The quickest transfer of a spatial dynamic object into a circular domain

The spatial problem of the time-optimal transfer of a point mass by a limited force onto a terminal set in the form of a circle without fixing the final velocity is investigated. The optimal modes of motion are constructed and investigated for arbitrary initial values of the three-dimensional position and velocity vectors using the maximum principle. The governing relations are obtained in the form of fourth-order and eighth-order algebraic equations for the minimum time of motion, which enable the dependence on the initial data to be investigated constructively. The qualitative features of the solution due to a jump discontinuity in the minimum time of motion, which lead to jumps in the control vector, are established. The problem is solved approximately by perturbation methods for the cases of motion close to singular ones. A complete investigation of the control problem for the motion of an object in the plane of a circle and close to it is presented using an original numerical-analytical approach.     

Time-optimal steering of a point mass onto the surface of a sphere at zero velocity

The problem of the time-optimal steering of a point mass onto the surface of a sphere at zero velocity, by a control force of bounded magnitude is investigated. It is assumed that the surface is penetrable and that the point may “land” on the sphere either from the outside or from the inside. An optimal control, in the open-loop and feedback form of trajectories the optimal time and the Bellman function are constructed using Pontrya'gin's maximum principle. The multidimensional boundary-value problem is reduced, by introducing self-similar variables, to the numerical solution of an algebraic equation of degree four and a transcendental equation. It is shown that the boundary-value problem degenerates when the optimal trajectory is nearly linear; a solution of the synthesis problem is constructed in the degenerate case. The efficacy of the approach proposed here is illustrated by specific examples in which families of trajectories are computed, and by an analysis of control regimes.     

The time-optimal transfer of a perturbed dynamical object to a given position

The problem of transferring a dynamical object (a point mass) of arbitrary dimensional to a required position in a time-optimal manner by means of a force of limited modulus is solved. The velocity of the object at the final instant is not specified. It is assumed that an arbitrary known perturbation, with a magnitude strictly less than that of the control, acts on the controlled system. For clarity when analysing the optimal controlled motion, considerable attention is paid to the case of a steady perturbation. A constructive procedure for finding the optimal response time and the control is developed for arbitrary permissible values of the governing parameters. The Bellman function and the feedback control are constructed over the whole of the phase space. The structural properties of the solution are established and an asymptotic analysis is carried out by small-parameter methods. The extremal directions of the perturbation vector and the corresponding response time and optimal control are found. A modification of the time-optimal problem to the case of a non-stationary perturbation is presented and the basic properties of the optimal solution are investigated.     

The antiplane problem of the motion of a point shear load over a two-layer elastic base at a constant velocity

The problem of modelling the motion of a force disturbance in an elastic medium that is heterogeneous over its depth is investigated. It is in an antiplane formulation in a moving system of coordinates that all possible versions of the ratio of the velocity of motion of the surface point shear load to the velocities of the shear waves in the layers of the two-layer elastic base are examined. Cases of a subsonic regime (SBR) in the upper and lower layers, of a supersonic regime (SPR) in the upper layer and an SBR in the lower layer, and of an SBR in the upper layer and an SPR in the lower layer are studied using the Fourier transform and the theory of residues. The last two cases are extremely interesting from the mathematical point of view, as here, on the boundary between the layers, the solutions of elliptic and hyperbolic equations meet, and previously unknown features arise in the displacements that,it seems, should also occur in the solution of the corresponding plane problem. The case of an SPR in the upper and lower layers is investigated using a special method for successive allowance for the incident, reflected and refracted shock wave fronts. In all cases, expressions are obtained for the displacements in the layers, and their characteristic features are investigated.     

双正则函数向量的带Haseman位移带共轭值的线性及非线性边值问题

首先利用积分方程的方法和Arzela-Ascoli定理讨论了实Clifford分析中双正则函数向量的带Haseman位移带共轭值的非线性边值问题解的存在性及其积分表达式,其次利用压缩映射原理解决了其线性边值问题解的存在唯一性及其积分表达式.     

