In this work, processes represented by linear stochastic dynamic system are investigated and by
considering optimal control problem, principle of optimality is proven. Also, for existence of optimal
control and corresponding optimal trajectory, proofs of theorems of necessity and sufficiency condition are
attained.
On the principle of optimality for linear stochastic dynamic systemijfcstjournal
In this work, processes represented by linear stochastic dynamic system are investigated and by
considering optimal control problem, principle of optimality is proven. Also, for existence of optimal
control and corresponding optimal trajectory, proofs of theorems of necessity and sufficiency condition are
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This paper derives new results for the design of sliding mode controller for the hybrid synchronization of identical hyperchaotic Chen systems (Jia, Dai and Hui, 2010). The synchronizer results derived in this paper for the hybrid synchronization of identical hyperchaotic Chen systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve hybrid synchronization of the
identical hyperchaotic Chen systems. Numerical simulations are shown to illustrate and validate the hybrid synchronization schemes derived in this paper for the identical hyperchaotic Chen systems.
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This paper investigates the hybrid chaos synchronization of identical hyperchaotic Newton-Leipnik systems (Ghosh and Bhattacharya, 2010) by sliding mode control. The stability results derived in this paper for the hybrid chaos synchronization of identical hyperchaotic Newton-Leipnik systems are established using Lyapunov stability theory. Hybrid synchronization of hyperchaotic Newton-Leipnik systems is achieved through the complete synchronization of first and third states of the systems and the anti-synchronization of second and fourth states of the master and slave systems. Since the Lyapunov exponents are not required for these calculations, the sliding mode control is very effective and convenient to achieve hybrid chaos synchronization of the identical hyperchaotic Newton-Leipnik systems. Numerical simulations are shown to validate and demonstrate the effectiveness of the synchronization schemes derived in this paper.
Research on 4-dimensional Systems without Equilibria with ApplicationTELKOMNIKA JOURNAL
Recently chaos-based encryption has been obtained more and more attention. Chaotic systems
without equilibria may be suitable to be used to design pseudorandom number generators (PRNGs)
because there does not exist corresponding chaos criterion theorem on such systems. This paper
proposes two propositions on 4-dimensional systems without equilibria. Using one of the propositions
introduces a chaotic system without equilibria. Using this system and the generalized chaos
synchronization (GCS) theorem constructs an 8-dimensional discrete generalized chaos synchronization
(8DBDGCS) system. Using the 8DBDGCS system designs a 216-word chaotic PRNG. Simulation results
show that there are no significant correlations between the key stream and the perturbed key streams
generated via the 216-word chaotic PRNG. The key space of the chaotic PRNG is larger than 21275. As
an application, the chaotic PRNG is used with an avalanche-encryption scheme to encrypt an RGB image.
The results demonstrate that the chaotic PRNG is able to generate the avalanche effects which are similar
to those generated via ideal chaotic PRNGs.
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...Wireilla
In this paper, the approximate controllability of impulsive linear fuzzy stochastic differential equations with nonlocal conditions in Banach space is studied. By using the Banach fixed point theorems, stochastic analysis, fuzzy process and fuzzy solution, some sufficient conditions are given for the approximate controllability of the system.
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...ijfls
In this paper, the approximate controllability of impulsive linear fuzzy stochastic differential equations with nonlocal conditions in Banach space is studied. By using the Banach fixed point
theorems, stochastic analysis, fuzzy process and fuzzy solution, some sufficient conditions are given for
the approximate controllability of the system.
On the principle of optimality for linear stochastic dynamic systemijfcstjournal
In this work, processes represented by linear stochastic dynamic system are investigated and by
considering optimal control problem, principle of optimality is proven. Also, for existence of optimal
control and corresponding optimal trajectory, proofs of theorems of necessity and sufficiency condition are
attained.
TIME-ABSTRACTING BISIMULATION FOR MARKOVIAN TIMED AUTOMATAijseajournal
Markovian timed automata (MTA) has been proposed as an expressive formalism for specification of realtime
properties in Markovian stochastic processes. In this paper, we define bisimulation relation for
deterministic MTA. This definition provides a basis for developing effective algorithms for deciding
bisimulation relation for such automata.
Sliding Mode Controller Design for Hybrid Synchronization of Hyperchaotic Che...ijcsa
This paper derives new results for the design of sliding mode controller for the hybrid synchronization of identical hyperchaotic Chen systems (Jia, Dai and Hui, 2010). The synchronizer results derived in this paper for the hybrid synchronization of identical hyperchaotic Chen systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve hybrid synchronization of the
identical hyperchaotic Chen systems. Numerical simulations are shown to illustrate and validate the hybrid synchronization schemes derived in this paper for the identical hyperchaotic Chen systems.
Hybrid Chaos Synchronization of Hyperchaotic Newton-Leipnik Systems by Slidin...ijctcm
This paper investigates the hybrid chaos synchronization of identical hyperchaotic Newton-Leipnik systems (Ghosh and Bhattacharya, 2010) by sliding mode control. The stability results derived in this paper for the hybrid chaos synchronization of identical hyperchaotic Newton-Leipnik systems are established using Lyapunov stability theory. Hybrid synchronization of hyperchaotic Newton-Leipnik systems is achieved through the complete synchronization of first and third states of the systems and the anti-synchronization of second and fourth states of the master and slave systems. Since the Lyapunov exponents are not required for these calculations, the sliding mode control is very effective and convenient to achieve hybrid chaos synchronization of the identical hyperchaotic Newton-Leipnik systems. Numerical simulations are shown to validate and demonstrate the effectiveness of the synchronization schemes derived in this paper.
Research on 4-dimensional Systems without Equilibria with ApplicationTELKOMNIKA JOURNAL
Recently chaos-based encryption has been obtained more and more attention. Chaotic systems
without equilibria may be suitable to be used to design pseudorandom number generators (PRNGs)
because there does not exist corresponding chaos criterion theorem on such systems. This paper
proposes two propositions on 4-dimensional systems without equilibria. Using one of the propositions
introduces a chaotic system without equilibria. Using this system and the generalized chaos
synchronization (GCS) theorem constructs an 8-dimensional discrete generalized chaos synchronization
(8DBDGCS) system. Using the 8DBDGCS system designs a 216-word chaotic PRNG. Simulation results
show that there are no significant correlations between the key stream and the perturbed key streams
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SLIDING MODE CONTROLLER DESIGN FOR GLOBAL CHAOS SYNCHRONIZATION OF COULLET SY...ijistjournal
This paper derives new results for the design of sliding mode controller for the global chaos synchronization of identical Coullet systems (1981). The synchronizer results derived in this paper for the complete chaos synchronization of identical hyperchaotic systems are established using sliding control theory and Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Coullet systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Coullet systems.
SLIDING MODE CONTROLLER DESIGN FOR GLOBAL CHAOS SYNCHRONIZATION OF COULLET SY...ijistjournal
This paper derives new results for the design of sliding mode controller for the global chaos synchronization of identical Coullet systems (1981). The synchronizer results derived in this paper for the complete chaos synchronization of identical hyperchaotic systems are established using sliding control theory and Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Coullet systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Coullet systems.
