scholarly journals Overview on the Pointwise Constrained Liapunov Vectorial Convexity Theorem

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Clara Carlota ◽  
Sílvia Chá ◽  
António Ornelas
Keyword(s):  
Optimal Control ◽  
Open Problem ◽  
Differential Inclusions ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Real Life ◽  
Convexity Theorem ◽  
Realistic Case ◽  
Life Problems ◽  
Pointwise Constraints

In applications of the Calculus of Variations, Optimal Control and Differential Inclusions, very important real-life problems are nonconvex vectorial and subject to pointwise constraints. The classical Liapunov convexity theorem is a crucial tool allowing researchers to solve nonconvex vectorial problems involving single integrals. However, the possibility of extending such theorem so as to deal with pointwise constraints has remained an open problem for two decades, in the more realistic case using variable vectorial velocities. We have recently solved it, in the sense of proving necessary conditions and sufficient conditions for solvability of such problem. A quick overview of our results is presented here, the main point being that, somehow, convex constrained nonuniqueness a.e. implies nonconvex constrained existence.

scholarly journals The nonsmoth optimal control problem for ensamble of trajectories of dynamic system under conditions of indeterminaci

2020 ◽  
Vol 5 ◽  
pp. 38-42
Author(s):  
Otakulov Salim ◽  
Rahimov Boykxuroz Shermuhamedovich ◽  
Haydarov Tulkinjon Turgunbayevich
Keyword(s):  
Optimal Control ◽  
Dynamic System ◽  
Control Problem ◽  
Differential Inclusions ◽  
Sufficient Conditions ◽  
Value Analysis ◽  
Minimax Control ◽  
Informational Model ◽  
The One ◽  
Necessary And Sufficient

In the paper we consider the one model of dynamic system under conditions of indeterminacy – linear controllable differential inclusions. For the informational model of the control system the minimax control problem for ensemble trajectories is researched. This control problem is study with a methods nonsmooth and multi-value analysis. The necessary and sufficient conditions of optimality are obtained.

scholarly journals Certain Properties of Vague Graphs with a Novel Application

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1647
Author(s):  
Yongsheng Rao ◽  
Saeed Kosari ◽  
Zehui Shao
Keyword(s):  
Social Systems ◽  
Sufficient Conditions ◽  
Real Life ◽  
Laplacian Matrix ◽  
Fuzzy Graph ◽  
Fuzzy Graphs ◽  
Life Problems ◽  
Real World Problem ◽  
Laplacian Energy ◽  
Definition Of

Fuzzy graph models enjoy the ubiquity of being present in nature and man-made structures, such as the dynamic processes in physical, biological, and social systems. As a result of inconsistent and indeterminate information inherent in real-life problems that are often uncertain, for an expert, it is highly difficult to demonstrate those problems through a fuzzy graph. Resolving the uncertainty associated with the inconsistent and indeterminate information of any real-world problem can be done using a vague graph (VG), with which the fuzzy graphs may not generate satisfactory results. The limitations of past definitions in fuzzy graphs have led us to present new definitions in VGs. The objective of this paper is to present certain types of vague graphs (VGs), including strongly irregular (SI), strongly totally irregular (STI), neighborly edge irregular (NEI), and neighborly edge totally irregular vague graphs (NETIVGs), which are introduced for the first time here. Some remarkable properties associated with these new VGs were investigated, and necessary and sufficient conditions under which strongly irregular vague graphs (SIVGs) and highly irregular vague graphs (HIVGs) are equivalent were obtained. The relation among strongly, highly, and neighborly irregular vague graphs was established. A comparative study between NEI and NETIVGs was performed. Different examples are provided to evaluate the validity of the new definitions. A new definition of energy called the Laplacian energy (LE) is presented, and its calculation is shown with some examples. Likewise, we introduce the notions of the adjacency matrix (AM), degree matrix (DM), and Laplacian matrix (LM) of VGs. The lower and upper bounds for the Laplacian energy of a VG are derived. Furthermore, this study discusses the VG energy concept by providing a real-time example. Finally, an application of the proposed concepts is presented to find the most effective person in a hospital.

scholarly journals Reliability Properties of the NDL Family of Discrete Distributions with Its Inference

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1139
Author(s):  
Mohammed Mohammed Ahmed Almazah ◽  
Badr Alnssyan ◽  
Abdul Hadi N. Ahmed ◽  
Ahmed Z. Afify
Keyword(s):  
Least Squares ◽  
Negative Binomial ◽  
Weighted Least Squares ◽  
Sufficient Conditions ◽  
Real Life ◽  
Discrete Distributions ◽  
Real World Data ◽  
Life Problems ◽  
Anderson Darling ◽  
Von Mises

