Class Of Piecewise Constant Functions Research Articles

On the L 1-relaxed area of graphs of BV piecewise constant maps taking three values

Abstract Given a bounded open connected set Ω ⊂ ℝ 2 {\Omega\subset\mathbb{R}^{2}} with Lipschitz boundary, we consider the class of piecewise constant maps u taking three fixed values α , β , γ ∈ ℝ 2 {\alpha,\beta,\gamma\in\mathbb{R}^{2}} , vertices of an equilateral triangle; for any u in this class, using a weak notion of Jacobian determinant valid for BV functions, we give a precise description of Det ⁡ ( ∇ ⁡ u ) {\operatorname{Det}(\nabla\kern 0.569055ptu)} and show that the relaxed graph area of u is bounded from above by a quantity related to the flat norm of Det ⁡ ( ∇ ⁡ u ) {\operatorname{Det}(\nabla\kern 0.569055ptu)} . The provided upper bound allows to show the validity of a De Giorgi conjecture regarding the relaxed area functional when one restricts to this class of piecewise constant functions.

Controlled Process of Crystallization in Weld Pool

In this report the problem of control of solidification crack formation in welded plates is considered. In this problem the welding source is determined in dependence on the preset configuration and curvature of the rear part weld pool. A double ellipsoid model of weld pool with preset semi-axes may be used for the first approximation of preset weld pool configuration. It is an inverse problem which may be more efficiently solved as optimal control problem. The Function of welding source as a controlling function obtained in the result of solution is determined in a class of piecewise continuous functions which is more common class and the continuous-smooth functions are special partial case of common class. Recent methods of optimal control which use for solution of optimal control problems require to present the controlling functions in class of piecewise constant functions. Laser influence, electron beam, plasma, arc and submerged arc are the welding sources with high concentrated energy. A mathematical models of these welding sources may be introduced in class of piecewise continuous function with an efficient accuracy.

The method of parameterization in the quadratic optimal control problem

In the framework of control parameterization method the optimization problem with respect to linear phase system with quadratic functional is considered. Approximation of the control is obtained in the class of piecewise constant functions. It is formed as a linear combination of a special set of support functions. Coefficients of this combination are variables of the finite-dimensional problem. To effectively solve this problem explicit expressions for the functional with respect to parameters of approximations are obtained. As a result the quadratic mathematical programming problem is formulated.

Construction of reachability and controllability sets in a special linear control problem

The article considers the problem of constructing reachability and controllability sets for a control problem. The motion of an object is described by a linear system of ordinary differential equations, and control is selected from the class of piecewise-constant functions. Straight boundaries are also set on the controls. The article provides definitions of reacha- bility and controllability sets. It is shown that the problems of constructing these sets are equivalent and can be reduced to the problem of linear mapping of a multidimensional cube. The properties of these sets are also given. In addition, the existing approaches to solving the problem are analyzed. Since they are all too computationally complex, the question of creating a more efficient algorithm arises. The work proposes an algorithm for constructing the required sets as a system of linear inequalities. A proof of the theorem showing the correctness of the algorithm is provided. The complexity of the presented approach is estimated.

SYNTHESIS OF ZONAL CONTROL OF LUMPED SOURCES FOR THE HEAT CONDUCTION PROCESS

The paper studies the problem of synthesis of control of lumped sources for an object with distributed parameters based on discrete observation of the phase state at specific object points. We propose an approach in which the whole phase space at the observed points is preliminarily divided in some way into given subsets (zones). The synthesized controls are selected from the class of piecewise-constant functions, and their current values are determined by a subset of the phase space containing the population of current states of the object at the observed points, at which controls take constant values. Such synthesized controls are called zonal. We give a numerical technique for obtaining optimal values of zonal controls using efficient first-order optimization methods. To this purpose, we derive formulas for the gradient of the objective function in the space of zonal controls.