HYBRID SLIDING SYNCHRONIZER DESIGN OF IDENTICAL HYPERCHAOTIC XU SYSTEMS ijitjournal
In this paper, new results have been obtained via sliding mode control for the hybrid chaos synchronization
of identical hyperchaotic Xu systems (Xu, Cai and Zheng, 2009). In hybrid synchronization of master and
slave systems, the odd states are completely synchronized, while the even states are anti-synchronized. The
stability results derived in this paper for the hybrid synchronization of identical hyperchaotic Xu systems
are established using Lyapunov stability theory. MATLAB simulations have been shown for the numerical
results to illustrate the hybrid synchronization schemes derived for the identical hyperchaotic Xu systems.
SLIDING MODE CONTROLLER DESIGN FOR SYNCHRONIZATION OF SHIMIZU-MORIOKA CHAOTIC...ijistjournal
This paper investigates the global chaos synchronization of identical Shimizhu-Morioka chaotic systems (Shimizu and Morioka, 1980) by sliding mode control. The stability results derived in this paper for the complete synchronization of identical Shimizu-Morioka chaotic systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Shimizu-Morioka chaotic systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Shimizu-Morioka systems.
SLIDING MODE CONTROLLER DESIGN FOR SYNCHRONIZATION OF SHIMIZU-MORIOKA CHAOTIC...ijistjournal
This paper investigates the global chaos synchronization of identical Shimizhu-Morioka chaotic systems (Shimizu and Morioka, 1980) by sliding mode control. The stability results derived in this paper for the complete synchronization of identical Shimizu-Morioka chaotic systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Shimizu-Morioka chaotic systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Shimizu-Morioka systems.
On the State Observer Based Stabilization of T-S Systems with Maximum Converg...CSCJournals
This paper presents improved relaxed stabilization conditions and design procedures of state observers based controllers for continuous nonlinear systems in T-S model representation. First, the T-S model approach for nonlinear systems and some stabilization results are recalled. New stabilization conditions are obtained by relaxing those derived in previous works in this field. The asymptotic and exponential stabilization are considered with the maximization of the convergence rate. A design procedure for stabilizing T-S observer based controller using the concept of PDC (Parallel Distributed Compensation) and the improved relaxed stabilization conditions is proposed.
Adaptive Projective Lag Synchronization of T and Lu Chaotic Systems IJECEIAES
In this paper, the synchronization problem of T chaotic system and Lu chaotic system is studied. The parameter of the drive T chaotic system is considered unknown. An adaptive projective lag control method and also parameter estimation law are designed to achieve chaos synchronization problem between two chaotic systems. Then Lyapunov stability theorem is utilized to prove the validity of the proposed control method. After that, some numerical simulations are performed to assess the performance of the proposed method. The results show high accuracy of the proposed method in control and synchronization of chaotic systems.
Designing SDRE-Based Controller for a Class of Nonlinear Singularly Perturbed...Waqas Tariq
Designing a controller for nonlinear systems is difficult to be applied. Thus, it is usually based on a linearization around their equilibrium points. The state dependent Riccati equation control approach is an optimization method that has the simplicity of the classical linear quadratic control method. On the other hand, the singular perturbation theory is used for the decomposition of a high-order system into two lower-order systems. In this study, the finite-horizon optimization of a class of nonlinear singularly perturbed systems based on the singular perturbation theory and the state dependent Riccati equation technique together is addressed. In the proposed method, first, the Hamiltonian equations are described as a state-dependent Hamiltonian matrix, from which, the reduced-order subsystems are obtained. Then, these subsystems are converted into outerlayer, initial layer correction and final layer correction equations, from which, the separated state dependent Riccati equations are derived. The optimal control law is, then, obtained by computing the Riccati matrices.
ACTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC ZHEN...ijscai
This paper deals with a new research problem in the chaos literature, viz. hybrid synchronization of a
pair of chaotic systems called the master and slave systems. In the hybrid synchronization design of
master and slave systems, one part of the systems, viz. their odd states, are completely synchronized (CS),
while the other part, viz. their even states, are completely anti-synchronized (AS) so that CS and AS coexist in the process of synchronization. This research work deals with the hybrid synchronization of
hyperchaotic Zheng systems (2010) and hyperchaotic Yu systems (2012). The main results of this hybrid
synchronization research work have been proved using Lyapunov stability theory. Numerical examples of
the hybrid synchronization results are shown along with MATLAB simulations for the hyperchaotic
Zheng and hyperchaotic Yu systems.
THE LEFT AND RIGHT BLOCK POLE PLACEMENT COMPARISON STUDY: APPLICATION TO FLIG...ieijjournal1
It is known that if a linear-time-invariant MIMO system described by a state space equation has a number
of states divisible by the number of inputs and it can be transformed to block controller form, we can
design a state feedback controller using block pole placement technique by assigning a set of desired Block
poles. These may be left or right block poles. The idea is to compare both in terms of system’s response.
THE LEFT AND RIGHT BLOCK POLE PLACEMENT COMPARISON STUDY: APPLICATION TO FLIG...ieijjournal
It is known that if a linear-time-invariant MIMO system described by a state space equation has a number of states divisible by the number of inputs and it can be transformed to block controller form, we can design a state feedback controller using block pole placement technique by assigning a set of desired Block poles. These may be left or right block poles. The idea is to compare both in terms of system’s response.
An Exponential Observer Design for a Class of Chaotic Systems with Exponentia...ijtsrd
In this paper, a class of generalized chaotic systems with exponential nonlinearity is studied and the state observation problem of such systems is explored. Using differential inequality with time domain analysis, a practical state observer for such generalized chaotic systems is constructed to ensure the global exponential stability of the resulting error system. Besides, the guaranteed exponential decay rate can be correctly estimated. Finally, several numerical simulations are given to demonstrate the validity, effectiveness, and correctness of the obtained result. Yeong-Jeu Sun "An Exponential Observer Design for a Class of Chaotic Systems with Exponential Nonlinearity" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-1 , December 2020, URL: https://www.ijtsrd.com/papers/ijtsrd38233.pdf Paper URL : https://www.ijtsrd.com/engineering/electrical-engineering/38233/an-exponential-observer-design-for-a-class-of-chaotic-systems-with-exponential-nonlinearity/yeongjeu-sun
Anti-Synchronization Of Four-Scroll Chaotic Systems Via Sliding Mode Control IJITCA Journal
In this paper, new results are derived for the anti-synchronization of identical Liu-Chen four-scroll chaotic systems (Liu and Chen, 2004) and identical Lü-Chen-Cheng four-scroll chaotic systems (Lü,Chen and Cheng, 2004) by sliding mode control. The stability results derived in this paper for the antisynchronization of identical four-scroll chaotic systems are established using sliding mode control and Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve anti-synchronization of the identical four-scroll chaotic systems. Numerical simulations are shown to illustrate and validate the antisynchronization schemes derived in this paper for the identical four-scroll systems
ACTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC XU AN...Zac Darcy
The synchronization of chaotic systems treats a pair of chaotic systems, which are usually called as master
and slave systems. In the chaos synchronization problem, the goal of the design is to synchronize the states
of master and slave systems asymptotically. In the hybrid synchronization design of master and slave
systems, one part of the systems, viz. their odd states, are completely synchronized (CS), while the other
part, viz. their even states, are completely anti-synchronized (AS) so that CS and AS co-exist in the process
of synchronization. This research work deals with the hybrid synchronization of hyperchaotic Xi systems
(2009) and hyperchaotic Li systems (2005). The main results of this hybrid research work are established
with Lyapunov stability theory. MATLAB simulations of the hybrid synchronization results are shown for
the hyperchaotic Xu and Li systems.