On the orbital stability of planar periodic motions of a rigid body in the Bobylev-Steklov case

We deal with the problem of orbital stability of pendulum-like periodic motions of a heavy rigid body with a fixed point. We suppose that a mass geometry corresponds to the Bobylev-Steklov case. The stability problem is solved in nonlinear setting. In the case of small amplitude oscillations and rotations with large angular velocities the small parameter can be introduced and the problem can be investigated analytically. In the case of unspecified oscillation amplitude or rotational angular velocity the problem is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients.     

Motion with a constant velocity modulus in a central gravitational field

The problem of the motion of a particle (point mass) with a constant velocity modulus in a Newtonian central gravitational field is investigated by two methods: using Lagrange's equations with a multiplier, and using the equations of dynamics proposed earlier [1] for systems with non-holonomic constraints that are non-linear with respect to velocities. A phase diagram of the motion is constructed. The structure of the trajectories as a function of the initial conditions is investigated. Formulae in the form of quadratures are obtained for calculating the time of motion along the trajectory and the angular distance of flight. A qualitative analysis of the properties of improper integrals expressing the angular distance is presented. These properties are illustrated by the results of a numerical investigation. The possibility of carrying out elementary manoeuvres in the vicinity of an attracting centre are analysed.     

The problem of the time-optimal control of spacecraft reorientation

The use of Pontryagin's maximum principle to solve spacecraft motion control problems is demonstrated. The problem of the optimal control of the spatial reorientation of a spacecraft (as a rigid body) from an arbitrary initial angular position to an assigned final angular position in the minimum rotation time is investigated in detail. The case in which velocity parameters of the motion are constrained is considered. An analytical solution of the problem is obtained in closed form using the method of quaternions, and mathematical expressions for synthesizing the optimal control programme are given. The kinematic problem of spacecraft reorientation is solved completely. A design scheme for solving the maximum principle boundary-value problem for arbitrary turning conditions and inertial characteristics of the spacecraft is given. A solution of the problem of the optimal control of spatial reorientation for a dynamically symmetrical spacecraft is presented in analytical form (to expressions in elementary functions). The results of mathematical modelling of the motion of a spacecraft under optimal control, which confirm the practical feasibility of the control algorithm developed, are given. Estimates have shown that the turn time of modern spacecraft with a constrained magnitude of the angular momentum can be reduced by 15–25% compared with conventional reorientation methods. The greatest effect is achieved for turns through large angles (90° or more) when the final rotation vector is equidistant from the longitudinal axis and the transverse plane of the spacecraft.     

Bringing a non-linear manoeuvringobject to the optimal position in the shortest time

A different game problem with two players (cars), in which one player (car) pursues the other, is considered. The roles of theplayers are fixed, and the functional to be minimized (for player I) and maximixed (for player II) is the maximum value of a given scalar non-negative function (the performance index) of the phase vector along the trajectory of the dynamical system over a fairly long time interval. A zero value of the performance index corresponds to the situation in which the pursuer is behind the evader at a given distance from it, and the velocity vectors are codirectional and lie on the same straight line. A detailed investigation is presented of the special case in which the car being pursued is at rest, and the pursuer is moving in the plane at a velocity of constant magnitude subject to a certain constraint on its turning radius. The game ends when the car is moving in a circle of given radius, in which case its velocity vector must point toward the centre of the circle. The relations of the Pontryagin maximum principle characterizing optimal open-loop controls are written out and analysed. The main result of the paper is the synthesis of an optimal feedback control.     

The energy-optimal motion of a vibration-driven robot in a resistive medium

The rectilinear motion of a two-mass system in a resistive medium is considered. The motion of the system as a whole occurs by longitudinal periodic motion of one body (the internal mass) relative to the other body (the shell). The problem consists of finding the periodic law of motion of the internal mass that ensures velocity-periodic motion of the shell at a specified average velocity and minimum energy consumption. The initial problem reduces to a variational problem with isoperimetric conditions in which the required function is the velocity of the shell. It is established that, with optimal motion, the shell velocity is a piecewise-constant time function taking two values (a positive value and a negative value). The magnitudes of these velocities and the overall size of the intervals in which they are taken are uniquely defined, while the optimal motion itself is non-uniquely defined. The simplest optimal motion, for which the period is divided into two sections – one with a positive velocity and the other with a negative velocity of motion of the shell – is investigated in detail. It is shown that, among all the optimal motions, this simplest motion is characterized by the maximum amplitude of oscillations of the internal mass relative to the shell. © Elsevier Ltd. All rights reserved.     

