PRINCIPLE AND THEOREMS OF REVERSIONAL CALCULATION

2021 ◽  
Vol 3 ◽  
pp. 12-20
Author(s):  
A.I. BOKHONSKY ◽  
N.I. VARMINSKAYA ◽  
A.I. RYZHKOV
Keyword(s):  
Control Function ◽  
Minimum Energy ◽  
Sufficient Conditions ◽  
Analytical Control ◽  
Optimal Controls ◽  
Initial State ◽  
Numerical Equality ◽  
Special Cases ◽  
Controlled Motion ◽  
State Of Rest

A reverse optimality principle (ROP) is formulated and an algorithm for its use for constructing optimal portable object movements is presented. Using an example, sufficient conditions for the extremality of the restored criterion functional are verified when constructing an optimal control of the «acceleration–deceleration» type. The following theorems were formulated and proved: on the numerical equality of integrals with different integral functions, on the minimum energy to achieve the goal of optimally controlled motion in the form of «acceleration–deceleration». Basing on the generalization of the results for optimal controls designing of the «acceleration–deceleration» type of motion, whence the known special cases follow, universal analytical control function (translational acceleration) was found. Analytically and numerically was confirmed the existence of the limiting minimum control energy at which the movement of an object from the initial state of rest to a new state of rest is possible at a fixed distance and time of motion.

scholarly journals On the solvability of the tracking problem in the optimization of the thermal process by moving point controls

2021 ◽  
Vol 102 (2) ◽  
pp. 67-73
Author(s):  
A. Kerimbekov ◽  
◽  
A.T. Ermekbaeva ◽  
E. Seidakmat kyzy ◽  
◽  
...  
Keyword(s):  
Control Function ◽  
Complete Solution ◽  
Sufficient Conditions ◽  
Point Sources ◽  
Thermal Processes ◽  
Optimal Controls ◽  
Tracking Problem ◽  
Point Control ◽  
Linear Integral ◽  
Linear Integral Equations

In the present article we investigate problems of tracking in the moving point control of thermal processes described by Fredholm integro-differential equations in partial derivatives with the Fredholm integral operator, in the case when the functions of point sources are nonlinear with respect to the control function. It is found that optimal controls are defined as solutions to a system of linear integral equations, and an algorithm for constructing its solution is developed. Sufficient conditions for the unique solvability of the tracking problem are found and an algorithm for constructing a complete solution to the nonlinear optimization problem was indicated.

Causal inference via string diagram surgery

2021 ◽  
pp. 1-22
Author(s):  
Bart Jacobs ◽  
Aleks Kissinger ◽  
Fabio Zanasi
Keyword(s):  
Causal Inference ◽  
Probabilistic Reasoning ◽  
Causal Effect ◽  
Sufficient Conditions ◽  
Causal Effects ◽  
Stochastic Matrices ◽  
Counterfactual Reasoning ◽  
Special Cases ◽  
String Diagram ◽  
Set Up

Abstract Extracting causal relationships from observed correlations is a growing area in probabilistic reasoning, originating with the seminal work of Pearl and others from the early 1990s. This paper develops a new, categorically oriented view based on a clear distinction between syntax (string diagrams) and semantics (stochastic matrices), connected via interpretations as structure-preserving functors. A key notion in the identification of causal effects is that of an intervention, whereby a variable is forcefully set to a particular value independent of any prior propensities. We represent the effect of such an intervention as an endo-functor which performs ‘string diagram surgery’ within the syntactic category of string diagrams. This diagram surgery in turn yields a new, interventional distribution via the interpretation functor. While in general there is no way to compute interventional distributions purely from observed data, we show that this is possible in certain special cases using a calculational tool called comb disintegration. We demonstrate the use of this technique on two well-known toy examples: one where we predict the causal effect of smoking on cancer in the presence of a confounding common cause and where we show that this technique provides simple sufficient conditions for computing interventions which apply to a wide variety of situations considered in the causal inference literature; the other one is an illustration of counterfactual reasoning where the same interventional techniques are used, but now in a ‘twinned’ set-up, with two version of the world – one factual and one counterfactual – joined together via exogenous variables that capture the uncertainties at hand.

Full Implementation under Ambiguity

2021 ◽  
Vol 13 (1) ◽  
pp. 148-178
Author(s):  
Huiyi Guo ◽  
Nicholas C. Yannelis
Keyword(s):  
Social Choice ◽  
Expected Utility ◽  
Sufficient Conditions ◽  
Choice Functions ◽  
Maxmin Expected Utility ◽  
Special Cases ◽  
Social Choice Functions ◽  
Bayesian Implementation ◽  
Pareto Efficient ◽  
Choice Set

This paper introduces the maxmin expected utility framework into the problem of fully implementing a social choice set as ambiguous equilibria. Our model incorporates the Bayesian framework and the Wald-type maxmin preferences as special cases and provides insights beyond the Bayesian implementation literature. We establish necessary and almost sufficient conditions for a social choice set to be fully implementable. Under the Wald-type maxmin preferences, we provide easy-to-check sufficient conditions for implementation. As applications, we implement the set of ambiguous Pareto-efficient and individually rational social choice functions, the maxmin core, the maxmin weak core, and the maxmin value. (JEL D71, D81, D82)

scholarly journals Generalized Knaster-Kuratowski-Mazurkiewicz type theorems and applications to minimax inequalities

2017 ◽  
Vol 20 (K2) ◽  
pp. 131-140
Author(s):  
Linh Manh Ha
Keyword(s):  
Variational Inequalities ◽  
Equilibrium Problems ◽  
Applied Mathematics ◽  
Sufficient Conditions ◽  
Topological Spaces ◽  
Minimax Theorems ◽  
Minimax Inequalities ◽  
Special Cases ◽  
Kkm Mappings ◽  
Definition Of

