Classical Calculus Of Variations Research Articles

Фундаментальний аналіз та оптимізація (мінімізація) динамічних навантажень у пружних елементах механізмів підйому вантажу кранів. Частина І

The fundamental analysis and optimization (minimization) of dynamic loads in elastic elements (ropes) of load lifting mechanisms of cranes used in urban development is carried out. The basis of the analysis and methodology of optimization of motion modes of the mentioned mechanisms is the approach implemented in the classical calculus of variations, namely: the integral functional is minimized, which reflects undesirable properties of the mechanical system. Optimized modes of motion of the crane load lifting mechanism, which is represented in the paper by a mechanical system with a linear function of the position of its elements (the load on the rope and the drive mechanism). The significant influence of motion modes of exactly drive mechanisms on dynamic loads in elastic elements (ropes) of the load lifting mechanism of a city-building crane is shown. Usually separate typical modes of motion optimize only one certain property of the specified mechanism. In many practical cases, it is necessary to select modes of motion that optimize a set of properties of a machine or mechanism (respectively, a crane or a load lifting mechanism of a city-building crane). It is for the selection of such modes of motion of machines and mechanisms that dynamic criteria for their evaluation are needed. All existing criteria for evaluating machines and mechanisms can be conditionally divided into two classes: 1) positional (maximum values of kinematic characteristics of links, dynamic loads, amplitudes and frequencies of oscillations, dynamics coefficients, etc.); 2) integral (rms values of kinematic characteristics of links and dynamic loads, energy costs, etc.). In the work the fundamental analysis of the mentioned criteria is carried out and it is shown that for estimation of motion modes of hoisting machines, as well as mechanisms of cargo lifting cranes (boom, bridge, gantry, city cranes) it is necessary to use integral criteria, because they are the ones that estimate motion during the whole cycle. Integral criteria for evaluating urban construction cranes and load lifting mechanisms are presented, which fully reflect their dynamic properties, motion characteristics and, in particular, minimize dynamic loads on elastic elements (ropes). The results obtained in this study can be further used to clarify and improve the existing engineering methods of calculation of hoisting cranes used in urban planning, as well as their load lifting mechanisms, both at the design stages and in the modes of proper operation.

ANALYSIS OF DYNAMICS AND OPTIMISATION OF TRANSIENT MOTION MODES OF HOISTING MECHANISMS OF OVERHEAD URBAN CRANES

The research of operating modes of hoisting mechanisms and machines, in particular, overhead and city-building cranes is carried out. Based on dynamic analysis of operating modes of mechanisms of the mentioned machines the character of load occurrence in their elastic elements (ropes) is established. The synthesis of optimum transient modes of operation (start/braking) of city-building and overhead cranes is carried out. The models of L.D. Landau and N.A. Lobov for physical and mechanical analysis of motion of the specified machines are used, on the basis of which with attraction of methods of classical calculus of variations the laws of motion of a cargo and a cargo carriage of similar cranes which optimize (minimize) inevitably arising pendulum oscillations of a cargo on a rope (as in the period of start-up of the crane, and at its braking, up to full stop) are received analytically. The evaluation of motion modes in the work is carried out by using a criterion with sub-integral functions in the form of "energy" of accelerations of higher orders (third, fourth, etc.). Exactly such criteria are used in the selection of modes of motion of lifting mechanisms of cranes of bridge/bridge-building type, which have in their composition elastic elements (ropes). The above approach makes it possible to achieve continuity of functions of velocities and accelerations of various links of the system, which leads to a significant reduction in their oscillations. The analysis of modes of motion of the system, which is described by the main coordinate of motion and elastic coordinates, allowed one to establish the regularity of using the proposed criterion to minimize the elastic vibrations of individual links (ropes) of the crane hoisting mechanism. The results of this study can be further used to clarify and improve existing engineering methods for calculating loads in flexible/elastic elements (ropes) of bridge and city-building cranes both in the design of hoisting mechanisms and in the modes of their real operation when mechatronic control systems are used in controlling the movement of the load carriage.