Optimal program control in the class of quadratic splines for linear systems

This article describes an algorithm for solving the optimal control problem in the case when the considered process is described by a linear system of ordinary differential equations. The initial and final states of the system are fixed and straight two-sided constraints for the control functions are defined. The purpose of optimization is to minimize the quadratic functional of control variables. The control is selected in the class of quadratic splines. There is some evolution of the method when control is selected in the class of piecewise constant functions. Conveniently, due to the addition/removal of constraints in knots, the control function can be piecewise continuous, continuous, or continuously differentiable. The solution algorithm consists in reducing the control problem to a convex mixed-integer quadratically-constrained programming problem, which could be solved by using well-known optimization methods that utilize special software.

On the uniqueness of a solution to an inverse problem of scattering by an inhomogeneous solid with a piecewise Hölder refractive index in a special function class

The problem of reconstructing a piecewise Hölder continuous function describing the refractive index of an inhomogeneous obstacle scattering a monochromatic wave is considered. The boundary value scattering problem is reduced to a system of integral equations. The equivalence of the integral and differential formulations of the problem is proved. A two-step method for solving the inverse problem is proposed. A linear integral equation of the first kind is solved at the first step. Sufficient conditions for the uniqueness of its solution in the class of piecewise constant functions are obtained. At the second step, the unknown refractive index is explicitly expressed in terms of the solution obtained at the first step.

The two-step method for determining a piecewise-continuous refractive index of a 2D scatterer by near field measurements

The problem of determining an unknown piecewise-continuous refractive index of an inhomogeneous scatterer is considered. The boundary value problem is reduced to the Lippmann–Schwinger integral equation. The solution of the inverse problem is obtained in two steps. At the first step, the integral equation of the first kind is solved in the inhomogeneity region using measured values of the total field outside the region. It is proved that the solution to the integral equation of the first kind is unique in the class of piecewise constant functions. At the second step, the unknown refractive index is explicitly calculated via the found solution of the integral equation and the given total field. The proposed method was implemented and verified by solving a test inverse problem with a given refractive index. Efficiency of the two-step method was approved by comparison between the exact solution and the approximate ones.

On the Uniqueness of a Solution to an Inverse Problem of Scattering by an Inhomogeneous Solid with a Piecewise Hölder Refractive Index in a Special Function Class

The problem of reconstructing a piecewise Holder continuous function describing the refractive index of an inhomogeneous obstacle scattering a monochromatic wave is considered. The boundary value scattering problem is reduced to a system of integral equations. The equivalence of the integral and differential formulations of the problem is proved. A two-step method for solving the inverse problem is proposed. A linear integral equation of the first kind is solved at the first step. Sufficient conditions for the uniqueness of its solution in the class of piecewise constant functions are obtained. At the second step, the unknown refractive index is explicitly expressed in terms of the solution obtained at the first step.

Direct and Inverse Problems of Electromagnetic Control

A mathematic model is suggested of the direct and universe problems of flaw detection. The direct problem consists in determining the complex amplitude of EMF in the through-type-induction- transducer–air gap–ferromagnetic cylinder system. In the direct problem, the distribution laws of magnetic permeability and electrical conduction are assumed to be known. A formula is obtained for calculating the complex amplitude of EMF in the class of piecewise-constant dependences of electromagnetic parameters. The inverse problem of flaw detection for a ferromagnetic cylinder consists in determining the magnetic permeability based on the measured values of the moduli of the EMF amplitudes in the system on a fixed frequency grid. An approximate solution of the inverse problem is searched in the class of piecewise-constant functions. Tikhonov’s method of regularization is used to solve the inverse problem. The results of numerical and physical modeling are presented.

A Study of Subtractive Digital Dither in Single-Stage and Multi-Stage Quantizers

The effect of subtractive digital dither on single-stage and multi-stage quantizers is examined. In the single stage case, it is proved that the conditional density of the quantization error becomes asymptotically independent of the input and uniformly distributed as the number of digital dither levels increases. For a fixed number of levels, the conditional expectation of a class of piecewise constant functions is shown to be independent of the input. A Fourier series representation of quantization error densities is used to construct a linear periodically time varying model of multistage quantization and dithering systems. For a homogeneous multi-stage quantization system, it is shown that when the least common multiple of the numbers of digital dither levels at successive stages is large, the statistical dependence of the quantization noise on the input disappears almost entirely. Computer simulations confirm this analysis.

