-valued chiral-field-type equation is the generating (prototype) equation for the Davey-Stewartson and Ishimori equations. It coincides with the nonlinear equation for the Davey-Stewartson eigenfunction ψ_DS. The initial-value problem for this equation is solved by the techniques for the and the nonlocal Riemann-Hilbert problem. The classes of exact solutions with the functional parameters and exponential-rational solutions are constructed by the method. The static lump solution in the case α = i and the exponentially localized solution at α = i are found. Other similar examples of nonlinear integrable equations in 2 + 1 and 1 + 1 dimensions are discussed.     

Optimal control for quasi-linear systems with small parameters

We consider the controlled systems where the non-linear term is multiplied by a small scalar parameter ε. In the class of these quasi-linear systems, we shall determine the control and optimal trajectory which minimizes the index of performance represented by quadratics functionals. The initial and final conditions are specified and the final time is free. The presence of the small parameter leads to an approximate solution of the formulated problem of optimum. Thus, the zeroth-order solution is obtained for ε=0. The first order solution results by using the sweep method which determines the perturbation of the control and of the state variable on the optimal neighboring trajectory.     

Estimating the complexity of a class of path-following methods for solving linear programs by curvature integrals

In this paper we study a particular class of primal-dual path-following methods which try to follow a trajectory of interior feasible solutions in primal-dual space toward an optimal solution to the primal and dual problem. The methods investigated are so-called first-order methods: each iteration consists of a long step along the tangent of the trajectory, followed by explicit recentering steps to get close to the trajectory again. It is shown that the complexity of these methods, which can be measured by the number of points close to the trajectory which have to be computed in order to achieve a desired gain in accuracy, is bounded by an integral along the trajectory. The integrand is a suitably weighted measure of the second derivative of the trajectory with respect to a distinguished path parameter, so the integral may be loosely called a curvature integral.     

The discrete maximum principle and the ɛ-technique

A complete proof of the -maximum principle for discrete-time system is given. In proving the -maximum principle, the general linearization of the system equations about the optimum trajectory is avoided. Therefore, the requirements for the system equations are different from those of earlier works. It is shown that the -maximum principle under some mild conditions does approach the general discrete maximum principle and that the -maximum principle is always in a strong form. Thus, if is sufficiently small, the -problem can approximate the solution of the original problem and the difficulties inherent in abnormal problems can be avoided. It is also pointed out that the indeterminancy in the singular control problem can be avoided by using the -technique.This research was supported in part by AFOSR Grant No. AF-AFOSR-F44620-68-C-0023 and NSF Grant No. GK-5608.     

The Asymptotic Solution of a Connection Problem of a Second Order Ordinary Differential Equation

The solutions of the equation are discussed in the limit ρ → 0. The solutions which oscillate about ? |t| as t → ∞ have asymptotic expansions whose leading terms are where Ã₊, , Ã_?, and are constants. The connection problem is to determine the asymptotic expansion at + ∞. In other words, we wish to find (Ã₊, ) as functions of Ã_? and The nonlinear solutions with Ã_± not small are analyzed by using the method of averaging. It is shown that this method breaks down for small amplitudes. In this case a solution can be obtained on [0, ∞) as a small amplitude perturbation about the exact nonoscillating solution W(t) whose asymptotic expansion is This is a solution of (1) which corresponds to Ã₊ ≡ 0 in (2). A quantity which determines the scale of the small amplitude response is ?W'(0). This quantity is found to be exponentially small. The determination of this constant is shown to reduce to a solution of the equation for the first Painlevé transcendent. The asymptotic behavior of the required solution is determined by solving an integral equation.     

Controller design to optimize a composite performance measure

This paper studies a mixed objective problem of minimizing a composite measure of thel ₁, ₂, andl -norms together with thel -norm of the step response of the closed loop. This performance index can be used to generate Pareto-optimal solutions with respect to the individual measures. The problem is analyzed for discrete-time, single-input single-output (SISO), linear time-invariant systems. It is shown via Lagrange duality theory that the problem can be reduced to a convex optimization problem with a priori known dimension. In addition, continuity of the unique optimal solution with respect to changes in the coefficients of the linear combination that defines the performance measure is estabilished.This research was supported by the National Science Foundation under Grants No. ECS-92-04309, ECS-92-16690 and ECS-93-08481.     