Variatsionnye metody resheniya zadach, svyazannykh s iskusstvennym intellektom

We introduce the concepts of a learning algorithm, an objective function, a recognition system, a class of patterns, a training set, a reward algorithm, a finitely convergent algorithm, an adaptive control system, a control goal, control tactics, adaptation time, etc., related to the problem of artificial intelligence in the processes of learning and adaptation. The general problem of self-learning (unsupervised learning)—about the separation of sets—in terms of the classical calculus of variations is posed. The generality of the problem is due to the introduction of an additional time variable into the analysis. The problem is solved by determining extremal conditions under which the minimization of the overall average risk functional is achieved. Problems corresponding to nonfixed and fixed time intervals are considered. For these two cases, expressions are found for calculating variations in cost functionals. Necessary conditions are indicated for determining the extremal values of the self-learning process (separation of classes of a set of patterns) in time.

ANALYTICAL SUBSTANTIATION OF DETERMINATION OF INTEGRAL CHARACTERISTICS OF ROTOR SYSTEMS OF URBAN CRANES

The work is devoted to the actual problem of analytical substantiation of determination of integral characteristics of rotor systems of city-building cranes, knowledge of which is necessary for carrying out further balancing of their rotors on coasting (or in the process of start-up). It allows to refuse the use of contact vibration sensors and other special equipment for monitoring the balancing process. Besides, there is a possibility to optimize the driving torque (during start-up) and braking torque during coasting of the rotor system. The purpose of the work is to develop an algorithm for determining the integral characteristics of the rotor system at start-up/run-out at a special ratio of moment coefficients - components of the total moment of resistance to rotation. Approaches and models of mathematical physics and classical calculus of variations are used. The results obtained in the paper will be useful for clarification and improvement of engineering methods of calculation of rotor systems of construction/urban cranes operating in transient modes (start-up, coasting, reversing, etc.), both at the stages of their design and in the modes of real operation.

A variational theory for integral functionals involving finite-horizon fractional gradients

The center of interest in this work are variational problems with integral functionals depending on nonlocal gradients with finite horizon that correspond to truncated versions of the Riesz fractional gradient. We contribute several new aspects to both the existence theory of these problems and the study of their asymptotic behavior. Our overall proof strategy builds on finding suitable translation operators that allow to switch between the three types of gradients: classical, fractional, and nonlocal. These provide useful technical tools for transferring results from one setting to the other. Based on this approach, we show that quasiconvexity, which is the natural convexity notion in the classical calculus of variations, gives a necessary and sufficient condition for the weak lower semicontinuity of the nonlocal functionals as well. As a consequence of a general varGamma -convergence statement, we obtain relaxation and homogenization results. The analysis of the limiting behavior for varying fractional parameters yields, in particular, a rigorous localization with a classical local limit model.

DYNAMIC OPTIMIZATION OF LOADS, OCCURRING DURING THE BRAKING OF AN OVERHEAD CRANE TRAVEL GEAR: AN ANALYTICAL APPROACH

The paper proposes a model for determining the dynamic loads arising during thebraking of an overhead crane travel mechanism. By the methods of mathematical physics, classical variationalcalculus at application of the apparatus of the decision of the usual linear differential equations thelaws of movement of the mechanism of movement of the specified crane at which dynamic loadings in themechanism are minimized are analytically defined. This result gives an opportunity to carry out constructivemeasures for decrease in level of dynamic loading of the crane.In order to reduce the dynamic loads during the braking of the bridge crane movement mechanism,additional measures have been proposed, namely: 1) regulation of crane braking force; 2) reduction of bogieand crane weight by: a) application of steel structures in the form of trusses instead of solid wall ones; b) useof bent profiles for manufacturing of steel structures; c) application of lighter alloys; 3) reduction of the momentof inertia of the electric motor rotor of the crane travel mechanism by: a) using several motors with totalpower equal to the power of one drive motor, which reduces the moment of inertia of the rotor; b) usingmotors with lightweight rotors; 4) using couplings with variable stiffness.The results obtained in the work can be further used to refine and improve the existing engineeringmethods for calculating the braking systems of overhead crane travel mechanisms to minimize the dynamicloads in these mechanisms both at the stages of their design and in the modes of real operation.

