\documentclass[11pt,twoside]{article}\makeatletter

\IfFileExists{xcolor.sty}%
  {\RequirePackage{xcolor}}%
  {\RequirePackage{color}}
\usepackage{colortbl}
\usepackage{wrapfig}
\usepackage{ifxetex}
\ifxetex
  \usepackage{fontspec}
  \usepackage{xunicode}
  \catcode`⃥=\active \def⃥{\textbackslash}
  \catcode`❴=\active \def❴{\{}
  \catcode`❵=\active \def❵{\}}
  \def\textJapanese{\fontspec{Noto Sans CJK JP}}
  \def\textChinese{\fontspec{Noto Sans CJK SC}}
  \def\textKorean{\fontspec{Noto Sans CJK KR}}
  \setmonofont{DejaVu Sans Mono}
  
\else
  \IfFileExists{utf8x.def}%
   {\usepackage[utf8x]{inputenc}
      \PrerenderUnicode{–}
    }%
   {\usepackage[utf8]{inputenc}}
  \usepackage[english]{babel}
  \usepackage[T1]{fontenc}
  \usepackage{float}
  \usepackage[]{ucs}
  \uc@dclc{8421}{default}{\textbackslash }
  \uc@dclc{10100}{default}{\{}
  \uc@dclc{10101}{default}{\}}
  \uc@dclc{8491}{default}{\AA{}}
  \uc@dclc{8239}{default}{\,}
  \uc@dclc{20154}{default}{ }
  \uc@dclc{10148}{default}{>}
  \def\textschwa{\rotatebox{-90}{e}}
  \def\textJapanese{}
  \def\textChinese{}
  \IfFileExists{tipa.sty}{\usepackage{tipa}}{}
\fi
\def\exampleFont{\ttfamily\small}
\DeclareTextSymbol{\textpi}{OML}{25}
\usepackage{relsize}
\RequirePackage{array}
\def\@testpach{\@chclass
 \ifnum \@lastchclass=6 \@ne \@chnum \@ne \else
  \ifnum \@lastchclass=7 5 \else
   \ifnum \@lastchclass=8 \tw@ \else
    \ifnum \@lastchclass=9 \thr@@
   \else \z@
   \ifnum \@lastchclass = 10 \else
   \edef\@nextchar{\expandafter\string\@nextchar}%
   \@chnum
   \if \@nextchar c\z@ \else
    \if \@nextchar l\@ne \else
     \if \@nextchar r\tw@ \else
   \z@ \@chclass
   \if\@nextchar |\@ne \else
    \if \@nextchar !6 \else
     \if \@nextchar @7 \else
      \if \@nextchar (8 \else
       \if \@nextchar )9 \else
  10
  \@chnum
  \if \@nextchar m\thr@@\else
   \if \@nextchar p4 \else
    \if \@nextchar b5 \else
   \z@ \@chclass \z@ \@preamerr \z@ \fi \fi \fi \fi
   \fi \fi  \fi  \fi  \fi  \fi  \fi \fi \fi \fi \fi \fi}
\gdef\arraybackslash{\let\\=\@arraycr}
\def\@textsubscript#1{{\m@th\ensuremath{_{\mbox{\fontsize\sf@size\z@#1}}}}}
\def\Panel#1#2#3#4{\multicolumn{#3}{){\columncolor{#2}}#4}{#1}}
\def\abbr{}
\def\corr{}
\def\expan{}
\def\gap{}
\def\orig{}
\def\reg{}
\def\ref{}
\def\sic{}
\def\persName{}\def\name{}
\def\placeName{}
\def\orgName{}
\def\textcal#1{{\fontspec{Lucida Calligraphy}#1}}
\def\textgothic#1{{\fontspec{Lucida Blackletter}#1}}
\def\textlarge#1{{\large #1}}
\def\textoverbar#1{\ensuremath{\overline{#1}}}
\def\textquoted#1{‘#1’}
\def\textsmall#1{{\small #1}}
\def\textsubscript#1{\@textsubscript{\selectfont#1}}
\def\textxi{\ensuremath{\xi}}
\def\titlem{\itshape}
\newenvironment{biblfree}{}{\ifvmode\par\fi }
\newenvironment{bibl}{}{}
\newenvironment{byline}{\vskip6pt\itshape\fontsize{16pt}{18pt}\selectfont}{\par }
\newenvironment{citbibl}{}{\ifvmode\par\fi }
\newenvironment{docAuthor}{\ifvmode\vskip4pt\fontsize{16pt}{18pt}\selectfont\fi\itshape}{\ifvmode\par\fi }
\newenvironment{docDate}{}{\ifvmode\par\fi }
\newenvironment{docImprint}{\vskip 6pt}{\ifvmode\par\fi }
\newenvironment{docTitle}{\vskip6pt\bfseries\fontsize{22pt}{25pt}\selectfont}{\par }
\newenvironment{msHead}{\vskip 6pt}{\par}
\newenvironment{msItem}{\vskip 6pt}{\par}
\newenvironment{rubric}{}{}
\newenvironment{titlePart}{}{\par }