Modelling of the controlled motion ofa point on a plane

A solution of the problem of feedback control of the motion of a point on a plane is presented. The equations of the controlprogramme (the objective) are set up as a system of differential equations with a given set of singular trajectories in the domain of admissible positions of the controlled point, as well as a given topological structure of the partition into trajectories. These equations define the vector field of velocities of the programmed motions of the point and are used to find the corresponding control forces.     

Solution of a Multidimensional Tropical Optimization Problem Using Matrix Sparsification

A complete solution is proposed for the problem of minimizing a function defined on vectors with elements in a tropical (idempotent) semifield. The tropical optimization problem under consideration arises, for example, when we need to find the best (in the sense of the Chebyshev metric) approximate solution to tropical vector equations and occurs in various applications, including scheduling, location, and decision-making problems. To solve the problem, the minimum value of the objective function is determined, the set of solutions is described by a system of inequalities, and one of the solutions is obtained. Thereafter, an extended set of solutions is constructed using the sparsification of the matrix of the problem, and then a complete solution in the form of a family of subsets is derived. Procedures that make it possible to reduce the number of subsets to be examined when constructing the complete solution are described. It is shown how the complete solution can be represented parametrically in a compact vector form. The solution obtained in this study generalizes known results, which are commonly reduced to deriving one solution and do not allow us to find the entire solution set. To illustrate the main results of the work, an example of numerically solving the problem in the set of three-dimensional vectors is given.     

Characterizations of the Solution Sets of Convex Programs and Variational Inequality Problems

For a convex program in a normed vector space with the objective function admitting the Gateaux derivative at an optimal solution, we show that the solution set consists of the feasible points lying in the hyperplane whose normal vector equals the Gateaux derivative. For a general continuous convex program, a feasible point is an optimal solution iff it lies in a hyperplane with a normal vector belonging to the subdifferential of the objective function at this point. In several cases, the solution set of a variational inequality problem is shown to coincide with the solution set of a convex program with its dual gap function as objective function, while the mapping involved can be used to express the above normal vectors.The research was supported by the National Science Council of the Republic of China. The authors are grateful to the referees for valuable comments and constructive suggestions.     

On a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow

Summary. The aim of this work is to study a decoupled algorithm of a fixed point for solving a finite element (FE) problem for the approximation of viscoelastic fluid flow obeying an Oldroyd B differential model. The interest for this algorithm lies in its applications to numerical simulation and in the cost of computing. Furthermore it is easy to bring this algorithm into play. The unknowns are the viscoelastic part of the extra stress tensor, the velocity and the pressure. We suppose that the solution is sufficiently smooth and small. The approximation of stress, velocity and pressure are resp. discontinuous, continuous, continuous FE. Upwinding needed for convection of , is made by discontinuous FE. The method consists to solve alternatively a transport equation for the stress, and a Stokes like problem for velocity and pressure. Previously, results of existence of the solution for the approximate problem and error bounds have been obtained using fixed point techniques with coupled algorithm. In this paper we show that the mapping of the decoupled fixed point algorithm is locally (in a neighbourhood of ) contracting and we obtain existence, unicity (locally) of the solution of the approximate problem and error bounds. Received July 29, 1994 / Revised version received March 13, 1995     

On the dynamic Markov-Dubins problem: From path planning in robotics and biolocomotion to computational anatomy