The natural discrete Lindley (NDL) distribution is an intuitive idea that uses discrete analogs to well-known continuous distributions rather than using any of the published discretization techniques. The NDL is a flexible extension of both the geometric and the negative binomial distributions. In the present article, we further investigate new results of value in the areas of both theoretical and applied reliability. To be specific, several closure properties of the NDL are proved. Among the results, sufficient conditions that maintain the preservation properties under useful partial orderings, convolution, and random sum of random variables are introduced. Eight different methods of estimation, including the maximum likelihood, least squares, weighted least squares, Cramér–von Mises, the maximum product of spacing, Anderson–Darling, right-tail Anderson–Darling, and percentiles, have been used to estimate the parameter of interest. The performance of these estimators has been evaluated through extensive simulation. We have also demonstrated two applications of NDL in modeling real-life problems, including count data. It is worth noting that almost all the methods have resulted in very satisfactory estimates on both simulated and real-world data.

scholarly journals The characterization of regular ordered Gamma semigroups in terms of (E,EVq_k)-fuzzy Gamma ideals

2017 ◽  
Vol 13 (4) ◽  
pp. 576-580
Author(s):  
Ibrahim Gambo ◽  
Nor Haniza Sarmin ◽  
Hidayat Ullah Khan ◽  
Muhammad Faiz Khan
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Sufficient Conditions ◽  
Real Life ◽  
Completely Regular ◽  
Life Problems ◽  
Left Regular ◽  
Necessary And Sufficient

The advancement in the fascinating area of fuzzy set theory has become area of much interest, generalization of the existing fuzzy subsystems of other algebraic structures is very important to tackle more current real life problems. In this paper, we give more generalized form of regular ordered gamma semigroups in terms of (E,EVq_k)-fuzzy gamma ideals. Particularly, we characterized left regular, right regular, simple and completely regular ordered gamma semigroups in terms of this new notion. Some necessary and sufficient conditions for ordered gamma semigroup to be completely regular are provided in this paper.

scholarly journals Pontryagin’s Maximum Principle for Optimal Control of Stochastic SEIR Models

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Ruimin Xu ◽  
Rongwei Guo
Keyword(s):  
Optimal Control ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Hamiltonian Function ◽  
Model System ◽  
Seir Model ◽  
Verification Theorem ◽  
Explicit Formulation ◽  
Adjoint Variables ◽  
Sufficient Conditions For Optimality

In this paper, we study the necessary conditions as well as sufficient conditions for optimality of stochastic SEIR model. The most distinguishing feature, compared with the well-studied SEIR model, is that the model system follows stochastic differential equations (SDEs) driven by Brownian motions. Hamiltonian function is introduced to derive the necessary conditions. Using the explicit formulation of adjoint variables, desired necessary conditions for optimal control results are obtained. We also establish a sufficient condition which is called verification theorem for the stochastic SEIR model.

scholarly journals An existence theorem for optimal stochastic programming

1968 ◽  
Vol 8 (1) ◽  
pp. 114-118 ◽  
Author(s):  
A. W. J. Stoddart
Keyword(s):  
Optimal Control ◽  
Stochastic Programming ◽  
Existence Theorem ◽  
Stochastic Systems ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Direct Methods ◽  
Optimal Program ◽  
Sufficient Conditions For Optimality ◽  
Theory Of Optimal Control

In [4], Hanson has obtained necessary conditions and sufficient conditions for optimality of a program in stochastic systems. However, in many cases, especially in a general treatment, a program satisfying these conditions cannot be determined explicitly, so that the question of existence of an optimal program in such systems is significant. In this paper, we obtain conditions sufficient for existence of an optimal program by applying the direct methods of the calculus of variations [9], [6] and the theory of optimal control [7], [5].

The Psychologist in the Medical School: An Example of Solving Real-Life Problems

1970 ◽  
Author(s):  
Matisyohu Weisenberg ◽  
Carl Eisdorfer ◽  
C. Richard Fletcher ◽  
Murray Wexler
Keyword(s):  
Medical School ◽  
Real Life ◽  
Life Problems

Optimal control of higher order differential inclusions with functional constraints

2020 ◽  
Vol 26 ◽  
pp. 37 ◽  
Author(s):  
Elimhan N. Mahmudov
Keyword(s):  
Optimal Control ◽  
Optimality Conditions ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Higher Order ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Functional Constraints ◽  
Complementary Slackness ◽  
Complementary Slackness Conditions

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

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