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.     

Calculating the electric charges that shield from an external coaxial electric field on the surface of a conducting axially symmetric body

A method of solving a one-dimensional integral equation for finding charges on the surface of a conducting axially symmetric body is given. For the case of an ellipsoid of rotation in an electric field with polynomial values on the axis of symmetry, an exact solution is obtained. The axis of symmetry of the body and the axis of the external field coincide. A numerical algorithm based on a combination of a projective method and a method of iterative regularization for solving a Fredholm equation of the first kind is proposed. The projectors are chosen as B-splines. The charges calculated for an ellipsoid of rotation are close to the analytical ones.     

The existence of smooth solutions in a problem of the optimal control of the rotation of an axisymmetric rigid body

The problem of the existence of a solution in the problem of the optimal control of the rotation of an axisymmetric rigid body for the arbitrary case of angular velocity boundary conditions is studied. A square integrable functional, which is consistent with the symmetry of the rotating body and characterizes the power consumption, is chosen as the criterion. The principal moment of the applied external forces serves as the control and the time of termination of a manoeuvre can be both specified as well as free. In the case of a specified termination time, it is shown that the solution (control) belongs to the class of infinitely-differentiable functions of time. The reasoning is based on the use of the singularities of the structure of the differential equations and the possibility of reducing the initial problem to two successive variational problems. The existence of a solution of the first of these problems in the class of square integrable functions is proved using the Cauchy–Bunyakovskii inequality. The second problem reduces to a search for the minimum of a functional which is weakly lower semi-continuous on a weakly compact set and the existence of its solution in the same class of functions follows from the Weierstrass theorem. The required conclusion concerning the smoothness of the solution of the optimal control problem is obtained from the necessary conditions of Pontryagin's maximum principle. In the case of a free termination time, one of the minimizing sequence can be constructed and it can be shown that, in the general case, there is no solution in the class of measurable controls.     

Analytical solutions in the problem of the optimal control of the rotation of an axisymmetric body

The problem of the optimal control of the rotation of an axisymmetric rigid body is investigated. An integral functional, characterizing the power consumption to carry out a manoeuvre is chosen as the criterion, and the boundary conditions for the angular velocity vector are arbitrary. The principal moment of the applied external forces serves as the control. The necessary conditions of the maximum principle are used to solve the problem in the case of a fixed completion time. New non-trivial first integrals are established for the canonical system of direct and conjugate differential equations obtained, which enable the set of all extremals to be parametrized. Hence, the optimal-control problem is reduced to a problem of non-linear mathematical programming. It is shown that there cannot be more than two different solutions in the latter, and a family of boundary conditions is established when the optimum rotation is determined in a uniquely explicit form.     

The wave field generated by a centre of rotation in an unbounded elastic medium with a semi-infinite conical crack

The dynamic stress intensity factor at the edge of a semi-infinite conical crack when the medium is loaded by a non-stationary centre of rotation is determined. A centre of rotation is understood to be a set of four forces of equal magnitude that act in the same plane and form pairs having the same direction of rotation.¹ If the magnitude of these forces is time dependent, i.e., their application is non-stationary, they form a non-stationary centre of rotation. The solution of the problem required the use of methods of integral transformations and discontinuous solutions, which reduced the problem to an integral differential equation in Laplace transform space. The combined use of the orthogonal polynomial method and time discretization to solve the equation enabled a formula for the stress intensity factor to be obtained.     

A family of analytic extremals in problems of the optimal control of the rotation of a body

The problem of the optimal control of the rotation of an absolutely rigid body about the centre of mass is investigated. The main purpose of the control is to vary the angular velocity vector from its initial value to the required terminal value in a finite time so that the manoeuvre would require the smallest power consumption, which is characterized by an integral quadratic functional. The principal torque produced by the external forces applied to the body serves as the control. The change in orientation is not taken into account, i.e., the problem of the overspeed–braking control of the body, is studied. A new class of analytic extremals based on the use of space-time deformations of the solutions of the dynamical Euler equations for the free rotation of a rigid body is described. Sufficient conditions for the existence of such extremals for all types of symmetries are presented.     