MODELING AND OPTIMIZATION OF THE PROCESSES OF MOVEMENT AND ACCELERATION OF THE OVERHEAD CRANE TROLLEY IN THE MODE OF DAMPING UNCONTROLLED LOAD OSCILLATIONS

The paper deals with the modeling and optimization of the processes of movement and acceleration of a bridge crane trolley in the mode of damping uncontrolled oscillations of the load. For the dynamic system of a flat pendulum with vibration damping, which describes the oscillations of a bridge crane load on a flexible rope suspension in a separate vertical plane, it is proposed to use third-order time splines that model the motion and acceleration of the load suspension point in the horizontal direction of the trolley's movement. To determine the time dependence of the angle of deviation of the crane from the gravitational vertical, it is proposed to use the methods of classical calculus of variations (Euler-Poisson equation), which allow optimizing (minimizing) the value of this angle in the process of accelerating a trolley with a load suspended from the ropes of an overhead crane. An analytical solution to the problem of damping residual uncontrollable oscillations of the overhead crane load, which usually occur after full acceleration or braking of the load suspension point on the trolley, is obtained. To derive the dependencies, an analytical approach was used to calculate the value of the angle of deviation of the overhead crane's cargo rope from the gravitational vertical, depending on the acceleration and displacement of the load suspension point. The problem of loosening of a load moved by an overhead crane is considered and solved in a new way that allows to completely avoiding pendulum spatial oscillations of the load on a rope suspension. The mathematical apparatus of linear algebra (Kramer's rule, in particular) is used, which allows us to establish analytically the law of time motion of a rope with a load, the angle of deviation of which from the vertical takes minimum values in the process of acceleration of the cargo trolley.

DYNAMIC LOADS IN ELASNIC ELEMENTS/ROPES OF CONSTRUCTION CRANES ANALYSIS AND OPTIMIZATION

In the study, an analysis of the dynamic model of construction cranes’ load lifting mechanism, proposed by professor Loveikyn V.S. and others, was carried out. In order to determine the dynamic loads in the elastic rope, a dynamic model is used, in which all sections of the shafts are considered rigid, and only the cable is elastic. Bringing the engine rotor masses, brake pulley, gears, drum and load to the rope branchers that are wound on the drum, allows the original complex multi-mass tech-nical system to be reduced to a simplified (two-mass) crane lifting mechanism dynamic model, which is comprehensively and in detail analyzed in transition-type modes (for example, in the start-up mode). 
 An established and substantiated criterion for the movement quality of the hoisting crane load mechanism, at which rope system dynamic stresses are minimized in the starting mode, as well as the dynamic coefficient for different ways of lifting the load: 1) “from the weight”; 2) “with a pick up”/ “from the base” − typical during cranes real work in cargo lifting/lowering operations (loading and unloading). 
 Classical variational calculus methods usage (Euler-Poisson equation) allows to determine the conditions, according to which the above-mentioned criterion for the movement quality of the load lifting mechanism can be implemented.
 The movements laws of the load cargo and the drive crane mechanism are also defined, as well as law of dynamic coefficient change in time, for which there is no oscillation nature of time dependence, and the specified time functions have a mono-tonically growing character over time.
 The results obtained in the work allow (when using mechatronic control systems for the cargo lifting process) prevent overloading of the crane rope system and possible emergency situations, which lead to breaks in the elastic elements of the load lifting mechanisms (i.e. ropes), and also clarify and improve the existing engineering methods of similar complex technical systems calculations in their real operation modes.

Variational problem of adaptive optimal control. Theoretical and applied computer analysis

The problem of adaptive optimal control of a dynamical system, which belongs to the class of conditional variational problems with moving boundaries, is considered. A variational and computer study of the controlled adaptive motion of a material point is carried out for the problem of the energy quality functional minimizing with a moving, not predetermined right transboundary and in the case when the mass of the point changes depending on the unfixed final time. The problem is solved using the schemes and procedures of the classical calculus of variations, as well as adaptive estimation techniques, including the derivation of the variation of the auxiliary quality functional, the corresponding Euler equations, and the adaptive estimation algorithm. When solving a general conditional variational problem, the obtained closed system of differential equations was studied for the formation of an adaptive optimal control system for a dynamic plant with a given performance functional. The results of the unconditional formulation of the problem are generalized to the case of additional differential (nonholonomic) and holonomic constraints. In a variational adaptive optimal control problem, the transversality condition is formulated in terms of the local programming condition. The developed variational scheme of adaptive optimal synthesis can be used in the calculation and design of controlled dynamic systems. This optimization scheme is also promising for use in systems where operating time is non-fixed in advance. The results achieved in this paper concern obtaining specific equations, expressions, and formulas relative to the model example under study and finding graphs of the main time functions that determine the nature of the movement of the control object and the quality of the corresponding transients. The proposed adaptive optimal control algorithms for purposeful movement of the studied material point were tested in digital mode and showed their effectiveness which makes them promising for further use in more complex nonlinear adaptive systems of dynamic optimal control.

