Synthesis of a control in the problem of the time-optimal transfer of a point mass to a specified position with zero velocity

The problem of the time-optimal control of the motion of a point mass by means of a force of bounded modulus is considered. It is required that the point be transferred from an arbitrary state of motion to the origin of the system of coordinates with zero velocity. By introducing self-similar conjugate variables, the solution of the two-point problem can be successfully reduced to a search for the optimal root of a certain function, specified analytically. A complete solution of the control problem in the form of a synthesis is obtained using mathematical modelling methods. The feedback coefficients along the unit vectors of the position and velocity vectors are found and a control algorithm and a Bellman function are constructed. Examples using practical initial data are presented.     

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.     

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.     

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.     

Reduction of an elastic system to a prescribed state by use of a boundary force

A solution is constructed for the problem of optimal control of the motion of a distributed elastic system, using a lumped boundary force. The system's state is described by a constant-coefficient hyperbolic equation. A general case of arbitrary initial and final distributions is examined. Questions of control by lumped and distributed forces are discussed.     

Optimal control in a model of the motion of a viscoelastic medium with objective derivative

In this paper we consider the Jeffreys model of the motion of a viscoelastic incompressible medium with the Yaumann derivative. Within this model we study the optimal control problem for the right-hand sides of the initial boundary value problem. We prove the existence of the optimal strong solution.     

On an integrable case of perturbed keplerian motion

A general solution of a differential vector equation of perturbed Keplerian motion is derived for the case when the position vector and perturbing acceleration vector are collinear. A variable change is employed, in which the new independent variable is expressed in terms of the initial values of the phase variables and time, using the elliptical Jacobi function. The two-point boundary value problem for the initial equation is reduced to the Cauchy problem, A parametric representation is obtained for the regularized trajectory of motion of a material point under the action of a central force.     

Optimal control for steady flows of the Jeffreys fluids with slip boundary condition

Some optimal control problem is studied for the stationary motion equations for the Jeffreys medium with the slip condition of the Navier type on the boundary. The control parameter is the external force. The existence of a weak solution is proven minimizing a given cost functional. Some properties of the solution set of the optimization problem are established.     

Optimal control of the viscous Camassa–Holm equation

This paper studies the problem of optimal control of the viscous Camassa–Holm equation. The existence and uniqueness of weak solution to the viscous Camassa–Holm equation are proved in a short interval. According to variational method, optimal control theories and distributed parameter system control theories, we can deduce that the norm of solution is related to the control item and initial value in the special Hilbert space. The optimal control of the viscous Camassa–Holm equation under boundary condition is given and the existence of optimal solution to the viscous Camassa–Holm equation is proved.     

Similarity and dimensional analysis in the plane problem of the propagation of a vertical hydraulic fracture crack in an elastic medium

The problem of the growth of a vertical hydraulic fracture crack in an unbounded elastic medium under the pressure produced by a viscous incompressible fluid is studied qualitatively and by numerical methods. The fluid motion is described in the approximation of lubrication theory. Near the crack tip a fluid-free domain may exist. To find the crack length, Irwin’s fracture criterion is used. The symmetry groups of the equations describing the hydraulic fracture process are studied for all physically meaningful cases of the degeneration of the problem with respect to the control parameters. The condition of symmetry of the system of equations under the group of scaling and time-shift transformations enables the self-similar variables and the form of the time dependence of the quantities involved in the problem to be found. It is established that at non-zero rock pressure the well-known solution of Spence and Sharp is an asymptotic form of the initial-value problem, whereas the solution of Zheltov and Khristianovich is a limiting self-similar solution of the problem. The problem of the formation of a hydraulic fracture crack taking into account initial data is solved using numerical methods, and the problem of arriving at asymptotic mode is investigated. It is shown that the solution has a self-similar asymptotic form for any initial conditions, and the convergence of the exact solutions to the asymptotic forms is non-uniform in space and time.     

Controllability and Time Optimal Control for Low Reynolds Numbers Swimmers

The aim of this paper is to tackle the self-propelling at low Reynolds number by using tools coming from control theory. More precisely we first address the controllability problem: “Given two arbitrary positions, does it exist “controls” such that the body can swim from one position to another, with null initial and final deformations?”. We consider a spherical object surrounded by a viscous incompressible fluid filling the remaining part of the three dimensional space. The object is undergoing radial and axi-symmetric deformations in order to propel itself in the fluid. Since we assume that the motion takes place at low Reynolds number, the fluid is governed by the Stokes equations. In this case, the governing equations can be reduced to a finite dimensional control system. By combining perturbation arguments and Lie brackets computations, we establish the controllability property. Finally we study the time optimal control problem for a simplified system. We derive the necessary optimality conditions by using the Pontryagin maximum principle. In several particular cases we are able to compute the explicit form of the time optimal control and to investigate the variation of optimal solutions with respect to the number of inputs.     