On the controllability of a robot arm

Considered is the rotation of a robot arm or rod in a horizontal plane about an axis through the arm's fixed end and driven by a motor whose torque is controlled. The model was derived and investigated computationally by Sakawa and co-authors in [7] for the case that the arm is described as a homogeneous Euler beam. The resulting equation of motion is a partial differential equation of the type of a wave equation which is linear with respect to the state, if the control is fixed, and non-linear with respect to the control. Considered is the problem of steering the beam, within a given time interval, from the position of rest for the angle zero into the position of rest under a certain given angle. At first we show that, for every L²-control, there is exactly one (weak) solution of the initial boundary value problem which describes the vibrating system without the end condition. Then we show that the problem of controllability is equivalent to a non-linear moment problem. This, however, is not exactly solvable. Therefore, an iteration method is developed which leads to an approximate solution of sufficient accuracy in two steps. This method is numerically implemented and demonstrated by an example. © 1998 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Optimal control of backward stochastic heat equation with Neumann boundary control and noise

This paper considers a stochastic control problem in which the dynamic system is a controlled backward stochastic heat equation with Neumann boundary control and boundary noise and the state must coincide with a given random vector at terminal time. Through defining a proper form of the mild solution for the state equation, the existence and uniqueness of the mild solution is given. As a main result, a global maximum principle for our control problem is presented. The main result is also applied to a backward linear-quadratic control problem in which an optimal control is obtained explicitly as a feedback of the solution to a forward–backward stochastic partial differential equation.     

A new approach to the classical Stokes flow problem: Part I Methodology and first-order analytical results

The problem of determining the axisymmetric Stokes flow past an arbitrary body, the boundary shape of which can be represented by an analytic function, is examined by developing an exact method. An appropriate nonorthogonal coordinate system is introduced, and it is shown that the Hilbert space to which the stream function belongs is spanned by the set of Gegenbauer polynomials based on the physical argument that the drag on a body should be finite. The partial differential equation of the original problem is then reduced to two simultaneous vector differential equations. By the truncation of this infinite-dimensional system to the one-dimensional subspace, an explicit analytic solution to the Stokes equation valid for all bodies in question is obtained as a first approximation.     

Oscillatory integrals generated by the Ostrovsky equation

The Ostrovsky equation governs the propagation of long nonlinear surface waves in the presence of rotation. It is related to the Korteweg-de Vries (KdV) and the Kadomtsev-Petviashvili models. KdV can be obtained from the equation in question when the rotation parameter γ equals zero. A fundamental solution of the Cauchy problem for the linear Ostrovsky equation is presented in the form of an oscillatory Fourier integral. Another integral representation involving Airy and Bessel functions is derived for it. It is shown that its asymptotic expansion as γ → 0 contains the KdV fundamental solution as the zero term. The Airy transform is used to establish some of its properties. Higher-order asymptotics for γ → 0 on a bounded time interval are obtained for both the fundamental solution and the solution of the linear Cauchy problem for the Ostrovsky equation.     

Numerical analysis of a contact problem including bone remodeling

In this work, a contact problem between an elastic body and a deformable obstacle is numerically studied. The bone remodeling of the material is also taken into account in the model and the contact is modeled using the normal compliance contact condition. The variational problem is written as a nonlinear variational equation for the displacement field, coupled with a first-order ordinary differential equation to describe the physiological process of bone remodeling. An existence and uniqueness result of weak solutions is stated. Then, fully discrete approximations are introduced based on the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are obtained, from which the linear convergence of the algorithm is derived under suitable regularity conditions. Finally, some 2D numerical results are presented to demonstrate the behavior of the solution.     