A class of filter problems,II: Solution for absolutely continuous kernels

A variational problem generated by a class of filter problems in which the autocorrelation function of the noise is assumed to be expressible as a convolution of a functionK with itself is considered. The variational problem involves the minimization of the square of theL ₁[0, ] norm of a functiony plus the square of theL ₂[0, ] norm of a functionG. The functionsy andG are related by a renewal equation which also involvesK. Existence and uniqueness of the solution was established earlier. In this paper, the solution is shown to have compact support.This research was supported by NSF Grants Nos. MCS-79-27137 and MCS-78-01106.     

Convergence of the Lambert-McLeod trajectory solver and of the celf method

Summary A trajectory problem is an initial value problemd y/dt=f(y),y(0)= where the interest lies in obtaining the curve traced by the solution (the trajectory), rather than in finding the actual correspondanc between values of the parametert and points on that curve. We prove the convergence of the Lambert-McLeod scheme for the numerical integration of trajectory problems. We also study the CELF method, an explicit procedure for the integration in time of semidiscretizations of PDEs which has some useful conservation properties. The proofs rely on the concept of restricted stability introduced by Stetter. In order to show the convergence of the methods, an idea of Strang is also employed, whereby the numerical solution is compared with a suitable perturbation of the theoretical solution, rather than with the theoretical solution itself.     

Optimal attitude control of a rigid body

Being mainly interested in the control of satellites, we investigate the problem of maneuvering a rigid body from a given initial attitude to a desired final attitude at a specified end time in such a way that a cost functional measuring the overall angular velocity is minimized.This problem is solved by applying a recent technique of Jurdjevic in geometric control theory. Essentially, this technique is just the classical calculus of variations approach to optimal control problems without control constraints, but formulated for control problems on arbitrary manifolds and presented in coordinate-free language. We model the state evolution as a differential equation on the nonlinear state spaceG=SO(3), thereby completely circumventing the inevitable difficulties (singularities and ambiguities) associated with the use of parameters such as Euler angles or quaternions. The angular velocities _k about the body's principal axes are used as (unbounded) control variables. Applying Pontryagin's Maximum Principle, we lift any optimal trajectorytg^*(t) to a trajectory onT ^*G which is then revealed as an integral curve of a certain time-invariant Hamiltonian vector field. Next, the calculus of Poisson brackets is applied to derive a system of differential equations for the optimal angular velocitiest _k ^* (t); once these are known the controlling torques which need to be applied are determined by Euler's equations.In special cases an analytical solution in closed form can be obtained. In general, the unknown initial values _k ^* (t₀) can be found by a shooting procedure which is numerically much less delicate than the straightforward transformation of the optimization problem into a two-point boundary-value problem. In fact, our approach completely avoids the explicit introduction of costate (or adjoint) variables and yields a differential equation for the control variables rather than one for the adjoint variables. This has the consequence that only variables with a clear physical significance (namely angular velocities) are involved for which gooda priori estimates of the initial values are available.     

Regularity Properties of Optimal Controls with Application to Discrete Approximation

This paper is directed to the analysis of regularity properties of optimal solutions for a nonlinear control problem with convex control constraints. Since the problem formulation is given typically in L -terms, we introduce first the essential limit set as a tool for the local investigation of L -functions. Under second-order conditions of the coercivity type on the solution, a structural result is obtained characterizing the local behavior of the optimal control by means of Lipschitz continuous functions. Further, the consequences for certain discrete approximations are discussed; for uniquely solvable problems, we show the Lipschitz continuity of the optimal control.     

Asymptotic Analysis and Solution of a Finite-Horizon H ∞ Control Problem for Singularly-Perturbed Linear Systems with Small State Delay

A finite-horizon H state-feedback control problem for singularly-perturbed linear time-dependent systems with a small state delay is considered. Two approaches to the asymptotic analysis and solution of this problem are proposed. In the first approach, an asymptotic solution of the singularly-perturbed system of functional-differential equations of Riccati type, associated with the original H problem by the sufficient conditions of the existence of its solution, is constructed. Based on this asymptotic solution, conditions for the existence of a solution of the original H problem, independent of the small parameter of singular perturbations, are derived. A simplified controller with parameter-independent gain matrices, solving the original H problem for all sufficiently small values of this parameter, is obtained. In the second approach, the original H problem is decomposed into two lower-dimensional parameter-independent H subproblems, the reduced-order (slow) and the boundary-layer (fast) subproblems; controllers solving these subproblems are constructed. Based on these controllers, a composite controller is derived, which solves the original H problem for all sufficiently small values of the singular perturbation parameter. An illustrative example is presented.     