\newcolumntype{L}[1]{){\raggedright\arraybackslash}p{#1}}
\newcolumntype{C}[1]{){\centering\arraybackslash}p{#1}}
\newcolumntype{R}[1]{){\raggedleft\arraybackslash}p{#1}}
\newcolumntype{P}[1]{){\arraybackslash}p{#1}}
\newcolumntype{B}[1]{){\arraybackslash}b{#1}}
\newcolumntype{M}[1]{){\arraybackslash}m{#1}}
\definecolor{label}{gray}{0.75}
\def\unusedattribute#1{\sout{\textcolor{label}{#1}}}
\DeclareRobustCommand*{\xref}{\hyper@normalise\xref@}
\def\xref@#1#2{\hyper@linkurl{#2}{#1}}
\begingroup
\catcode`\_=\active
\gdef_#1{\ensuremath{\sb{\mathrm{#1}}}}
\endgroup
\mathcode`\_=\string"8000
\catcode`\_=12\relax

\usepackage[a4paper,twoside,lmargin=1in,rmargin=1in,tmargin=1in,bmargin=1in,marginparwidth=0.75in]{geometry}
\usepackage{framed}

\definecolor{shadecolor}{gray}{0.95}
\usepackage{longtable}
\usepackage[normalem]{ulem}
\usepackage{fancyvrb}
\usepackage{fancyhdr}
\usepackage{graphicx}
\usepackage{marginnote}

\renewcommand{\@cite}[1]{#1}


\renewcommand*{\marginfont}{\itshape\footnotesize}

\def\Gin@extensions{.pdf,.png,.jpg,.mps,.tif}

  \pagestyle{fancy}

\usepackage[pdftitle={A New Look into the Controllability and Observability of Lyapunov type Matrix Dynamical Systems on Measure Chains},
 pdfauthor={}]{hyperref}
\hyperbaseurl{}

	 \paperwidth210mm
	 \paperheight297mm
              
\def\@pnumwidth{1.55em}
\def\@tocrmarg {2.55em}
\def\@dotsep{4.5}
\setcounter{tocdepth}{3}
\clubpenalty=8000
\emergencystretch 3em
\hbadness=4000
\hyphenpenalty=400
\pretolerance=750
\tolerance=2000
\vbadness=4000
\widowpenalty=10000

\renewcommand\section{\@startsection {section}{1}{\z@}%
     {-1.75ex \@plus -0.5ex \@minus -.2ex}%
     {0.5ex \@plus .2ex}%
     {\reset@font\Large\bfseries}}
\renewcommand\subsection{\@startsection{subsection}{2}{\z@}%
     {-1.75ex\@plus -0.5ex \@minus- .2ex}%
     {0.5ex \@plus .2ex}%
     {\reset@font\Large}}
\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}%
     {-1.5ex\@plus -0.35ex \@minus -.2ex}%
     {0.5ex \@plus .2ex}%
     {\reset@font\large}}
\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}%
     {-1ex \@plus-0.35ex \@minus -0.2ex}%
     {0.5ex \@plus .2ex}%
     {\reset@font\normalsize}}
\renewcommand\subparagraph{\@startsection{subparagraph}{5}{\parindent}%
     {1.5ex \@plus1ex \@minus .2ex}%
     {-1em}%
     {\reset@font\normalsize\bfseries}}