ENHANCING ENGLISH WRITING SKILLS THROUGH INTERNET-PLUS TOOLS IN THE PERSPECTI...ijfcstjournal
This investigation delves into incorporating a hybridized memetic strategy within the framework of English
composition pedagogy, leveraging Internet Plus resources. The study aims to provide an in-depth analysis
of how this method influences students’ writing competence, their perceptions of writing, and their
enthusiasm for English acquisition. Employing an explanatory research design that combines qualitative
and quantitative methods, the study collects data through surveys, interviews, and observations of students’
writing performance before and after the intervention. Findings demonstrate a beneficial impact of
integrating the memetic approach alongside Internet Plus tools on the writing aptitude of English as a
Foreign Language (EFL) learners. Students reported increased engagement with writing, attributing it to
the use of Internet plus tools. They also expressed that the memetic approach facilitated a deeper
understanding of cultural and social contexts in writing. Furthermore, the findings highlight a significant
improvement in students’ writing skills following the intervention. This study provides significant insights
into the practical implementation of the memetic approach within English writing education, highlighting
the beneficial contribution of Internet Plus tools in enriching students' learning journeys.
A SURVEY TO REAL-TIME MESSAGE-ROUTING NETWORK SYSTEM WITH KLA MODELLINGijfcstjournal
Messages routing over a network is one of the most fundamental concept in communication which requires
simultaneous transmission of messages from a source to a destination. In terms of Real-Time Routing, it
refers to the addition of a timing constraint in which messages should be received within a specified time
delay. This study involves Scheduling, Algorithm Design and Graph Theory which are essential parts of
the Computer Science (CS) discipline. Our goal is to investigate an innovative and efficient way to present
these concepts in the context of CS Education. In this paper, we will explore the fundamental modelling of
routing real-time messages on networks. We study whether it is possible to have an optimal on-line
algorithm for the Arbitrary Directed Graph network topology. In addition, we will examine the message
routing’s algorithmic complexity by breaking down the complex mathematical proofs into concrete, visual
examples. Next, we explore the Unidirectional Ring topology in finding the transmission’s
“makespan”.Lastly, we propose the same network modelling through the technique of Kinesthetic Learning
Activity (KLA). We will analyse the data collected and present the results in a case study to evaluate the
effectiveness of the KLA approach compared to the traditional teaching method.
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It is known that if a linear-time-invariant MIMO system described by a state space equation has a number
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The synchronization of chaotic systems treats a pair of chaotic systems, which are usually called as master
and slave systems. In the chaos synchronization problem, the goal of the design is to synchronize the states
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systems, one part of the systems, viz. their odd states, are completely synchronized (CS), while the other
part, viz. their even states, are completely anti-synchronized (AS) so that CS and AS co-exist in the process
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Foreign Language (EFL) learners. Students reported increased engagement with writing, attributing it to
the use of Internet plus tools. They also expressed that the memetic approach facilitated a deeper
understanding of cultural and social contexts in writing. Furthermore, the findings highlight a significant
improvement in students’ writing skills following the intervention. This study provides significant insights
into the practical implementation of the memetic approach within English writing education, highlighting
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refers to the addition of a timing constraint in which messages should be received within a specified time
delay. This study involves Scheduling, Algorithm Design and Graph Theory which are essential parts of
the Computer Science (CS) discipline. Our goal is to investigate an innovative and efficient way to present
these concepts in the context of CS Education. In this paper, we will explore the fundamental modelling of
routing real-time messages on networks. We study whether it is possible to have an optimal on-line
algorithm for the Arbitrary Directed Graph network topology. In addition, we will examine the message
routing’s algorithmic complexity by breaking down the complex mathematical proofs into concrete, visual
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Software architecture is the structural solution that achieves the overall technical and operational
requirements for software developments. Software engineers applied software architectures for their
software system developments; however, they worry the basic benchmarks in order to select software
architecture styles, possible components, integration methods (connectors) and the exact application of
each style.
The objective of this research work was a comparative analysis of software architecture styles by its
weakness and benefits in order to select by the programmer during their design time. Finally, in this study,
the researcher has been identified architectural styles, weakness, and Strength and application areas with
its component, connector and Interface for the selected architectural styles.
SYSTEM ANALYSIS AND DESIGN FOR A BUSINESS DEVELOPMENT MANAGEMENT SYSTEM BASED...ijfcstjournal
A design of a sales system for professional services requires a comprehensive understanding of the
dynamics of sale cycles and how key knowledge for completing sales is managed. This research describes
a design model of a business development (sales) system for professional service firms based on the Saudi
Arabian commercial market, which takes into account the new advances in technology while preserving
unique or cultural practices that are an important part of the Saudi Arabian commercial market. The
design model has combined a number of key technologies, such as cloud computing and mobility, as an
integral part of the proposed system. An adaptive development process has also been used in implementing
the proposed design model.
AN ALGORITHM FOR SOLVING LINEAR OPTIMIZATION PROBLEMS SUBJECTED TO THE INTERS...ijfcstjournal
Frank t-norms are parametric family of continuous Archimedean t-norms whose members are also strict
functions. Very often, this family of t-norms is also called the family of fundamental t-norms because of the
role it plays in several applications. In this paper, optimization of a linear objective function with fuzzy
relational inequality constraints is investigated. The feasible region is formed as the intersection of two
inequality fuzzy systems defined by frank family of t-norms is considered as fuzzy composition. First, the
resolution of the feasible solutions set is studied where the two fuzzy inequality systems are defined with
max-Frank composition. Second, some related basic and theoretical properties are derived. Then, a
necessary and sufficient condition and three other necessary conditions are presented to conceptualize the
feasibility of the problem. Subsequently, it is shown that a lower bound is always attainable for the optimal
objective value. Also, it is proved that the optimal solution of the problem is always resulted from the
unique maximum solution and a minimal solution of the feasible region. Finally, an algorithm is presented
to solve the problem and an example is described to illustrate the algorithm. Additionally, a method is
proposed to generate random feasible max-Frank fuzzy relational inequalities. By this method, we can
easily generate a feasible test problem and employ our algorithm to it.
LBRP: A RESILIENT ENERGY HARVESTING NOISE AWARE ROUTING PROTOCOL FOR UNDER WA...ijfcstjournal
Underwater detector network is one amongst the foremost difficult and fascinating analysis arenas that
open the door of pleasing plenty of researchers during this field of study. In several under water based
sensor applications, nodes are square measured and through this the energy is affected. Thus, the mobility
of each sensor nodes are measured through the water atmosphere from the water flow for sensor based
protocol formations. Researchers have developed many routing protocols. However, those lost their charm
with the time. This can be the demand of the age to supply associate degree upon energy-efficient and
ascendable strong routing protocol for under water actuator networks. During this work, the authors tend
to propose a customary routing protocol named level primarily based routing protocol (LBRP), reaching to
offer strong, ascendable and energy economical routing. LBRP conjointly guarantees the most effective use
of total energy consumption and ensures packet transmission which redirects as an additional reliability in
compare to different routing protocols. In this work, the authors have used the level of forwarding node,
residual energy and distance from the forwarding node to the causing node as a proof in multicasting
technique comparisons. Throughout this work, the authors have got a recognition result concerning about
86.35% on the average in node multicasting performances. Simulation has been experienced each in a
wheezy and quiet atmosphere which represents the endorsement of higher performance for the planned
protocol.