状态和控制均含时滞的受外界扰动的线性系统的最优控制

主要研究一类受外界持续扰动且状态和控制含不同时滞的线性系统的最优控制,首先通过变量代换,将系统化为控制不含时滞的滞后型微分系统.接着使用最优控制的极大值原理的必要条件,得到含超前和滞后项的两点边值问题.为了得到最优控制律的解析解,引进一个灵敏参数ε,得到两点边值问题序列,通过迭代法,得到最优控制律的解析解.并对外界扰动状态构造降维观测器,来实现最优控制律的物理可实现性.最后实例验证了上述方法的有效性.     

On the zero dynamics of one class of distributed discrete systems

The problem of optimal control of a linear dynamical system under set-membership uncertainty is studied: it is required to steer the system to the terminal set with a guarantee and to maximize the guaranteed value of the quality criterion. The sets of the initial and current a preposteriori distributions of the states of the dynamical system are introduced; they are used to determine a positional solution of the problem of optimal a preposteriori³ observation with the help of inaccurate measurements of input and output signals of the observation object by two measuring devices. The obtained solution is used for determining a positional solution of the optimal control problem under uncertainty. Depending on the amount of the information used, optimal closable and closed output feedbacks are determined. The method of quasiimplementation of optimal feedbacks by means of optimal estimators and a regulator producing real-time control actions is described. The results are illustrated by examples.     

Time-optimal steering of a point mass to a specified position with the required velocity

The planar problem of the time-optimal steering of a point mass to the origin of coordinates with a required velocity by means of a force of limited modulus is considered. The optimal control and the optimal time are investigated for different characteristic values of the positional radius vector and arbitrary values of the magnitude and direction of the initial velocity. The results are extended to the case of boundary conditions which are imposed on the magnitude and direction of the initial velocity in the form of inequalities and are transferred to a problem with control for a fixed time.     

Asymptotics of the optimal time in a time-optimal problem with two small parameters

A time-optimal problem of control of a small-mass point by a force of bounded magnitude in an unresisting medium is considered. An asymptotic expansion of the optimal time and optimal control is constructed with respect to two independent small parameters: the mass of the point and the perturbation of the initial conditions. It is shown that the asymptotics of the optimal time in this problem is complicated even for cases of general position.     

Optimal feedback control for a thermoviscoelastic model of Voigt fluid motion

A system of equations describing the motion of viscous fluids with polymer additives is considered. The system involves a temperature-dependent viscosity and a velocity-dependent external force, which is used as a control. The existence of an optimal feedback control for the system is proved.     

The quickest transfer of a multidimensional dynamic object onto the surface of an ellipsoid

The time-optimal control of motion of a multidimensional dynamic object is investigated. Both open-loop and feedback controls are considered. It is assumed that the force vector is constrained by a non-degenerate ellipsoid. The terminal set is described by the surface of an ellipsoid that allows degeneracy, and the terminal velocity is not fixed. The initial position of the object can be either outside or inside the ellipsoid. Using the maximum principle, the necessary and sufficient conditions for the optimality of the control are established in the form of a system of polynomial equations, the orders of which depend on the dimension of the problem and the degree of degeneracy of the terminal ellipsoid. Iterative procedures and a technique for continuation with respect to the parameters of the approximating ellipsoid in the admissable region of change of the phase vector are proposed. Control problems for the limiting shapes of the terminal ellipsoid are investigated. The qualitative effect of the discontinuity of the functional and the control defined as functions of the phase vector is established.     

The optimal quasi-stationary motion of a vibration-driven robot in a viscous medium

We consider a rectilinear quasi-stationary motion of a two-mass system in a viscous medium. The motion of the system as a whole occurs due to periodic movements of the internal mass relatively to the shell. The problem is to describe the law of motion of the internal mass that provides the minimum energy consumption with a specified average velocity of the shell. We propose an algorithm for solving the problem with any law of the resistance of the medium. We obtain the energy-optimal law of motion of a spherical shell in a viscous liquid.     

Optimal control of the viscous generalized Camassa–Holm equation

In this paper, we study the optimal control problem for the viscous generalized Camassa–Holm equation. We deduce the existence and uniqueness of weak solution to the viscous generalized Camassa–Holm equation in a short interval by using Galerkin method. Then, by using optimal control theories and distributed parameter system control theories, the optimal control of the viscous generalized Camassa–Holm equation under boundary condition is given and the existence of optimal solution to the viscous generalized Camassa–Holm equation is proved.     

Optimal control of a viscous heat-conducting gas flow

Under consideration is the problem of controlling the one-dimensional flow of a polytropic viscous heat-conducting ideal gas along an interval with a stationary boundary. The initial velocity and the velocity at the permeable stationary boundary are chosen as controls. We prove the existence of an optimal control, derive some necessary optimality conditions, and show that the solution set is compact.