Optimal control of a concentrated system on the class of piecewise constant functions under uncertainty in the parameters and initial conditions

The authors analyze optimal control problems for objects described by systems of ordinary differential equations on the class of piecewise constant control functions with uncertain initial information about the parameters of the initial conditions and about object parameters. Piecewise constant values of the controls and, what is most important, the boundaries of the intervals of constancy of the controls are optimized in the problem. Given the number of the intervals of constancy, the necessary optimality conditions and formulas for the gradient of the objective functional are obtained. These formulas allow using efficient first-order optimization methods. For the case where the number of intervals of constancy is not specified, an algorithm of its optimization is proposed.

Discrete image modelling for piecewise constant segmentation by artificial neural networks

The problem of an image best approximation within the class of piecewise constant functions is considered. This allows a simpler data representation with a lower number of grey levels while retaining all information relevant to the particular application considered. The approximant can be found by solving a segmentation problem. The search for a solution is solved efficiently by training an artificial neural network (ANN) on a suitable set of templates by a standard procedure. The samples of the training alphabet fit the signal's local behaviour in the homogeneous image subregions and in the regions crossed by the edges. Therefore the original image domain is partitioned into disjoint 2D intervals (tiling), and for each one of them, the network selects the alphabet element closest to the corresponding image component. The main motivation of this work consists in devising a methodology suitable for real-time applications; indeed, the ANN tool is attractive for a hardware implementation.

Solution of optimal control problem in class of piecewise-constant functions

A numerical method of solution of problems of optimal control of objects that may be described by systems of ordinary differential equations in the class of piecewise-constant controls is proposed in the article. At the same time, both the piecewise-constant values of the controls as well as the constancy intervals of these values are optimized. Analytic formulas of the gradient of the functional with respect to the optimized parameters are obtained. The formulas for the gradient of the functional obtained for an initial continuous optimal control problem and the corresponding discretized optimal control problem are compared and results of numerical experiments are presented.

Dynamic method of determining aerodynamic characteristics of models from on experimental results obtained in short-duration wind tunnels

A technique is developed to recover the loads acting in model testing in short-duration wind tunnels. The model together with the load-measuring device is assumed to be a linear dynamic system. Normal responses of the system to unit loads are determined experimentally. The solution of a system of integral Volterra equations of the first kind is found in the class of piecewise-constant functions. Examples are given on solving a test problem with a known exact solution and a problem of determining the aerodynamic characteristics of the HB-2 reference model on the basis of loads obtained in wind-tunnel measurements.

Therapy Optimization for Viral Infections using Fuzzy Performance Index

This Work concerns the application of optimization theory in the planning of pharmacological treatment of viral infections. The dynamics of the disease is modeled using Marchuk's formulation the model being extended to include the effect of treatment and to allow representation of the pathological consequences of the infection on the patient The proposed performance index is based on fuzzy sets to reflect a compromise between non-exaet subjective quantities such as patient well being as compared with quantities such as cost of medication that can be objectively measured The admissible controls are assumed to belong to the class of piecewise constant functions. The numerical results show a considerable improvement of the optimized drug administration policy, when compared to non-optimized policies, in terms of reduction in the achieved performance index.

Thin weakly curved wings with maximum aerodynamic quality

Lifting wings that only slightly disturb the supersonic gas flow are considered. The plan shape and thickness distribution of the wing and the free-stream parameters are given. The flow problem is solved within the framework of the Prandtl model. The outer potential flow is determined in accordance with the linear theory. The turbulent boundary layer is found by the method of plane sections with allowance for the three-dimensional inviscid flow pattern. A numerical model of the flow is constructed in the class of piecewise-constant functions on characteristic calculation grids [1]. The variational problem of finding the weakly curved middle surface of the wing giving maximum aerodynamic quality is reduced, by analogy with [2], to a problem of nonlinear programming and is solved by the gradient projection method.