STRUCTURAL DYNAMICS AND EVOLUTION OF CAPSULE ENDOSCOPY (PILL CAMERA) TECHNOLO...ijfcstjournal
This research paper examined and re-evaluates the technological innovation, theory, structural dynamics
and evolution of Pill Camera(Capsule Endoscopy) technology in redirecting the response manner of small
bowel (intestine) examination in human. The Pill Camera (Endoscopy Capsule) is made up of sealed
biocompatible material to withstand acid, enzymes and other antibody chemicals in the stomach is a
technology that helps the medical practitioners especially the general physicians and the
gastroenterologists to examine and re-examine the intestine for possible bleeding or infection. Before the
advent of the Pill camera (Endoscopy Capsule) the colonoscopy was the local method used but research
showed that some parts (bowel) of the intestine can’t be reach by mere traditional method hence the need
for Pill Camera. Countless number of deaths from stomach disease such as polyps, inflammatory bowel
(Crohn”s diseases), Cancers, Ulcer, anaemia and tumours of small intestines which ordinary would have
been detected by sophisticated technology like Pill Camera has become norm in the developing nations.
Nevertheless, not only will this paper examine and re-evaluate the Pill Camera Innovation, theory,
Structural dynamics and evolution it unravelled and aimed to create awareness for both medical
practitioners and the public.
AN OPTIMIZED HYBRID APPROACH FOR PATH FINDINGijfcstjournal
Path finding algorithm addresses problem of finding shortest path from source to destination avoiding
obstacles. There exist various search algorithms namely A*, Dijkstra's and ant colony optimization. Unlike
most path finding algorithms which require destination co-ordinates to compute path, the proposed
algorithm comprises of a new method which finds path using backtracking without requiring destination
co-ordinates. Moreover, in existing path finding algorithm, the number of iterations required to find path is
large. Hence, to overcome this, an algorithm is proposed which reduces number of iterations required to
traverse the path. The proposed algorithm is hybrid of backtracking and a new technique(modified 8-
neighbor approach). The proposed algorithm can become essential part in location based, network, gaming
applications. grid traversal, navigation, gaming applications, mobile robot and Artificial Intelligence.
EAGRO CROP MARKETING FOR FARMING COMMUNITYijfcstjournal
The Major Occupation in India is the Agriculture; the people involved in the Agriculture belong to the poor
class and category. The people of the farming community are unaware of the new techniques and Agromachines, which would direct the world to greater heights in the field of agriculture. Though the farmers
work hard, they are cheated by agents in today’s market. This serves as a opportunity to solve
all the problems that farmers face in the current world. The eAgro crop marketing will serve as a better
way for the farmers to sell their products within the country with some mediocre knowledge about using
the website. This would provide information to the farmers about current market rate of agro-products,
their sale history and profits earned in a sale. This site will also help the farmers to know about the market
information and to view agricultural schemes of the Government provided to farmers.
EDGE-TENACITY IN CYCLES AND COMPLETE GRAPHSijfcstjournal
It is well known that the tenacity is a proper measure for studying vulnerability and reliability in graphs.
Here, a modified edge-tenacity of a graph is introduced based on the classical definition of tenacity.
Properties and bounds for this measure are introduced; meanwhile edge-tenacity is calculated for cycle
graphs and also for complete graphs.
COMPARATIVE STUDY OF DIFFERENT ALGORITHMS TO SOLVE N QUEENS PROBLEMijfcstjournal
This Paper provides a brief description of the Genetic Algorithm (GA), the Simulated Annealing (SA)
Algorithm, the Backtracking (BT) Algorithm and the Brute Force (BF) Search Algorithm and attempts to
explain the way as how the Proposed Genetic Algorithm (GA), the Proposed Simulated Annealing (SA)
Algorithm using GA, the Backtracking (BT) Algorithm and the Brute Force (BF) Search Algorithm can be
employed in finding the best solution of N Queens Problem and also, makes a comparison between these
four algorithms. It is entirely a review based work. The four algorithms were written as well as
implemented. From the Results, it was found that, the Proposed Genetic Algorithm (GA) performed better
than the Proposed Simulated Annealing (SA) Algorithm using GA, the Backtracking (BT) Algorithm and
the Brute Force (BF) Search Algorithm and it also provided better fitness value (solution) than the
Proposed Simulated Annealing Algorithm (SA) using GA, the Backtracking (BT) Algorithm and the Brute
Force (BF) Search Algorithm, for different N values. Also, it was noticed that, the Proposed GA took more
time to provide result than the Proposed SA using GA.
PSTECEQL: A NOVEL EVENT QUERY LANGUAGE FOR VANET’S UNCERTAIN EVENT STREAMSijfcstjournal
In recent years, the complex event processing technology has been used to process the VANET’s temporal
and spatial event streams. However, we usually cannot get the accurate data because the device sensing
accuracy limitations of the system. We only can get the uncertain data from the complex and limited
environment of the VANET. Because the VANET’s event streams are consist of the uncertain data, so they
are also uncertain. How effective to express and process these uncertain event streams has become the core
issue for the VANET system. To solve this problem, we propose a novel complex event query language
PSTeCEQL (probabilistic spatio-temporal constraint event query language). Firstly, we give the definition
of the possible world model of VANET’s uncertain event streams. Secondly, we propose an event query
language PSTeCEQL and give the syntax and the operational semantics of the language. Finally, we
illustrate the validity of the PSTeCEQL by an example.
CLUSTBIGFIM-FREQUENT ITEMSET MINING OF BIG DATA USING PRE-PROCESSING BASED ON...ijfcstjournal
Now a day enormous amount of data is getting explored through Internet of Things (IoT) as technologies
are advancing and people uses these technologies in day to day activities, this data is termed as Big Data
having its characteristics and challenges. Frequent Itemset Mining algorithms are aimed to disclose
frequent itemsets from transactional database but as the dataset size increases, it cannot be handled by
traditional frequent itemset mining. MapReduce programming model solves the problem of large datasets
but it has large communication cost which reduces execution efficiency. This proposed new pre-processed
k-means technique applied on BigFIM algorithm. ClustBigFIM uses hybrid approach, clustering using kmeans algorithm to generate Clusters from huge datasets and Apriori and Eclat to mine frequent itemsets
from generated clusters using MapReduce programming model. Results shown that execution efficiency of
ClustBigFIM algorithm is increased by applying k-means clustering algorithm before BigFIM algorithm as
one of the pre-processing technique.
A MUTATION TESTING ANALYSIS AND REGRESSION TESTINGijfcstjournal
Software testing is a testing which conducted a test to provide information to client about the quality of the
product under test. Software testing can also provide an objective, independent view of the software to
allow the business to appreciate and understand the risks of software implementation. In this paper we
focused on two main software testing –mutation testing and mutation testing. Mutation testing is a
procedural testing method, i.e. we use the structure of the code to guide the test program, A mutation is a
little change in a program. Such changes are applied to model low level defects that obtain in the process
of coding systems. Ideally mutations should model low-level defect creation. Mutation testing is a process
of testing in which code is modified then mutated code is tested against test suites. The mutations used in
source code are planned to include in common programming errors. A good unit test typically detects the
program mutations and fails automatically. Mutation testing is used on many different platforms, including
Java, C++, C# and Ruby. Regression testing is a type of software testing that seeks to uncover
new software bugs, or regressions, in existing functional and non-functional areas of a system after
changes such as enhancements, patches or configuration changes, have been made to them. When defects
are found during testing, the defect got fixed and that part of the software started working as needed. But
there may be a case that the defects that fixed have introduced or uncovered a different defect in the
software. The way to detect these unexpected bugs and to fix them used regression testing. The main focus
of regression testing is to verify that changes in the software or program have not made any adverse side
effects and that the software still meets its need. Regression tests are done when there are any changes
made on software, because of modified functions.