Oscillatory integrals generated by the Ostrovsky equation

The Ostrovsky equation governs the propagation of long nonlinear surface waves in the presence of rotation. It is related to the Korteweg-de Vries (KdV) and the Kadomtsev-Petviashvili models. KdV can be obtained from the equation in question when the rotation parameter γ equals zero. A fundamental solution of the Cauchy problem for the linear Ostrovsky equation is presented in the form of an oscillatory Fourier integral. Another integral representation involving Airy and Bessel functions is derived for it. It is shown that its asymptotic expansion as γ → 0 contains the KdV fundamental solution as the zero term. The Airy transform is used to establish some of its properties. Higher-order asymptotics for γ → 0 on a bounded time interval are obtained for both the fundamental solution and the solution of the linear Cauchy problem for the Ostrovsky equation. Received: November 23, 2004; revised: March 13, 2005 Research is supported by US Department of Defense, under grant No. DAAD19-03-1-0204     

Calculation of the rupture life of mechanical rubber goods

The stress-strain relation for an aging material is obtained from an analysis of a four-element model of a viscoelastic body with variable coefficients. In this formulation the problem of calculating the rupture life is divided into four steps: a) solution of the boundary-value problem of the theory of elasticity of an incompressible material; b) calculation of the stationary thermal field; c) solution of the rheological equation at the danger point; d) solution of the criterial equation for the local fracture time. An example of the calculation of the high-temperature rupture life of a rubber cord under constant load is given. The agreement with experiment is satisfactory.Riga Polytechnic Institute. Translated from Mekhanika Polimerov, No. 1, pp. 91–95, January–February, 1976.     

二维升力体的非线性振动

本文给出了二维升力体非线性振动的解析解答。     

Development of the visual image in electrophotography

The development of the visual image in an electrophotographic copying machine is studied on basis of a mathematical model proposed by Spence. The resulting moving boundary problem is solved in time steps. An electrostatic problem, which has to be treated at each step, is formulated as an integral equation, which is solved numerically. The solution is found for various values of the parameters involved and the dependence of the solution on them is discussed.     

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 problem of the time-optimal turning of a manipulator

Two types of manipulator that perform three-dimensional motions are considered, and the control problem in which the manipulator rotation is performed in minimum time is studied. The rate of rotation of a rigid body about an axis rises as the moment of inertia about this axis falls. Manipulator control amounts to a problem of the rotation of a system of rigid bodies about an axis. In addition to the angle of rotation, there is a further controlled coordinate, whose variation can vary the moment of inertia about the axis. Assuming that the moment of inertia can be stantaneously “frozen” (that pulse control signals are possible), the in-time-optimal control modes were found in /1, 2/, (see also Akulenko, L.D. et al., “Optimization of the control modes of manipulation robots”, Preprint 218, In-t. Problem Mekhaniki Akad. Nauk SSSR, Moscow 1983). In these modes, the rotation, occurs in the entire time interval with minimum moment of inertia about the axis of rotation. The rotation when there are constraints on the control (pulse control signals are not permitted) was considered in /3/. Numerical studies there led to the false conclusion that, in the optimal motion, with a finite number of control switchings, the moment of inertia is also a minimum throughout the time interval. Below, for a set of extreme configurations, a control is constructed for the two types of manipulator, which satisfies the Pontryagin maximum principle, when there are constraints on the control signals. During its rotation the manipulator section then performs oscillations about a position corresponding to minimum moment of inertia about the axis of rotation. It is shown that the motion considered in /3/, which contains a singular mode with minimum moment of inertia, is not optimal. The motion which satisfies the maximum principle is compared with it. There can be a singular mode in the optimal motion /4/ only when the number of control switchings is infinite.     

An approximate solution of the axisymmetric contact problem for an elastic sphere

An axisymmetric, fractionally non-linear contact problem for an elastic sphere with a priori unknown boundary of the contactarea is considered. An integral equation for determining the density of the contact pressures is constructed taking account of the shear displacements of the boundary points of the elastic body. An approximate solution, which refines the equations of Hertz' theory, is constructed in the case of a small contact area.     

Bifurcation modes in a nonlinear optical system with distributed field rotation

We consider a model of a nonlinear optical system with distributed field rotation described by a functional-differential diffusion equation. An existence theorem is proved for periodical spatially nonhomogeneous traveling-wave solutions, which are generated from a spatially homogeneous stationary solution by an Andronov-Hopf (cycle-generating) bifurcation. A series expansion of the solution in powers of a small parameter is obtained and a stability condition is given. Simulation results are used to discuss the properties of the model. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996, pp. 89–99.