SOLVING THE PROBLEM OF LONGITUDINAL BENDING OF A ROD BY METHODS OF DISCRETE CALCULUS OF VARIATIONS

The question of the longitudinal bending of a rod consisting of rigid links connected by hinges is being studied. It is shown that, as in the classical version of a solid rod, the problem can be posed as a variational minimum energy problem, but given by a functional defined on a class of functions with a discrete domain of definition. The main provisions of the classical calculus of variations are transferred to functionals of this type: the formula of variation is found, a generalization of the main lemma of the calculus of variations is proved, an analogue of the Euler equation, which is a difference equation, is obtained. Applying the results obtained and the known properties of classical difference equations, we succeeded in solving an analogue of the Euler problem for two types of a hinge rod: for a rod consisting of links of the same length, and for an arbitrary choice of lengths of links. In both cases we find the critical Euler force, as well as the equation and the form of the deflection curve.

Necessary conditions for a minimum in classical calculus of variations in the presence of various types of degenerations

In the paper, a new approach is offered for studying an extremal in the problems of classical calculus of variations in the presence of various types of degenerations. This approach: firstly, is based on using Weierstrass type variations in two forms: in the form of variations on the right with respect to the given point, and in the form of variations on the left with respect to the same point; secondly, allows to get new and stronger results. The research is conducted under the assumption that along the considered extremal the Weierstrass and Legendre conditions as well as other necessary conditions for minimum degenerate, i.e. they are fulfilled as equalities at points or on some intervals. Two types of new necessary conditions are obtained: of equality type and of inequality type conditions for a strong and also a weak local minimum. Specific examples and a counterexample show that some of the necessary conditions for minimum obtained in this article are strengthening and refining of the corresponding known results in this direction.

T-SNE, forceful colorings, and mean field limits

t-SNE is one of the most commonly used force-based nonlinear dimensionality reduction methods. This paper has two contributions: the first is forceful colorings, an idea that is also applicable to other force-based methods (UMAP, ForceAtlas2,...). In every equilibrium, the attractive and repulsive forces acting on a particle cancel out: however, both the size and the direction of the attractive (or repulsive) forces acting on a particle are related to its properties: the force vector can serve as an additional feature. Secondly, we analyze the case of t-SNE acting on a single homogeneous cluster (modeled by affinities coming from the adjacency matrix of a random k-regular graph); we derive a mean-field model that leads to interesting questions in classical calculus of variations. The model predicts that, in the limit, the t-SNE embedding of a single perfectly homogeneous cluster is not a point but a thin annulus of diameter $\sim k^{-1/4} n^{-1/4}$. This is supported by numerical results. The mean field ansatz extends to other force-based dimensionality reduction methods.

Optimisation of Dynamic Loads of Rope Systems of Lifting Mechanisms of Bridge Cranes During Cargo Handling

Relevance. In this study, considerable attention is devoted to the analysis of dynamic loads that occur in the rope systems of lifting mechanisms of overhead cranes during start-up and braking, and the reduction of these loads.Purpose. To identify the magnitude and nature of changes in dynamic loads in the elements of lifting mechanisms of bridge cranes, a comprehensive dynamic analysis of the lifting mechanism of the bridge crane and its elastic elements (rope systems) was performed.Methods. The dynamic analysis of the above mechanisms and systems was performed on well-grounded mathematical models of bridge-type cranes (single- and double-mass).Results. The analysis of the obtained calculations of mathematical models of the mechanism of lifting the load of bridge cranes demonstrated that the dynamic loads applied to the structural elements and drive mechanisms are oscillatory and comparable to static loads. The analysis of the obtained calculations of mathematical models of the mechanism of lifting the load of bridge cranes demonstrated that the dynamic loads applied to the structural elements and drive mechanisms are oscillatory and comparable to static loads. To minimise the integral functionals, the methods of classical calculus of variations, mathematical physics and differential equations were used to model the dynamics of loading processes of rope systems and drives of bridge cranes, and the terminal (initial and final conditions of movement of such systems) were considered, which allowed solving the optimisation problem unambiguouslyThus, to reduce dynamic loads in structural elements (in particular, in ropes) during transients in such lifting mechanisms of bridge cranes, it is proposed to perform optimisation of the modes of movement of their drive mechanisms. An essential place in such optimisation is occupied by the choice of the optimisation criterion. Among such criteria, integral optimisation criteria were used. As such integral optimisation criteria, the RMS values of the current loads in the elastic elements (ropes) of overhead travelling cranes have been used.Conclusions. Such integral criteria are integral functionalities that usually reflect undesirable properties of machines and their mechanisms, thus, they are subject to minimisation