GREEN WSN- OPTIMIZATION OF ENERGY USE THROUGH REDUCTION IN COMMUNICATION WORK...ijfcstjournal
Advances in micro fabrication and communication techniques have led to unimaginable proliferation of
WSN applications. Research is focussed on reduction of setup operational energy costs. Bulk of operational
energy costs are linked to communication activities of WSN. Any progress towards energy efficiency has a
potential of huge savings globally. Therefore, every energy efficient step is an endeavour to cut costs and
‘Go Green’. In this paper, we have proposed a framework to reduce communication workload through: Innetwork compression and multiple query synthesis at the base-station and modification of query syntax
through introduction of Static Variables. These approaches are general approaches which can be used in
any WSN irrespective of application.
A NEW MODEL FOR SOFTWARE COSTESTIMATION USING HARMONY SEARCHijfcstjournal
Accurate and realistic estimation is always considered to be a great challenge in software industry.
Software Cost Estimation (SCE) is the standard application used to manage software projects. Determining
the amount of estimation in the initial stages of the project depends on planning other activities of the
project. In fact, the estimation is confronted with a number of uncertainties and barriers’, yet assessing the
previous projects is essential to solve this problem. Several models have been developed for the analysis of
software projects. But the classical reference method is the COCOMO model, there are other methods
which are also applied such as Function Point (FP), Line of Code(LOC); meanwhile, the expert`s opinions
matter in this regard. In recent years, the growth and the combination of meta-heuristic algorithms with
high accuracy have brought about a great achievement in software engineering. Meta-heuristic algorithms
which can analyze data from multiple dimensions and identify the optimum solution between them are
analytical tools for the analysis of data. In this paper, we have used the Harmony Search (HS)algorithm for
SCE. The proposed model which is a collection of 60 standard projects from Dataset NASA60 has been
assessed.The experimental results show that HS algorithm is a good way for determining the weight
similarity measures factors of software effort, and reducing the error of MRE.
AGENT ENABLED MINING OF DISTRIBUTED PROTEIN DATA BANKSijfcstjournal
Mining biological data is an emergent area at the intersection between bioinformatics and data mining
(DM). The intelligent agent based model is a popular approach in constructing Distributed Data Mining
(DDM) systems to address scalable mining over large scale distributed data. The nature of associations
between different amino acids in proteins has also been a subject of great anxiety. There is a strong need to
develop new models and exploit and analyze the available distributed biological data sources. In this study,
we have designed and implemented a multi-agent system (MAS) called Agent enriched Quantitative
Association Rules Mining for Amino Acids in distributed Protein Data Banks (AeQARM-AAPDB). Such
globally strong association rules enhance understanding of protein composition and are desirable for
synthesis of artificial proteins. A real protein data bank is used to validate the system.
International Journal on Foundations of Computer Science & Technology (IJFCST)ijfcstjournal
International Journal on Foundations of Computer Science & Technology (IJFCST) is a Bi-monthly peer-reviewed and refereed open access journal that publishes articles which contribute new results in all areas of the Foundations of Computer Science & Technology. Over the last decade, there has been an explosion in the field of computer science to solve various problems from mathematics to engineering. This journal aims to provide a platform for exchanging ideas in new emerging trends that needs more focus and exposure and will attempt to publish proposals that strengthen our goals. Topics of interest include, but are not limited to the following:
Because the technology is used largely in the last decades; cybercrimes have become a significant
international issue as a result of the huge damage that it causes to the business and even to the ordinary
users of technology. The main aims of this paper is to shed light on digital crimes and gives overview about
what a person who is related to computer science has to know about this new type of crimes. The paper has
three sections: Introduction to Digital Crime which gives fundamental information about digital crimes,
Digital Crime Investigation which presents different investigation models and the third section is about
Cybercrime Law.
DISTRIBUTION OF MAXIMAL CLIQUE SIZE UNDER THE WATTS-STROGATZ MODEL OF EVOLUTI...ijfcstjournal
In this paper, we analyze the evolution of a small-world network and its subsequent transformation to a
random network using the idea of link rewiring under the well-known Watts-Strogatz model for complex
networks. Every link u-v in the regular network is considered for rewiring with a certain probability and if
chosen for rewiring, the link u-v is removed from the network and the node u is connected to a randomly
chosen node w (other than nodes u and v). Our objective in this paper is to analyze the distribution of the
maximal clique size per node by varying the probability of link rewiring and the degree per node (number
of links incident on a node) in the initial regular network. For a given probability of rewiring and initial
number of links per node, we observe the distribution of the maximal clique per node to follow a Poisson
distribution. We also observe the maximal clique size per node in the small-world network to be very close
to that of the average value and close to that of the maximal clique size in a regular network. There is no
appreciable decrease in the maximal clique size per node when the network transforms from a regular
network to a small-world network. On the other hand, when the network transforms from a small-world
network to a random network, the average maximal clique size value decreases significantly
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Techniques to optimize the pagerank algorithm usually fall in two categories. One is to try reducing the work per iteration, and the other is to try reducing the number of iterations. These goals are often at odds with one another. Skipping computation on vertices which have already converged has the potential to save iteration time. Skipping in-identical vertices, with the same in-links, helps reduce duplicate computations and thus could help reduce iteration time. Road networks often have chains which can be short-circuited before pagerank computation to improve performance. Final ranks of chain nodes can be easily calculated. This could reduce both the iteration time, and the number of iterations. If a graph has no dangling nodes, pagerank of each strongly connected component can be computed in topological order. This could help reduce the iteration time, no. of iterations, and also enable multi-iteration concurrency in pagerank computation. The combination of all of the above methods is the STICD algorithm. [sticd] For dynamic graphs, unchanged components whose ranks are unaffected can be skipped altogether.
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On the Principle of Optimality for Linear Stochastic Dynamic System
1. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016
DOI:10.5121/ijfcst.2016.6105 57
ON THE PRINCIPLE OF OPTIMALITY FOR LINEAR
STOCHASTIC DYNAMIC SYSTEM
Yakup H. HACI, Muhammet CANDAN and Aykut OR
Department of Mathematics, Canakkale Onsekiz Mart University, Canakkale, Turkey
ABSTRACT
In this work, processes represented by linear stochastic dynamic system are investigated and by
considering optimal control problem, principle of optimality is proven. Also, for existence of optimal
control and corresponding optimal trajectory, proofs of theorems of necessity and sufficiency condition are
attained.
KEYWORDS
Optimal control problem, linear stochastic dynamic systems, optimal trajectory.
1. INTRODUCTION
Linear stochastic dynamic system (LSDS) is the one of the most developing field of discrete
system theory. Since LSDS theory has seen an important growth due to its interaction with fields
like base of computer science, graph theory, coding theoy, diagnosing problem in the discrete
events, imitation problem etc. Thus, it requires more detailed investigation.