\def\l@section#1#2{\addpenalty{\@secpenalty} \addvspace{1.0em plus 1pt}
 \@tempdima 1.5em \begingroup
 \parindent \z@ \rightskip \@pnumwidth 
 \parfillskip -\@pnumwidth 
 \bfseries \leavevmode #1\hfil \hbox to\@pnumwidth{\hss #2}\par
 \endgroup}
\def\l@subsection{\@dottedtocline{2}{1.5em}{2.3em}}
\def\l@subsubsection{\@dottedtocline{3}{3.8em}{3.2em}}
\def\l@paragraph{\@dottedtocline{4}{7.0em}{4.1em}}
\def\l@subparagraph{\@dottedtocline{5}{10em}{5em}}
\@ifundefined{c@section}{\newcounter{section}}{}
\@ifundefined{c@chapter}{\newcounter{chapter}}{}
\newif\if@mainmatter 
\@mainmattertrue
\def\chaptername{Chapter}
\def\frontmatter{%
  \pagenumbering{roman}
  \def\thechapter{\@roman\c@chapter}
  \def\theHchapter{\roman{chapter}}
  \def\thesection{\@roman\c@section}
  \def\theHsection{\roman{section}}
  \def\@chapapp{}%
}
\def\mainmatter{%
  \cleardoublepage
  \def\thechapter{\@arabic\c@chapter}
  \setcounter{chapter}{0}
  \setcounter{section}{0}
  \pagenumbering{arabic}
  \setcounter{secnumdepth}{6}
  \def\@chapapp{\chaptername}%
  \def\theHchapter{\arabic{chapter}}
  \def\thesection{\@arabic\c@section}
  \def\theHsection{\arabic{section}}
}
\def\backmatter{%
  \cleardoublepage
  \setcounter{chapter}{0}
  \setcounter{section}{0}
  \setcounter{secnumdepth}{2}
  \def\@chapapp{\appendixname}%
  \def\thechapter{\@Alph\c@chapter}
  \def\theHchapter{\Alph{chapter}}
  \appendix
}
\newenvironment{bibitemlist}[1]{%
   \list{\@biblabel{\@arabic\c@enumiv}}%
       {\settowidth\labelwidth{\@biblabel{#1}}%
        \leftmargin\labelwidth
        \advance\leftmargin\labelsep
        \@openbib@code
        \usecounter{enumiv}%
        \let\p@enumiv\@empty
        \renewcommand\theenumiv{\@arabic\c@enumiv}%
	}%
  \sloppy
  \clubpenalty4000
  \@clubpenalty \clubpenalty
  \widowpenalty4000%
  \sfcode`\.\@m}%
  {\def\@noitemerr
    {\@latex@warning{Empty `bibitemlist' environment}}%
    \endlist}

\def\tableofcontents{\section*{\contentsname}\@starttoc{toc}}
\parskip0pt
\parindent1em
\def\Panel#1#2#3#4{\multicolumn{#3}{){\columncolor{#2}}#4}{#1}}
\newenvironment{reflist}{%
  \begin{raggedright}\begin{list}{}
  {%
   \setlength{\topsep}{0pt}%
   \setlength{\rightmargin}{0.25in}%
   \setlength{\itemsep}{0pt}%
   \setlength{\itemindent}{0pt}%
   \setlength{\parskip}{0pt}%
   \setlength{\parsep}{2pt}%
   \def\makelabel##1{\itshape ##1}}%
  }
  {\end{list}\end{raggedright}}
\newenvironment{sansreflist}{%
  \begin{raggedright}\begin{list}{}
  {%
   \setlength{\topsep}{0pt}%
   \setlength{\rightmargin}{0.25in}%
   \setlength{\itemindent}{0pt}%
   \setlength{\parskip}{0pt}%
   \setlength{\itemsep}{0pt}%
   \setlength{\parsep}{2pt}%
   \def\makelabel##1{\upshape ##1}}%
  }
  {\end{list}\end{raggedright}}
\newenvironment{specHead}[2]%
 {\vspace{20pt}\hrule\vspace{10pt}%
  \phantomsection\label{#1}\markright{#2}%