Envelope Theorems in Economics: Historical Development and Modern Cost-Benefit Applications

This paper reviews some historical development and modern applications of the envelope theorems in economics from a static to a dynamic context. First, we show how the static version of the theorem surfaced in economics, which had eventually lead to the well-known Shephard’s lemma in microeconomics. Second, we present its dynamic version in terms of the classical calculus of variations and optimal control theory via the optimized Hamiltonian function. Third, we show some applications of the theorem for deriving dynamic cost-benefifit rules with special reference to environmental projects involving the green or comprehensive net national product (CNNP). Finally, we illustrate how to extend the cost-benefifit rules to a stochastic economic growth setting.

Jacobi fields in optimal control: Morse and Maslov indices

In this paper we discuss a general framework based on symplectic geometry for the study of second order conditions in constrained variational problems on curves. Using the notion of L-derivatives we construct Jacobi curves, which represent a generalisation of Jacobi fields from the classical calculus of variations, but which also works for non-smooth extremals. This construction includes in particular the previously known constructions for specific types of extremals. We state and prove Morse-type theorems that connect the negative inertia index of the Hessian of the problem to some symplectic invariants of Jacobi curves.

Linear-quadratic non-zero sum differential game for mean-field stochastic systems with asymmetric information

In this study, we consider a class of asymmetric information linear-quadratic non-zero sum stochastic differential game problems. The system dynamics are governed by a forward linear mean-field stochastic differential equation and the cost functional is quadratic. In order to facilitate some practical applications, the mean-field terms for the state process and control process are considered in both the system dynamics and cost functional. Using the classical calculus of variation and dual methods, the open-loop Nash equilibrium point can be expressed by introducing an auxiliary mean-field forward backward stochastic differential equation, which comprises one forward and two backward components. Due to some Riccati equations and ordinary differential equations that possess unique solutions, the Nash equilibrium point can also be represented in feedback form for several special cases under asymmetric information. The corresponding filtering equations are derived, and the existence and uniqueness of the solutions are proved. An investment problem in finance is discussed to demonstrate the good performance of the theoretical results.

Optimal Control Theory: Introduction to the Special Issue

Optimal control theory is a modern extension of the classical calculus of variations [...]

ON THE CONSTRUCTION OF ENERGY-EFFICIENT ALGORTIMS OF THE WALKING ROBOT RAISING AND BRAKING

An energetically effective motion of a walking robot on a hard ground is considered. With the help of the methods of the classical calculus of variations, laws of the optimal acceleration and deceleration energy of the robot are calculated based on the minimum criterion. The resulting motion laws correspond to the extremals of the investigated quality functional.

Mechanics of good trade execution in the framework of linear temporary market impact

We define the concept of good trade execution and we construct explicit adapted good trade execution strategies in the framework of linear temporary market impact. Good trade execution strategies are dynamic, in the sense that they react to the actual realisation of the traded asset price path over the trading period; this is paramount in volatile regimes, where price trajectories can considerably deviate from their expected value. Remarkably, however, the implementation of our strategies does not require the full specification of an SDE evolution for the traded asset price, making them robust across different models. Moreover, rather than minimising the expected trading cost, good trade execution strategies minimise trading costs in a pathwise sense, a point of view not yet considered in the literature. The mathematical apparatus for such a pathwise minimisation hinges on certain random Young differential equations that correspond to the Euler–Lagrange equations of the classical Calculus of Variations. These Young differential equations characterise our good trade execution strategies in terms of an initial value problem that allows for easy implementations.

Synthesis of optimal control force for tripod manipulator drives

A tripod-based parallel structure manipulator is considered in the article. The movement of the working body from a known position to a given final position is realized by changing the lengths of the tripod end points. Linear links with an electric DC drive are used as end points. The algorithm was developed for the synthesis of the electric motor control voltages when moving the parallel structure manipulator working body based on the tripod from the initial position to the specified final position of the fixed travel time from the minimum criterion characterizing heating of the electric motor. The mathematical model of the manipulator controlled movements is described by linear differential equations of the second order. The static characteristic of the motor is used. The resistance force is taken proportional to the working body movement speed. The problem is solved by the methods of classical calculus of variations. The results of numerical modeling are presented, confirming the effectiveness of the proposed algorithm for determining the working body movement optimal laws.