Pullback attractors for a model of weakly concentrated aqueous polymer solution motion with a rheological relation satisfying the objectivity principle

The qualitative dynamics of weak solutions to a nonautonomous model of polymer solution motion (with a rheological relation satisfying the objectivity principle) is studied using the theory of pullback attractors of trajectory spaces. For this purpose, the existence of weak solutions is proved for the model under study, a family of trajectory spaces is defined, trajectory and minimal pullback attractors are introduced, and their existence is proved.     

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.     

Singular Arcs in the Optimal Evasion Against a Proportional Navigation Vehicle

The two-dimensional optimal evasion problem against a proportional navigation pursuer is analyzed using a nonlinear model. The velocities of both players have constant modulus, but change in direction. The problem is to determine the time-minimum trajectory (disengagement) or time-maximum trajectory (evasion) of the evader while moving from the assigned initial conditions to the final conditions. A maximum principle procedure allows one to reduce the optimal control problem to the phase portrait analysis of a system of two differential equations. The qualitative features of the optimal process are determined.     

Traveling waves of an elliptic-hyperbolic model of phase transitions via varying viscosity-capillarity

We consider an elliptic-hyperbolic model of phase transitions and we show that any Lax shock can be approximated by a traveling wave with a suitable choice of viscosity and capillarity. By varying viscosity and capillarity coefficients, we can cover any Lax shock which either remains in the same phase, or admits a phase transition. The argument used in this paper extends the one in our earlier works. The method relies on LaSalle?s invariance principle and on estimating attraction region of the asymptotically stable of the associated autonomous system of differential equations. We will show that the saddle point of this system of differential equations lies on the boundary of the attraction region and that there is a trajectory leaving the saddle point and entering the attraction region. This gives us a traveling wave connecting the two states of the Lax shock. We also present numerical illustrations of traveling waves.     

Positional strengthenings of the maximum principle and sufficient optimality conditions

We derive nonlocal necessary optimality conditions, which efficiently strengthen the classical Pontryagin maximum principle and its modification obtained by B. Ka?kosz and S. ?ojasiewicz as well as our previous result of a similar kind named the “feedback minimum principle.” The strengthening of the feedback minimum principle (and, hence, of the Pontryagin principle) is owing to the employment of two types of feedback controls “compatible” with a reference trajectory (i.e., producing this trajectory as a Carath´eodory solution). In each of the versions, the strengthened feedback minimum principle states that the optimality of a reference process implies the optimality of its trajectory in a certain family of variational problems generated by cotrajectories of the original and compatible controls. The basic construction of the feedback minimum principle—a perturbation of a solution to the adjoint system—is employed to prove an exact formula for the increment of the cost functional. We use this formula to obtain sufficient conditions for the strong and global minimum of Pontryagin’s extremals. These conditions are much milder than their known analogs, which require the convexity in the state variable of the functional and of the lower Hamiltonian. Our study is focused on a nonlinear smooth Mayer problem with free terminal states. All assertions are illustrated by examples.     

Necessary conditions for functional differential inclusions

We consider an optimal control problem in which the dynamic equation and cost function depend on the recent past of the trajectory. The regularity assumed in the basic data is Lipschitz continuity with respect to the sup norm. It is shown that, for a given optimal solution, an adjoint arc of bounded variation exists that satisfies an associated Hamiltonian inclusion. From this result, known smooth versions of the Pontryagin maximum principle for hereditary problems can be easily derived. Problems with Euclidean endpoint constraints are also considered.     

A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control

Simple directly verifiable conditions are derived under whichthere exists a state trajectory satisfying a specified stateconstraint. The conclusions differ from the kind of informationprovided by viability and invariance-type theorems, insofaras an estimate is provided of the distance (in the supremumnorm) of the state trajectory from a specified state trajectory,in terms of the degree to which the specified state trajectoryviolates the state constraint. The constructions involved inthe existence proof are related to ones previously employedby Soner to establish continuity properties of a value functionarising in infinite-horizon state-constrained optimal control,but the accompanying analysis contains refinements to ensurea sub-Lipschitz property of the value function considered here.It is expected that this existence result will have a numberof implications for systems theory and optimal control. Herewe show how it leads to a non-degenerate maximum principle forstate-constrained optimal-control problems, in situations wherethe standard necessary conditions give no useful informationabout minimizers. Email: rampazzo{at}pdmat1.unipd.it Email: r.vinter{at}ic.ac.uk     

Irregular trajectories in vakonomic mechanical systems

In his works, V.V. Kozlov proposed a mathematical model for the dynamics of a mechanical system with nonintegrable constraints, which was called vakonomic. In contrast to the then conventional nonholonomic model, trajectories in the vakonomic model satisfy necessary conditions for a minimum in a variational problem with equality constraints. We consider the so-called irregular case of this variational problem, which was not covered by Kozlov, when the trajectory is a singular point of the constraints and the necessary minimum conditions based on the Lagrange principle make no sense. This situation is studied using the theory of abnormal problems developed by the first author. As a result, the classical necessary minimum conditions are strengthened and developed to this class of problems.     