The mathematical theory of optimal control consisting all the continuous and discrete systems is
strong studied more than 40 years. L. S. Pontryagin et al. first obtained basic results of the theory
of optimal control and then this theory was developed and simplificated by other academicians
[1-4]. Gaishun presented a detailed works regarding the optimal control problem to
multidimensional differential equation[5]. Gabasov et al. considered a linear quadratic optimal
control problem with geometric constraints on the control actions and gave some novel numerical
methods for both optimal program and positional solutions[6]. Also, they studied a problem of
linear optimal control under uncertainty[7]. We firstly investigated the optimal discrete processes
given by nonlinear multiparameter stochastic dynamic systems.[8-9]
The present article is devoted the determination of optimal control and corresponding trajectory
with necessary and sufficient condition theorems in the linear stochastic dynamic system. First
section introduces general definition and basic concepts of the LSDS. Then in the second
section,existence of optimal control and corresponding trajectory is discussed as necessary and
sufficient conditions. Finally, main idea and aim of this paper is presented.
2. LINEAR STOCHASTIC DYNAMIC SYSTEM
LSDS is generalization of multi-parametric finite sequential dynamic system but it contains a
probability variable. General form of this system is defined by[5]:
>
⋅
=< )
(
),
(
,
,
,
, 0
ν
ω F
p
s
Y
S
X
K k
v ,...
2
,
1
= (1)
2. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016
58
where [ ] [ ]m
r
GF
S
GF
X )
2
(
,
)
2
( =
= and [ ]q
GF
Y )
2
(
= are input, state and output index
(alphabet) respectively; 0
s is initial state vector, )
(ω
p is determitistic discrete probability
distribution. ( }
,...,
,
{ 2
1 p
ω
ω
ω
=
Ω is finite set, })
1
)
(
,
:
)
(
{
)
( =
Ω
∈
= ∑
Ω
∈
i
i
i
i p
p
p
ω
ω
ω
ω
ω
Characteristic Boolean vector function [12] denoted by { }
)
(
),...,
(
)
( 1
⋅
⋅
=
⋅ m
F
F
F ν
ν
ν which is also
known as transfer functions is equal;
))
(
)
(
)(
(
)
(
)
(
)
(
))
(
)
(
),
(
,
( c
c
x
c
c
s
c
c
s
c
c
x
c
s
c
F ω
ξ
ω ν
ν
ν
ν ⊕
Ψ
⊕
Φ
=
=
⊕ v= 1,...,k (2)
where the points = , , … , ∈ = { }
Z
c
c
c
c
c
c
c
Z
c
c i
L
k
k
k
L
k k
∈
≤
≤
≤
≤
∈ ,
,...,
, 0
1
1
0
1
1
are points in k
Z , determining position i
L , ( k
i ,...
2
,
1
= positive integer) is the duration of the
stage i of this process. Z is set of integers, for ∈ S , x ∈ X ; ),
(c
s )
(c
x are defined
over the set d
G as an m and dimensional state and input vectors. )
(c
s
ν
ξ is a shift
operator[5],[11] defined as:
)
(
)
( ν
ν
ξ e
c
s
c
s +
= , )
0
,...,
0
,
1
,
0
,...,
0
(
ν
ν =
e , v= 1,...,k (3)
{ }
k
c ,...
1
),
( =
Φ ν
ν , { }
k
,...,
1
, =
Ψ ν
ν are characteristic Boolean transition matrices of size m
m×
and r
m× respectively. ω is a random variable. )
2
(
GF is Galois field and all operations are on
)
2
(
GF in this study.
In LSDS, each random variable ω has a special case. In (2) ω is an input variable . However, ω
can be a set of all possible states in the synthesis of dynamic systems.
Moreover, the state of the system depends on the random variable ω which affects not only
parameters of the LSDS but also the input variables.
The discrete optimal processes given by LSDS are characterized by functional:
)}
(
{
)
( L
T
c
as
M
x
J ω
= (4)
Where {.}
ω
M is a mathematical expected value of ,
ω )
,...,
,
( 2
1 m
a
a
a
a = is given Boolean
vector and )
( L
T
c
s is transpose of )
( L
c
s . It is clearly seen that for X
c
x ∈
)
( , the number of
equations in (1) is more than the number of unknowns. Therefore, the considered problem can be
determined as an admissible optimal control problem [4], [6-7].
Definition. If there is a unique solution in (2) for X
c
x ∈
)
( , then )
(c
x is called an admissible
control[8-9].
Let X̂ be the set of admissible administrator functions defined from d
Ĝ { }
L
d c
G
= into
[ ]r
GF )
2
(
3. OPTIMAL CONTROL PROBLEM AND PRINCIPLE OF OPTIMALITY
For linear stochastic dynamic system , we can analyze the following problem:
3. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016
59
For a given linear stochastic sequential dynamic system, to move from an initial state 0
s to a
desired terminal state )
(
* L
c
s in L steps, a control ∈
)
(c
x X̂ must exist such that the functional
(4) has a minimum value. In other words;
))
(
)
(
)(
(
)
(
)
(
)
( c
c
x
c
c
s
c
c
s ω
ξ ν
ν
ν ⊕
Ψ
⊕
Φ
= v= 1,...,k
X
c
x ˆ
)
( ∈ , d
G
c ˆ
∈
min
)}
(
{
)
( →
= L
T
c
as
M
x
J ω
Assume that condition of the unique solution [5],[8] in (2) is holds. Then sequential dynamic
system can written as below;
,
),
(
)
(
)
( d
G
c
c
c
c ∈
Φ
= ν
ν ξ
ϕ
ϕ v= 1,...,k, (5)
and by considering this system, if Boolean function is constructed as follows,
)),
(
)
(
)(
(
)
(
))
(
),
(
)
(
,
( c
c
x
c
c
c
c
c
x
c
h ω
ξ
ϕ
ξ
ϕ
ω ν
ν
ν
ν ⊕
Ψ
=
⊕ v= 1,...,k, (6)
Then by using this function we form the following sum:
∑ ∑
=
∆
⊕
=
)
,
( 1
,
2
))
(
),
(
)
(
,
(
ˆ
L
l
c
c
L
k
c
c
c
c
x
c
h
h
ν
ν
ν
ν ξ
ϕ
ω (7)
Where 2
∆ is the first Boolean difference
Theorem I Suppose that the system in (2) has an exact solution then the sum in (7) is path
independent.
Proof It is precise that if the equation
=
⊕
Ψ
⊕
⊕
Ψ ))
(
)
(
)(
(
)
(
))
(
)
(
)(
(
)
( '
' c
c
x
c
c
c
c
x
c
c v ν
ν
ν
ν
ν
ν
ν ξ
ω
ξ
ξ
ξ
ξ
ϕ
ω
ξ
ϕ (8)
)))
(
))
(
)(
(
)
(
))
(
)
(
)(
(
)
( '
'
'
'
'
' c
c
x
c
c
c
c
x
c
c v
v ν
ν
ν
ν
ν
ν
ξ
ω
ξ
ξ
ξ
ξ
ϕ
ω
ξ
ϕ ⊕
Ψ
⊕
⊕
Ψ
= .
,...,
2
,
1
, '
k
v
v =
holds, then Boolean sum in (7) is path independent. By using equivalent equations like:
,
),
(
)
(
)
( d
G
c
c
c
c ∈
Φ
= ν
ν ξ
ϕ
ϕ (9)
It is found[4]
(
)
( ' =
c
ν
ν
ξ
ξ
ϕ (
'
ν
Φ )
(
))
(
(
)) 1
1
c
c
c ϕ
ξ ν
ν
−
−
Φ ,
(
)
( ' =
c
ν
ν ξ
ξ
ϕ (
ν
Φ )
(
))
(
(
)) 1
1
'
' c
c
c ϕ
ξ ν
ν
−
−
Φ , .