Knaster-Kuratowski-Mazurkiewicz type theorems play an important role in nonlinear analysis, optimization, and applied mathematics. Since the first well-known result, many international efforts have been made to develop sufficient conditions for the existence of points intersection (and their applications) in increasingly general settings: Gconvex spaces [21, 23], L-convex spaces [12], and FCspaces [8, 9]. Applications of Knaster-Kuratowski-Mazurkiewicz type theorems, especially in existence studies for variational inequalities, equilibrium problems and more general settings have been obtained by many authors, see e.g. recent papers [1, 2, 3, 8, 18, 24, 26] and the references therein. In this paper we propose a definition of generalized KnasterKuratowski-Mazurkiewicz mappings to encompass R-KKM mappings [5], L-KKM mappings [11], T-KKM mappings [18, 19], and many recent existing mappings. Knaster-KuratowskiMazurkiewicz type theorems are established in general topological spaces to generalize known results. As applications, we develop in detail general types of minimax theorems. Our results are shown to improve or include as special cases several recent ones in the literature.

scholarly journals First and second order optimality conditions for the control of Fokker-Planck equations

2021 ◽  
Vol 27 ◽  
pp. 15
Author(s):  
M. Soledad Aronna ◽  
Fredi Tröltzsch
Keyword(s):  
Control Variable ◽  
Sufficient Conditions ◽  
Planck Equation ◽  
Second Order ◽  
Optimal Controls ◽  
Main Emphasis ◽  
The Cost ◽  
Necessary And Sufficient ◽  
Fokker Planck Equations ◽  
Fokker Planck

In this article we study an optimal control problem subject to the Fokker-Planck equation ∂tρ − ν∆ρ − div(ρB[u]) = 0 The control variable u is time-dependent and possibly multidimensional, and the function B depends on the space variable and the control. The cost functional is of tracking type and includes a quadratic regularization term on the control. For this problem, we prove existence of optimal controls and first order necessary conditions. Main emphasis is placed on second order necessary and sufficient conditions.

scholarly journals Minimum-Energy Bivariate Wavelet Frame with Arbitrary Dilation Matrix

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Fengjuan Zhu ◽  
Qiufu Li ◽  
Yongdong Huang
Keyword(s):  
Scaling Function ◽  
Minimum Energy ◽  
Sufficient Conditions ◽  
Reconstruction Algorithm ◽  
Wavelet Function ◽  
Necessary Condition ◽  
Wavelet Frame ◽  
Wavelet Frames ◽  
Numerical Examples ◽  
Dilation Matrix

In order to characterize the bivariate signals, minimum-energy bivariate wavelet frames with arbitrary dilation matrix are studied, which are based on superiority of the minimum-energy frame and the significant properties of bivariate wavelet. Firstly, the concept of minimum-energy bivariate wavelet frame is defined, and its equivalent characterizations and a necessary condition are presented. Secondly, based on polyphase form of symbol functions of scaling function and wavelet function, two sufficient conditions and an explicit constructed method are given. Finally, the decomposition algorithm, reconstruction algorithm, and numerical examples are designed.

scholarly journals On the General Consensus Protocol in Multiagent Networks with Double-Integrator Dynamics and Coupling Time Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Tao Dong ◽  
Xiaofeng Liao
Keyword(s):  
Time Delay ◽  
Matrix Theory ◽  
Sufficient Conditions ◽  
Consensus Algorithm ◽  
Directed Network ◽  
Consensus Protocol ◽  
Special Cases ◽  
Coupling Time ◽  
Almost All ◽  
Double Integrator

This paper considers the problem of the convergence of the consensus algorithm for multiple agents in a directed network where each agent is governed by double-integrator dynamics and coupling time delay. The advantage of this protocol is that almost all the existing linear local interaction consensus protocols can be considered as special cases of the present paper. By combining algebraic graph theory and matrix theory and studying the distribution of the eigenvalues of the associated characteristic equation, some necessary and sufficient conditions are derived for reaching the second-order consensus. Finally, an illustrative example is also given to support the theoretical results.

scholarly journals On an infinite dimensional linear-quadratic problem with fixed endpoints: The continuity question

2014 ◽  
Vol 24 (4) ◽  
pp. 723-733
Author(s):  
K.Maciej Przyłuski
Keyword(s):  
Minimum Energy ◽  
Sufficient Conditions ◽  
Control Problems ◽  
Linear Quadratic Control ◽  
Linear Quadratic ◽  
Minimum Norm Solution ◽  
Quadratic Problem ◽  
Infinite Dimensional ◽  
Necessary And Sufficient ◽  
Quadratic Control Problems

Abstract In a Hilbert space setting, necessary and sufficient conditions for the minimum norm solution u to the equation Su = Rz to be continuously dependent on z are given. These conditions are used to study the continuity of minimum energy and linear-quadratic control problems for infinite dimensional linear systems with fixed endpoints.

The Control of a Neural Complex Network

2014 ◽  
Vol 687-691 ◽  
pp. 444-446
Author(s):  
Fan Di Zhang
Keyword(s):  
Neural Network ◽  
Community Structure ◽  
Lyapunov Stability ◽  
Control Strategy ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Pinning Control ◽  
Projective Synchronization ◽  
Cluster Synchronization ◽  
Special Cases

In this paper, the synchronization of a neural network with community structure is investigated. Cluster projective generalizes previously existing synchronization schemes. The cluster projective synchronization is more general that includes projective synchronization and cluster synchronization, as its special cases. The cluster projective synchronization of these networks is discussed via some pinning control strategy. Several sufficient conditions for the network to achieve cluster projective synchronization are derived based on Lyapunov stability theory. Numerical simulations are used to demonstrate the effectiveness and feasibility of the proposed scheme.

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