  \pdfbookmark[2]{#2}{#1}%
  \hspace{-0.75in}{\bfseries\fontsize{16pt}{18pt}\selectfont#2}%
  }{}
      \def\TheFullDate{2014-01-15 (revised: 15 January 2014)}
\def\TheID{\makeatother }
\def\TheDate{2014-01-15}
\title{A New Look into the Controllability and Observability of Lyapunov type Matrix Dynamical Systems on Measure Chains}
\author{}\makeatletter 
\makeatletter
\newcommand*{\cleartoleftpage}{%
  \clearpage
    \if@twoside
    \ifodd\c@page
      \hbox{}\newpage
      \if@twocolumn
        \hbox{}\newpage
      \fi
    \fi
  \fi
}
\makeatother
\makeatletter
\thispagestyle{empty}
\markright{\@title}\markboth{\@title}{\@author}
\renewcommand\small{\@setfontsize\small{9pt}{11pt}\abovedisplayskip 8.5\p@ plus3\p@ minus4\p@
\belowdisplayskip \abovedisplayskip
\abovedisplayshortskip \z@ plus2\p@
\belowdisplayshortskip 4\p@ plus2\p@ minus2\p@
\def\@listi{\leftmargin\leftmargini
               \topsep 2\p@ plus1\p@ minus1\p@
               \parsep 2\p@ plus\p@ minus\p@
               \itemsep 1pt}
}
\makeatother
\fvset{frame=single,numberblanklines=false,xleftmargin=5mm,xrightmargin=5mm}
\fancyhf{} 
\setlength{\headheight}{14pt}
\fancyhead[LE]{\bfseries\leftmark} 
\fancyhead[RO]{\bfseries\rightmark} 
\fancyfoot[RO]{}
\fancyfoot[CO]{\thepage}
\fancyfoot[LO]{\TheID}
\fancyfoot[LE]{}
\fancyfoot[CE]{\thepage}
\fancyfoot[RE]{\TheID}
\hypersetup{citebordercolor=0.75 0.75 0.75,linkbordercolor=0.75 0.75 0.75,urlbordercolor=0.75 0.75 0.75,bookmarksnumbered=true}
\fancypagestyle{plain}{\fancyhead{}\renewcommand{\headrulewidth}{0pt}}

\date{}
\usepackage{authblk}

\providecommand{\keywords}[1]
{
\footnotesize
  \textbf{\textit{Index terms---}} #1
}

\usepackage{graphicx,xcolor}
\definecolor{GJBlue}{HTML}{273B81}
\definecolor{GJLightBlue}{HTML}{0A9DD9}
\definecolor{GJMediumGrey}{HTML}{6D6E70}
\definecolor{GJLightGrey}{HTML}{929497} 

\renewenvironment{abstract}{%
   \setlength{\parindent}{0pt}\raggedright
   \textcolor{GJMediumGrey}{\rule{\textwidth}{2pt}}
   \vskip16pt
   \textcolor{GJBlue}{\large\bfseries\abstractname\space}
}{%   
   \vskip8pt
   \textcolor{GJMediumGrey}{\rule{\textwidth}{2pt}}
   \vskip16pt
}

\usepackage[absolute,overlay]{textpos}

\makeatother 
      \usepackage{lineno}
      \linenumbers
      
\begin{document}

             \author[1]{Goteti V.R.L.  Sarma}

\renewcommand\Authands{ and }

\date{\small \em Received: 7 December 2013 Accepted: 4 January 2014 Published: 15 January 2014}

\maketitle


\begin{abstract}
        


In this paper we establish the concept of controllability and observability of time varying dynamical systems on measure chains. we obtain a set of necessary and sufficient conditions for the controllability and observability of Lyapunov type matrix dynamical system on measure chains.

\end{abstract}


\keywords{controllability and observability, varying dynamical systems, sufficient conditions for the controllability and observability.}

\begin{textblock*}{18cm}(1cm,1cm) % {block width} (coords) 
\textcolor{GJBlue}{\LARGE Global Journals \LaTeX\ JournalKaleidoscope\texttrademark}
\end{textblock*}

\begin{textblock*}{18cm}(1.4cm,1.5cm) % {block width} (coords) 
\textcolor{GJBlue}{\footnotesize \\ Artificial Intelligence formulated this projection for compatibility purposes from the original article published at Global Journals. However, this technology is currently in beta. \emph{Therefore, kindly ignore odd layouts, missed formulae, text, tables, or figures.}}
\end{textblock*}