,...,
2
,
1
, '
k
v
v = (10)
By substituting (10) into (8),
)
(
))
(
)
(
)(
(
))
(
(
))
(
(
)
(
))
(
)
(
)(
(
))
(
( '
'
1
1
1
c
c
c
x
c
c
c
c
c
c
x
c
c v
ϕ
ξ
ω
ξ
ξ
ξ
ϕ
ω ν
ν
ν
ν
ν
ν
ν
ν ⊕
Ψ
Φ
Φ
⊕
⊕
Ψ
Φ −
−
−
⊕
⊕
Ψ
Φ
= −
)
(
))
(
)
(
)(
(
))
(
( '
'
1
c
c
c
x
c
c v
ϕ
ω
ν
(
( ν
Φ )
(
))
(
)
(
)(
(
))
(
(
)) '
'
'
'
'
1
1
c
c
c
x
c
c
c ϕ
ξ
ω
ξ
ξ
ξ ν
ν
ν
ν
ν
ν
⊕
Ψ
Φ −
−
. (11)
0
0
)
( s
c
s =
4. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016
60
Multiplying the above equations from the both sides by
)
(
)
(
)
(
)
( '
'
' c
c
c
c ν
ν
ν
ν
ν
ν ξ
ξ Φ
Φ
=
Φ
Φ (12)
(11) forms,
)
(
))
(
)
(
)(
(
)
(
))
(
)
(
)(
(
)
( '
' c
c
c
x
c
c
c
c
x
c
c v
ϕ
ξ
ω
ξ
ξ
ϕ
ω
ξ ν
ν
ν
ν
ν
ν
⊕
Ψ
⊕
⊕
Ψ
Φ
= (
( ν
Φ )
(
))
(
))
(
)(
(
)
(
))
(
)
(
)(
(
) '
'
'
'
' c
c
c
x
c
c
c
c
x
c
c v ϕ
ξ
ω
ξ
ξ
ϕ
ω
ξ ν
ν
ν
ν
ν
⊕
Ψ
⊕
⊕
Ψ (13)
Now the equation (13) holds for every )
(c
ϕ , it is written by,
))
(
)
(
)(
(
))
(
)
(
)(
(
)
( '
' =
⊕
Ψ
⊕
⊕
Ψ
Φ c
c
x
c
c
c
x
c
c v ν
ν
ν
ν
ν
ν
ξ
ω
ξ
ξ
ω
ξ
))
(
)
(
)(
(
))
(
)
(
)(
(
)
( '
'
'
'
' c
c
x
c
c
c
x
c
c ν
ν
ν
ν
ν
ν
ν ξ
ω
ξ
ξ
ω
ξ ⊕
Ψ
⊕
⊕
Ψ
Φ
= (14)
The equation (14) is the exact solution of the system in (2). Since the solution is the exact,
the equation (8) holds and the Boolean sum in (7) is path independent.
Let )
,...,
,
(
ˆ 1
0 L
c
c
c
L be a piecewise curve connecting the point 0
c to the point L
c . In the th
l step
where L
l <
≤
0 , assigning a value t
ν to the piece of curve from l
c to the point 1
+
l
c . Then the
Boolean sum is in (7) has the following form:
∑ ∑
+
=
−
−
−
−
−
−
⊕
=
⊕
=
)
,
( 1
1
1
1
1
1
1
))
(
),
(
)
(
,
(
))
(
),
(
)
(
,
(
ˆ
L
l
t
t
c
c
L
l
t
t
t
t
t
t
t
t
t
c
c
c
x
c
h
c
c
c
x
c
h
h ϕ
ω
ϕ
ω ν
ν (15)
Hamilton- Pontryagin functional
),
(
)
(
))
(
),
(
( c
s
c
c
s
c
H ϕ
ϕ = d
G
c∈ (16)
can be involved, which is similar to Boolean functional.
Theorem II A control { }
)
(
),...,
(
),
(
)
,
( 1
0
1
0
0
1
0 −
+
−
= L
l
l
L
l
c
x
c
x
c
x
c
c
x and corresponding
trajectory { }
)
(
),...,
(
),
(
)
,
( 0
1
0
0
0 L
l
l
L
l
c
s
c
s
c
s
c
c
s +
= are optimal if and only if,
∑
+
=
−
−
−
⊕
=
L
l
t
t
t
t
t
l
l
c
c
x
c
c
h
c
s
c
H t
1
1
1
0
0
1
0
0
))
(
)
(
),
(
,
(
))
(
),
(
( ω
ϕ
ϕ ν (17)
Where )
(
0 t
c
ϕ , 1
,...,
1
, −
+
= L
l
l
t is obtained from the equation below;
,
),
(
)
(
)
( d
G
c
c
c
c ∈
Φ
= ν
ν ξ
ϕ
ϕ k
v ,....,
2
,
1
= (18)
With the aid of equation a
cL
=
)
(
0
ϕ .
Proof Firstly, if necessity condition is proven then, from (7) for L
t
l <
≤
+1 ,
)
(
)
(
))
(
),
(
( 0
0
0
0 t
t
t
t
c
s
c
c
s
c
H ϕ
ϕ =
))
(
)
(
)(
(
)
(
)
(
)
(
)
( 1
1
0
1
0
1
0
1
0 −
−
−
−
−
⊕
Ψ
⊕
Φ
= t
t
t
t
t
t
t
c
c
x
c
c
c
s
c
c t
t
ω
ϕ
ϕ ν
ν
))
(
)
(
)(
(
)
(
)
(
)
( 1
1
0
1
0
0
0 −
−
−
−
−
⊕
Ψ
⊕
= t
t
t
t
l
t
l
t
c
c
x
c
c
c
s
c t
ω
ϕ
ϕ ν
5. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016
61
))
(
),
(
)
(
,
(
))
(
),
(
( 0
1
1
0
1
0
0 t
t
t
t
v
l
t
l
t
c
c
c
x
c
h
c
s
c
H t
ϕ
ω
ϕ −
−
−
−
−
⊕
⊕
= . (19)
When the sum is taken from 1
+
l to L , the following equalities are;
))
(
,
)
(
)
(
,
(
))
(
),
(
(
))
(
),
(
( 0
1 1
1
1
0
1
1
0
1
0
1
0
0 t
L
l
t
L
l
t
t
t
t
t
t
L
l
t
t
t
c
c
c
x
c
h
c
s
c
H
c
s
c
H t
ϕ
ω
ϕ
ϕ ν
∑ ∑
∑ +
= +
=
−
−
−
−
−
+
=
⊕
⊕
=
(20)
and
∑
+
=
−
−
−
⊕
⊕
=
L
l
t
t
t
t
t
l
l
L
L
c
c
c
x
c
h
c
s
c
H
c
s
c
H t
1
0
1
1
0
1
0
0
0
0
))
(
),
(
)
(
,
(
))
(
),
(
(
))
(
),
(
( ϕ
ω
ϕ
ϕ ν (21)
are obtained.
On the contrary,
))
(
),
(
(
)
(
)
(
)
(
))
(
( 0
0
0
0
0
0 L
L
L
L
L
L
c
s
c
H
c
s
c
c
as
c
s ϕ
ϕ =
=
=
Φ (22)
Since )
(
0 L
c
s is an optimal state,
0
))
(
),
(
( 0
0
=
L
L
c
s
c
H ϕ (23)
is found.