Andrei Andreyevich Markov proposed in 1889 the problem (solved by Dubins in 1957) of finding the twice continuously differentiable (arc length parameterized) curve with bounded curvature, of minimum length, connecting two unit vectors at two arbitrary points in the plane. In this note we consider the following variant, which we call the dynamic Markov-Dubins problem (dM-D): to find the time-optimal C ² trajectory connecting two velocity vectors having possibly different norms. The control is given by a force whose norm is bounded. The acceleration may have a tangential component, and corners are allowed, provided the velocity vanishes there. We show that for almost all the two vectors boundary value conditions, the optimization problem has a smooth solution. We suggest some research directions for the dM-D problem on Riemannian manifolds, in particular we would like to know what happens if the underlying geodesic problem is completely integrable. Path planning in robotics and aviation should be the usual applications, and we suggest a pursuit problem in biolocomotion. Finally, we suggest a somewhat unexpected application to “dynamic imaging science”. Short time processes (in medicine and biology, in environment sciences, geophysics, even social sciences?) can be thought as tangent vectors. The time needed to connect two processes via a dynamic Markov-Dubins problem provides a notion of distance. Statistical methods could then be employed for classification purposes using a training set.     

Evasion of a fixed sphere by a dynamical object driven by a bounded force

A solution is obtained of the problem of synthesizing the control of the motion of a dynamical object (a point mass) evading a fixed spherical obstacle under the action of a bounded force. The set of all points for which evasion is possible is constructed in phase space (of arbitrary dimension), and control modes are constructed for bounded (fixed) and unbounded time intervals. The characteristics of the optimal motion, in particular, the time and minimum distance, are determined for specific initial data. The qualitative properties of the controlled motion are established.     

Optimal impulsive space trajectories based on linear equations

The problem of minimizing the total characteristic velocity of a spacecraft having linear equations of motion and finitely many instantaneous impulses that result in jump discontinuities in velocity is considered. Fixed time and fixed end conditions are assumed. This formulation is flexible enough to allow some of the impulses to be specifieda priori by the mission planner. Necessary and sufficient conditions for solution of this problem are found without using specialized results from control theory or optimization theory. Solution of the two-point boundary-value problem is reduced to a problem of solving a specific set of equations. If the times of the impulses are specified, these equations are at most quadratic. Although this work is restricted to linear equations, there are situations where it has potential application. Some examples are the computation of the velocity increments of a spacecraft near a real or fictitious satellite or space station in a circular or more general Keplerian orbit. Another example is the computation of maneuvers of a spacecraft near a libration point in the restricted three-body problem.This project was supported by the 1988 NASA/ASEE Faculty Fellowship Program at the California Institute of Technology and the Jet Propulsion Laboratory. The work was performed in the Advanced Projects Group, Section 312, Jet Propulsion Laboratory, Pasadena, California.     

Conditionally periodic motions in the attraction field of a rotating triaxial ellipsoid

Conditionally periodic solutions are constructed in the neighbourhood of previously derived steady-state solutions of the problem of the motion of a material point in the attraction field of a rotating triaxial ellipsoid, when the average motion of the material point and the ellipsoid angular velocity of rotation are commensurable.     

Rolling of a non-homogeneous ball over a sphere without slipping and twisting

Consider the problem of rolling a dynamically asymmetric balanced ball (the Chaplygin ball) over a sphere. Suppose that the contact point has zero velocity and the projection of the angular velocity to the normal vector of the sphere equals zero. This model of rolling differs from the classical one. It can be realized, in some approximation, if the ball is rubber coated and the sphere is absolutely rough. Recently, J. Koiller and K. Ehlers pointed out the measure and the Hamiltonian structure for this problem. Using this structure we construct an isomorphism between this problem and the problem of the motion of a point on a sphere in some potential field. The integrable cases are found.      

The stability of a non-autonomous functional-differential equation relative to part of the variables

The asymptotic stability and instability of the trivial solution of a functional-differential equation of delay type relative to part of the variables are investigated using limit equations and a Lyapunov function whose derivative is sign-definite. The theorems thus obtained are used to solve the problem of stabilizing mechanical control systems with delayed feedback. As examples, solutions of problems of the uniaxial and triaxial stabilization of the rotational motion of a rigid body with a delay in the control system are presented.