Integral curves from noisy diffusion MRI data with closed-form uncertainty estimates

We present a method for estimating the trajectories of axon fibers through diffusion tensor MRI (DTI) data that provides theoretically rigorous estimates of trajectory uncertainty. We develop a three-step estimation procedure based on a kernel estimator for a tensor field based on the raw DTI measurements, followed by a plug-in estimator for the leading eigenvectors of the tensors, and a plug-in estimator for integral curves through the resulting vector field. The integral curve estimator is asymptotically normal; the covariance of the limiting Gaussian process allows us to construct confidence ellipsoids for fixed points along the curve. Complete trajectories of fibers are assembled by stopping integral curve tracing at locations with multiple viable leading eigenvector directions and tracing a new curve along each direction. Unlike probabilistic tractography approaches to this problem, we provide a rigorous, theoretically sound model of measurement uncertainty as it propagates from the raw MRI data, to the tensor field, to the vector field, to the integral curves. In addition, trajectory uncertainty is estimated in closed form while probabilistic tractography relies on sampling the space of tensors, vectors, or curves. We show that our estimator provides more realistic trajectory uncertainty estimates than a more simplified prior approach for closed-form trajectory uncertainty estimation due to Koltchinskii et al. (Ann Stat 35:1576–1607, 2007) and a popular probabilistic tractography method due to Behrens et al. (Magn Reson Med 50:1077–1088, 2003) using theory, simulation, and real DTI scans.     

Properties of the Lagrange multipliers in the Pontryagin maximum principle for optimal control problems with state constraints

We study the Pontryagin maximum principle for an optimal control problem with state constraints. We analyze the continuity of a vector function µ (which is one of the Lagrange multipliers corresponding to an extremal by virtue of the maximum principle) at the points where the extremal trajectory meets the boundary of the set given by the state constraints. We obtain sufficient conditions for the continuity of µ in terms of the smoothness of the extremal trajectory.     

Robust control of chaos in Chua’s circuit based on internal model principle

The paper treats the question of robust control of chaos in Chua’s circuit based on the internal model principle. The Chua’s diode has polynomial non-linearity and it is assumed that the parameters of the circuit are not known. A robust control law for the asymptotic regulation of the output (node voltage) along constant and sinusoidal reference trajectories is derived. For the derivation of the control law, the non-linear regulator equations are solved to obtain a manifold in the state space on which the output error is zero and an internal model of the k-fold exosystem (k = 3 here) is constructed. Then a feedback control law using the optimal control theory or pole placement technique for the stabilization of the augmented system including the Chua’s circuit and the internal model is derived. In the closed-loop system, robust output node voltage trajectory tracking of sinusoidal and constant reference trajectories are accomplished and in the steady state, the remaining state variables converge to periodic and constant trajectories, respectively. Simulation results are presented which show that in the closed-loop system, asymptotic trajectory control, disturbance rejection and suppression of chaotic motion in spite of uncertainties in the system are accomplished.     

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.     

Hölder Continuity of Adjoint States and Optimal Controls for State Constrained Problems

We investigate Hölder regularity of adjoint states and optimal controls for a Bolza problem under state constraints. We start by considering any optimal solution satisfying the constrained maximum principle in its normal form and we show that whenever the associated Hamiltonian function is smooth enough and has some monotonicity properties in the directions normal to the constraints, then both the adjoint state and optimal trajectory enjoy Hölder type regularity. More precisely, we prove that if the state constraints are smooth, then the adjoint state and the derivative of the optimal trajectory are Hölder continuous, while they have the two sided lower Hölder continuity property for less regular constraints. Finally, we provide sufficient conditions for Hölder type regularity of optimal controls.     

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.     