\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
ecently the concept of time scales(or measure chains) is growing very rapidly into many areas of research. One reason being it includes both continuous and discrete systems as special cases. Hence many researchers \hyperref[b0]{[1]} - \hyperref[b3]{[4]} and \hyperref[b7]{[8]}- \hyperref[b10]{[11]} explored different concepts on time scales. Stephan Barnett \hyperref[b11]{[12]} studied the control theory for both the continuous and discrete cases for simple dynamical systems. Control theory has developed rapidly over the past two decades and is now established as an important area of contemporary Applied Mathematics. Many problems of great importance in the contemporary world require a quite different approach, the aim being to compel or control a system to behave in some desired fashion. Basically control theory has involved the study of analysis and control of any dynamical system. This theory has been successfully applied in a variety of branches in the disciplines of engineering and particularly it is receiving great impetus from Aerospace engineering. A fascinating fact is that all the widely different disciplines of applications depend on a common core of mathematical techniques of the modern control system theory.\par
In this paper we establish the concept of controllability and observability of dynamical systems on measure chains. The results presented in this chapter generalises the existing results on controllability and observability for continuous and discrete cases and includes them as a particular case. This paper is organised as follows; In section 2, we outline the salient features of time scales. In section 3, we briefly mention the necessary and sufficient conditions for the controllability and observability of vector dynamical systems on time scales.( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) S x t A t x t B t u t x t x y t C t x t 1 0 0 ? = + = = ? ? ? 
\section[{, ( )}]{, ( )}\par
Author: Department of Mathematics, University of Dodoma Dodoma, Tanzania. e-mail: gvrlsarma@rediffmail.com where A is n × n, B is n × m and C is r × n matrices whose elements are rd-continuous on a time scale T = [t 0 , t N ], the control u is (mx1) and x is an (nx1)vector. Section 4 presents the controllability and observability criteria of the most general matrix Lyapunov differential system on measure chainsS2 ) ( , ) ( ) ( ) ( )) ( ( ) ( ) ( ) ( ) ( ) ( ) ( 0 0 X t X t U t K t B t X t X t C t X t A t Y t X = + + ? ? ? = = ? ?\par
where A(t), B(t), K(t) and C(t) are square matrices of orders n×n, nxn, nxs and rxn respectively. The control U(t) is an sxn matrix and the out put Y(t) is an rxn matrix whose elements are rd-continuous on a measure chain T = [t 0 , t N ]. We firmly believe that these results will have a significant impact on control engineering problems. 
\section[{II.}]{II.} 
\section[{Salient Features of Time Scales}]{Salient Features of Time Scales}\par
A measure chain (or time scale) is an arbitrary closed subset of real numbers R and it is denoted by T throughout the paper. Time scales are not necessarily connected and this topological handicap is eliminated by introducing the notion of jump operators ? and ? as follows: Definition 2.1: Let T be a measure chain. For t ? T define the forward jump operator ? : T ? T by ? (t) = Inf \{s ? T : s > t \} and the backward jump operator ? : T ? T by ? (t) = Sup \{s ? T : s < t \} .\par
A point t ? T is said to be right dense, right scattered, left dense and left scattered according as ? (t) = t, ? (t) > t, ? (t) = t and ? (t) < t respectively. The grainnessµ : T ? [o, ?) is defined by µ (t) = ? (t) -t .\par
The set T k which is derived from the measure chain T as follows:\par
If T has a left scattered maximum m, thenT k = T -\{m\}, otherwise T k = T. Definition 2.2: Let f : T ? R , t ? T k . Then define f ? (t) to\par