A minimum trapping time problem in optimal control theory

AbstractIn this paper we consider a natural extension of the minimum time problem in optimal control theory which we refer to as the minimum trapping time problem. The minimum trapping time problem requires a fixed time interval [0, T], where T is finite. The aim is to determine a control for which the system trajectory not only reaches a specified target in minimum time but also remains trapped within the target until time T. Our aim is to devise a computational procedure for solving the minimum trapping time problem. The computational procedure we adopt uses control parametrisation in which the class of controls is approximated by a class of piecewise constant functions. The problem we are solving is therefore an approximation to the original minimum trapping time problem. Some properties for the approximate problem are then established. These lead to an extremely efficient iterative procedure for calculating the minimum trapping time.

Applications of methods for solving of special nonlinear programming problems to optimization of control systems

The algorithms of optimization of dynamic control systems in the class of impulse and piece-wise constant functions are considered in the report. They are based on the methods for solving of linear, piece-wise linear and quadratic programming problems worked out by the authors. I. Consider the time optimal control problem(1)x˙=Ax+Bu,xO=xo,d*≤ut≤d*,t∈T-O,ti,(2)ti→min,xti=xi,where x=xt=xit,…xnt is the vector of state of the system at the moment t;A∈Rn×n,B,xo,xi∈Rn;rankB,AB,…,An−1b=n;u=ut,t∈T, is the one-dimensional admissible contro from the class of piece-wise constant functions that are continuous to the right at the points of discontinuity. The algorithm for solving of problem(l-2)is based on the primal support method for solving of linear problem of terminal control with the unbound termination of the process and on the use of differencial equations for optimal support and optimal value of cost functional in the problem being considered. A numerical experiment was carried out to test the efficiency of the suggested method. The problems of the typeti→min,x˙i=x2,x˙2=x3,…,x˙n=u,xO=xO,xti=xi,ut≤i,t∈T=O,ti,were chosen as the test ones where the order of the system n =3,5, the initial approximation to the time optimal moment, the initial and terminal states varied by special rules. The accuracy of 10-15-10-15 was reached. 2. Consider the terminal control problemc′xto→max,x˙t=Atxt+Btut+ft,t∈T=tN,tK;B*i≤h′ixti≤Bi*,i=l,m¯;d*t≤ut≤d*t,in the class of the impulse controls u(t) , where c,xt,ft∈Rn,ut∈R2,t∈TUnder the numerical solution of the given problem the trajectory xt,t∈T of the system is approximately computed and the accuracy varies from iteration to iteration. The algorithm is finite without taking into consideration that the accuracy of the calculations approaches zero. At the first stage the “rough” calculations are conducted. As approaching to the optimum according to the definite rules the accuracy is raising. This allows to economize GPU time. The numerical experiment showed that the rate of algorithm increases approximately two times in comparison with the variant where the maximal accuracy is constantly used at iterations. 3. Three problems are considered for the system (I):I) Ju=minc′ixti+αi→max, 2)Ju=∑i∈Kc′ixti+αi→min, 3)Ju=maxi∈Kc′ixti+αi→min,with the constraintsHxti=g,f*t≤ut≤f*t,t∈T,where K is the finite set, ci are the vectors, i∈K;αi are scalars i∈K. The exact finite algorithms are worked out. They are programmed in FORTRAN-17. 4. The problem of quadratic functional on the trajectories of the system (2) is investigated.

On a differential-difference game of escape: PMM vol. 40, n≗ 6, 1976, pp. 995–1002

A nonlinear escape problem for conflict-controlled systems described by differential equations with a lagging argument is considered. The sufficient escape conditions which are realized in the class of piece wise-constant functions are obtained. The paper relates to the researches in [1 – 8] and is a continuation of [9, 10].