Heterogeneous credit portfolios and the dynamics of the aggregate losses

We study the impact of contagion in a network of firms facing credit risk. We describe an intensity based model where the homogeneity assumption is broken by introducing a random environment that makes it possible to take into account the idiosyncratic characteristics of the firms. We shall see that our model goes behind the identification of groups of firms that can be considered basically exchangeable. Despite this heterogeneity assumption our model has the advantage of being totally tractable. The aim is to quantify the losses that a bank may suffer in a large credit portfolio. Relying on a large deviation principle on the trajectory space of the process, we state a suitable law of large numbers and a central limit theorem useful for studying large portfolio losses. Simulation results are provided as well as applications to portfolio loss distribution analysis.     

Mathematical modeling and trajectory planning of mobile manipulators with flexible links and joints

This paper is concerned with mathematical modeling and optimal motion designing of flexible mobile manipulators. The system is composed of a multiple flexible links and flexible revolute joints manipulator mounted on a mobile platform. First, analyzing on kinematics and dynamics of the model is carried out then; open-loop optimal control approach is presented for optimal motion designing of the system. The problem is known to be complex since combined motion of the base and manipulator, non-holonomic constraint of the base and highly non-linear and complicated dynamic equations as a result of the flexible nature of both links and joints are taken into account. In the proposed method, the generalized coordinates and additional kinematic constraints are selected in such a way that the base motion coordination along the predefined path is guaranteed while the optimal motion trajectory of the end-effector is generated. This method by using Pontryagin’s minimum principle and deriving the optimality conditions converts the optimal control problem into a two point boundary value problem. A comparative assessment of the dynamic model is validated through computer simulations, and then additional simulations are done for trajectory planning of a two-link flexible mobile manipulator to demonstrate effectiveness and capability of the proposed approach.     

A variational principle for domino tilings

We formulate and prove a variational principle (in the sense of thermodynamics) for random domino tilings, or equivalently for the dimer model on a square grid. This principle states that a typical tiling of an arbitrary finite region can be described by a function that maximizes an entropy integral. We associate an entropy to every sort of local behavior domino tilings can exhibit, and prove that almost all tilings lie within (for an appropriate metric) of the unique entropy-maximizing solution. This gives a solution to the dimer problem with fully general boundary conditions, thereby resolving an issue first raised by Kasteleyn. Our methods also apply to dimer models on other grids and their associated tiling models, such as tilings of the plane by three orientations of unit lozenges.

     

Nonuniqueness set and the problem of time-optimal motions in a velocity field

We study the problem of time-optimal transfer to the origin for motions described by the law $\dot x = v(x) + u(t)$ , where v: ?² → ?² is a smooth vector field and the control u satisfies the inequality ‖u(t)‖ ≤ u ₀. We introduce the notion of nonuniqueness set for problems with continuous optimal controls. By using the Pontryagin maximum principle, we single out a family of trajectories that lead to the origin and may be optimal. The nonuniqueness set intersects this family and cuts away the nonoptimal part from each trajectory. We show that the nonuniqueness set for a plane-parallel velocity field with symmetric profile is a ray lying on the symmetry axis. In the case of a nonsymmetric profile, we construct a Cauchy problem whose trajectory is the nonuniqueness set.     

Justification of the adiabatic principle for hyperbolic Ginzburg-Landau equations

We study the adiabatic limit in hyperbolic Ginzburg-Landau equations which are the Euler-Lagrange equations for the Abelian Higgs model. By passing to the adiabatic limit in these equations, we establish a correspondence between the solutions of the Ginzburg-Landau equations and adiabatic trajectories in the moduli space of static solutions, called vortices. Manton proposed a heuristic adiabatic principle stating that every solution of the Ginzburg-Landau equations with sufficiently small kinetic energy can be obtained as a perturbation of some adiabatic trajectory. A rigorous proof of this result has been found recently by the first author.     

The Wave Function of the Lyman-Alpha Photon Part I. The Wave Function in the Angular and Linear Momentum Representations

To the best of the writer‘s knowledge no one has given the wave function of a photon emitted in an atomic, molecular, or nuclear transition. In the present paper we derive the wave function in the angular momentum and linear momentum representations for the photon emitted by a non-relativistic hydrogen atom, when the electron of the atom falls from the first excited state to the ground state. This is the simplest transition which produces a photon. A two level model for the atom is used, in which the lower level (the ground state energy) is associated with a non-degenerate wave function and the upper level (the energy of the first excited state) is associated with wave functions corresponding to the four-fold degeneracy of that state. We use a generalization of Dirac‘s method for finding the eigenfunctions in resonance scattering. We find the exact solution of the two-level problem using the exact matrix elements of the interaction. The calculations are finite without renormalization. In the next paper we shall introduce the [(x)\vec] \vec{x} -representation and thereby obtain the "position", "shape", and "trajectory" of the photon.