Finite element approximation of viscoelastic fluid flow: Existence of approximate solutions and error bounds

Summary We study a finite element approximation of viscoelastic fluid flow obeying an Oldroyd B type constitutive law. The approximate stress, velocity and pressure are respectivelyP ₁ discontinuous,P ₂ continuous,P ₁ continuous. We use the method of Lesaint-Raviart for the convection of the extra stress tensor. We suppose that the continuous problem admits a sufficiently smooth and sufficiently small solution. We show by a fixed point method that the approximate problem has a solution and we give an error bound.This work has been supported in part by the GDR CNRS 901 Rhéologie der polymères fondus.     

Existence and Limiting Behavior of Trajectories Associated with P0-equations

Given a continuous P₀-function F : Rⁿ Rⁿ, we describe a method of constructing trajectories associated with the P₀-equation F(x) = 0. Various well known equation-based reformulations of the nonlinear complementarity problem and the box variational inequality problem corresponding to a continuous P₀-function lead to P₀-equations. In particular, reformulations via (a) the Fischer function for the NCP, (b) the min function for the NCP, (c) the fixed point map for a BVI, and (d) the normal map for a BVI give raise to P₀-equations when the underlying function is P₀. To generate the trajectories, we perturb the given P₀-function F to a P-function F(x, ); unique solutions of F(x, ) = 0 as varies over an interval in (0, ) then define the trajectory. We prove general results on the existence and limiting behavior of such trajectories. As special cases we study the interior point trajectory, trajectories based on the fixed point map of a BVI, trajectories based on the normal map of a BVI, and a trajectory based on the aggregate function of a vertical nonlinear complementarity problem.     

Asymptotic profile of solutions to a non‐linear dissipative evolution system with conservation

We investigate the asymptotic profile to the Cauchy problem for a non‐linear dissipative evolution system with conservational form (1) provided that the initial data are small, where constants α, ν are positive satisfying ν²<4α(1 ? α), α<1. In (J. Phys. A 2005; 38 :10955–10969), the global existence and optimal decay rates of the solution to this problem have been obtained. The aim of this paper is to apply the heat kernel to examine more precise behaviour of the solution by finding out the asymptotic profile. Precisely speaking, we show that, when time t → ∞ the solution and solution in the L^p sense, where G(t, x) denotes the heat kernel and is determined by the initial data and the solution to a reformulated problem obtained in Section 3, β is related to ?₊ and ?_? which are determined by (41) in Section 4. The numerical simulation is presented in the end. The motivation of this work thanks to Nishihara (Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity. Z. Angew Math Phys 2006; 57 : 604–614). Copyright © 2007 John Wiley & Sons, Ltd.     

Uniqueness theorems for a nonlinear Tricomi problem and the related evolution problem

We consider an initial-boundary value problem for the non-linear evolution equation in a cylinder Q_t = Ω × (0, t), where T[u] = yu_xx + u_yy is the Tricomi operator and l(u) a special differential operator of first order. In [10] we proved the existence of a generalized solution of problem (1) and the existence of a generalized solution of the corresponding stationary boundary value problem (non-linear Tricomi problem) In this paper we give sufficient conditions for the uniqueness of these solutions.     

A cosine operator approach to modelingL 2(0,T; L 2 (Γ))—Boundary input hyperbolic equations

This paper investigates the regularity properties of the solution of a second-order hyperbolic equation defined over a bounded domain with boundary , under the action of a boundary forcing term inL ₂(0,T; L ₂()). Both Dirichlet and Neumann nonhomogeneous cases are considered. A functional analytic model based on cosine operator functions is presented, which provides an input-solution formula to be interpreted in appropriate topologies. With the help of this model, it is shown, for example, that the solution of the nonhomogeneous Dirichlet problem is inL ₂(0,T; L ₂()), when is either a parallelepiped or a sphere, while the solution of the nonhomogeneous Neumann problem is inL ₂(0,T; H ^3/4-e()) when is a parallelepiped and inL ₂(0,T; H ^2/3() when is a sphere. The Dirichlet case for general domains is studied by means of pseudodifferential operator techniques.This research was supported in part by the Air Force Office of Scientific Research under Grant AFOSR-78-3350 (1st author) and Grant AFOSR-77-3338 (2nd author).This research was performed while the author was visiting the Department of System Science, University of California, Los Angeles.     

An approximation method for the cauchy problem to the three-dimensional equation of vorticity transport

The vorticity problem (V₀) is shown to have (at least) locally in time a unique classical solution. For numerical purposes global solvability is desired. So by suitable operations we proceed to a family of modified vorticity problems (V_?), ? > 0, possessing a unique classical solution globally in time. For (V_?) a constructive approximation method is introduced. This procedure yields a sequence (ω) of approximate vorticity fields, converging to the global solution of (V_?) and to the local solution of (V₀).