Therefore, (17) is valid. This completes the proof of the necessity condition.
Sufficiency: The sufficient condition is proven in a similar form as of necessity and from
(21) the desired result is obtained.
Under assumption of equation (2) in equilibrium state, the following theorem can be
given;
Theorem III A control )
(
0
c
x and corresponding trajectory )
(
0
c
s are optimal in (1) if and only
if,
∑ ∑ ∑ ∑
= =
=
∆
⊕
×
∆
⊕
)
,
( 1 )
,
( 1
2
0
2
0
0
0
0 0
0
))
(
),
(
)
(
,
(
))
(
),
(
)
(
,
(
L L
c
c
k
c
c
k
c
c
c
c
x
c
h
c
c
c
c
x
c
h
ν ν
ν
ν
ν
ν
ν
ν ξ
ϕ
ω
ξ
ϕ
ω
(24)
Where the bar denotes the complement of this sum applied on the piecewise curve
connecting 0
c to L
c with all admissible controls.
Proof Necessity: By the exact solution of (2),
=
)
( L
c
s ∑ ∑
=
∆
⊕
Ψ
Φ
)
,
( 1
2
^
0
))
(
)
(
)(
(
)
,
(
L
c
c
K
L
c
c
c
x
c
c
c
ν
ν
ν
ν ω
ξ (25)
=
= )
(
)
( L
c
as
x
J ∑ ∑
=
∆
⊕
Ψ
Φ
)
,
( 1
2
^
0
))
(
)
(
)(
(
)
,
(
L
c
c
K
L
c
c
c
x
c
c
c
a
ν
ν
ν
ν ω
ξ (26)
Let )
,
(
)
(
^
c
c
a
c L
ν
ν ξ
ξ
ϕ Φ
= where 1
1
0
,....,
, −
= L
c
c
c
c and (
)
(
)
,
( 1
^
−
=
Φ S
c
S
c
c L
L
ν
ξ )
c
ν
ξ .
6. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016
62
)
(c
S is the fundemental matrix operator [5]of the system
0
0
)
(
),
(
)
(
)
(
s
c
s
c
s
c
c
s v
=
Φ
=
ν
ξ
(27)
at the point c. Thus a
cL
=
)
(
0
ϕ and for the optimal control, minimum is taken as;
∑ ∑
∑ ∑ =
=
∆
⊕
=
∆
⊕
)
,
( 1
2
0
)
,
( 1
2
0
0
0
0
0
))
(
),
(
)
(
,
(
min
))
(
),
(
)
(
,
(
L
L
c
c
k
c
c
k
c
c
c
c
x
c
h
c
c
c
c
x
c
h
ν
ν
ν
ν
ν
ν
ν
ν ξ
ϕ
ω
ξ
ϕ
ω
(28)
Considering the curve connecting 0
c to L
c over the all admissible controls )
(c
x , (16)
equals to Boolean equality,
∑ ∑
∑ ∑ =
=
∆
⊕
≥
∆
⊕
)
,
( 1
2
0
0
0
)
,
( 1
2
0
0
0
))
(
),
(
)
(
,
(
))
(
),
(
)
(
,
(
L
L
c
c
k
c
c
k
c
c
c
c
x
c
h
c
c
c
c
x
c
h
ν
ν
ν
ν
ν
ν
ν
ν ξ
ϕ
ω
ξ
ϕ
ω
(29)
Alternatively, the inequality (29) equals to;
∑ ∑
=
∆
⊕
)
,
( 1
2
0
0
))
(
),
(
)
(
,
(
L
c
c
k
c
c
c
c
x
c
h
ν
ν
ν
ν ξ
ϕ
ω ∑ ∑
=
=
∆
⊕
)
,
( 1
2
0
0
0
0
0
))
(
),
(
)
(
,
(
L
c
c
k
c
c
c
c
x
c
h
ν
ν
ν
ν ξ
ϕ
ω
(30)
Sufficiency: Assuming (30), ıf substituting )
,
(
)
(
^
c
c
a
c L
ν
ν ξ
ξ
ϕ Φ
= into (29),
∑ ∑
∑ ∑ =
=
∆
⊕
Ψ
Φ
≤
∆
⊕
Ψ
Φ
)
,
( 1
2
^
)
,
( 1
2
0
0
^
0
0
))
(
)
(
)(
(
)
,
(
))
(
)
(
)(
(
)
,
(
L
L
c
c
K
L
c
c
K
L
c
c
c
x
c
c
c
a
c
c
c
x
c
c
c
a
ν
ν
ν
ν
ν
ν
ν
ν ω
ξ
ω
ξ
(31)
it is clear that )
(c
x is an arbitrary admissible control and from (31) )
(
0
c
x is
determined as the optimal control.
4. CONCLUSIONS
It is shown that optimal control problem for processes with multi parametric linear stochastic
dynamical systems are studied. It is evident that there is an absolutely a control and
corresponding such that the considered system has a minimum value while connecting the initial
state with terminal state. Furthermore, theorems involving necessary and sufficient conditions for
optimality are proven. Thus, the results obtained in this paper will be contribute to application.
REFERENCES
[1] Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, Mishchenko., (1962) The Mathematical Theory of
Optimal Processes, Interscience Publishers, New York,
[2] Farajov, R. G., (1975) Linear Sequential Machines, Sov. Radio, (Russian).
[3] Gill, A., (1975) Linear Sequential Machines, Nauka, (Russian)
[4] Boltyanskii, V. G., (1978) Optimal Control of Discrete Systems, John Willey, New York, p.363.
[5] Gaishun, I. V., (1983) Completely Solvable Multidimensional Differential Equations, Nauka and
Tekhnika, Minsk, p.231.
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63
[6] Gabasov,R., Kirillova, F. M., Paulianok, N. S. ‘’Optimal Control of Linear Systems on Quadratic
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Multiparameter Binary Dynamic System”, Control and Cybernetics, Vol. 38, No. 3, pp625-633.
[9] Hacı Y., Candan M.,(2014) Optimal Control Problem for Processes Represented by Stochastic
Sequential Machine.’’ Internatinal Journal on Cybernetics and Informatics. Vol 3 no 4 pp. 21-26
[10] Yermolyev, Y. M., (1976) Stochastic Programming Methods, Nauka (in Russian), p.240.
[11] Burden, R. L., Dauglas, J. Numerical Analysis, PWS Publishing Company, Boston, 1985.
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AUTHOR
Yakup H. HACI1
received Ph.D at physics and mathematics in Azerbaijan NationalAcademy of Sciences at
the Institute of Cybernetics. He is currently working as professor at the department of Mathematics, in
Canakkale Onsekiz Mart University. His primary research interests are in the areas of the theory of multi
parametric binary sequential machines,linear and nonlinear optimization and discrete mathematics. He is an
author(co-author) of over 38 papers.
Muhammet CANDAN2
received his M.Sc degree from Canakkale Onsekiz Mart University, Canakkale,
Turkey, in 2012. He is currently Ph.D. candidate and working as a research assistant in Canakkale Onsekiz
Mart University. His research interests cover applied mathematics, finite dynamical systems, graph theory.
Aykut OR3
received his Ph.D at mathematics from Canakkale Onsekiz Mart University, Turkey, in 2014.
He is currently working as a research assistant in Canakkale Onsekiz Mart University. His mainly research
interests cover applied game theory, interval matrix, bimatix games.