be the number (provided it exists ) with the property that given any ?>0, there exists a neighbourhood ? of t such that Year 2014I (1.1) Global Journal of Researches in Engineering ( ) Volume XIV Issue II Version I Year 2014 6 I |[f(?(t)) -f(s)] -f ? (t) [?(t) -s] | ? ? |?(t) -s | for all s ? ? Then f ? (t) is called the delta derivative of f at t.\par
If T = R, the delta derivative is same as that of ordinary derivative and for T = Z, f ? (t) = f(t+1) -f(t) = ?f(t), which is the forward difference operator. Definition 2.3 : We say that f is delta differentiable on T k , if f ? (t) exists for all t ? T k . Result 2.1 : Assume f, g : T?R are delta differentiable functions at t?T k , then? f+g : T ?R is delta differentiable at t with (f+g) ? (t) = f ? (t) +g ? (t). ? For any constant k, kf : T? R is delta differentiable at t with (kf) ? (t) = k f ? (t). ? : T x T? R is delta differentiable at t with (fg) ? (t) = f ? (t) g(t) + f(?(t)) g ? (t) = f(t) g ? (t) + f ? (t) g(?(t)) . Definition 2.4: A function F : T k ? R is called an antiderivative of f : T k ? R, provided F ? (t) = f(t) holds for all t ? T k .\par
Then the delta integral of f is defined = F (t) -F (a)) ? t ?T.\par
Definition 2.5: Let f : T?T be a function. We say that f is rd -continuous if it is continuous in right dense points and if limit f (s) exists as s ? t -for all left-dense points t ? T.\par
Result 2.2: Rd -Continuous functions possess an anti derivative.\par
Proof: For the proof we refer \hyperref[b1]{[2]}. IV.Definition 
\section[{Controllability and Observability of Lyapunov Type Matrix Dynamical Systems on Measure Chains}]{Controllability and Observability of Lyapunov Type Matrix Dynamical Systems on Measure Chains}\par
In this section we consider Lyapunov type matrix dynamical system on measure chainsX t ? ( ) = A (t)X(t) X t t K t t X t X ( ( ))B( ) ( )U( ) , ( ) ? + = 0 0\par
and obtain necessary and sufficient conditions for the controllability and observability of S 2 . Throughout this section ? 1 (t) and ? 2 (t) represent fundamental matrix solutions of A(t)X(t) and B*(t) X(? (t))\par
Theorem 4.1 : The solution of the equation (4.1) is given by X t ( ) =?1(t) ? ? 1 1 0 0 2 1 0 ? ? ( ) ( ) * t X t ? 2 * ( ) t +?1(t) ? ? ? ? 1 1 2 1 2 0 ? ? ? ? ? ? ? ? ? ? ? ( ( ))K( )U( ) ( ) ( ) * * s s s s s t t t ? .= ?1(t,t0) X t s s s t s s t t t t 0 1 0 2 0 2 0 0 + ? ? ? ? ? ? ? ? ? ? ? ? ? ( , ( ))K( )U( ) ( , )\textbf{( , )}+ (4.1)\par
x ? (t) = respectively.\par
x ? (t) = Year 2014 I A New Look into the Controllability and Observability of Lyapunov type Matrix Dynamical Systems on Measure Chains X t ( ) =?1(t) ? ?1 1 0 0 2 1 0 ? ? ( ) ( ) * t X t ? 2 * ( ) t +?1(t) ? ? ? ? 1 1 2 1 2 0 ? ? ? ? ? ? ? ? ? ? ? ( ( ))K( )U( ) ( ) ( ) * * s s s s s t t t ? .= ?1(t,t0) X t s s s t s s t t t t 0 1 0 2 0 2 0 0 + ? ? ? ? ? ? ? ? ? ? ? ? ? ( , ( ))K( )U( ) ( , ) ( , ) * * ? where ? i (t,t 0 ) = ? i (t)? i -1 (t 0 ) (i=1,2)\par
Proof : Please refer \hyperref[b5]{[6]} Theorem 4.2 : The Lyapunov type matrix dynamical system on measure chains S 2 is completely controllable if and only if the symmetric controllability matrixW 1 (t 0 ,t N )= ? ? N t t o s s t s K s K s t 0 )) ( , ( ) ( ) ( )) ( , ( * 1 * 0 1 ? ? ? ? is non singular.\par
Then the control U (t) defined byU(t= -K * (t) ( ) [ ] ? ? ? ? ? 1 0 1 1 0 0 1 0 2 0 2 1 0 * * * , ( ) ( , ) ( , ) ( , ) ( , ) t t W t t X t t X t t t t N N f N ? ? ? defined for t 0 < t < t N transfers X(t 0 ) = X 0 to X(t N ) = Xf.\par
Proof : First we suppose that W 1 (t 0 ,t N ) is non -singular.\par
Then U (t) defined as above exists. We know that any solution of (4.1) has the formX t ( ) =? 1 (t,t 0 ) X t s s s t s s t t t t 0 1 0 2 0 2 0 0 + ? ? ? ? ? ? ? ? ? ? ? ? ? ( , ( ))K( )U( ) ( , )\textbf{( , ) * *}\par
? put t = t N and substitute U(t) defined as above we will get X t N ( ) = Xf and hence S 2 is controllable.\par
Conversely, suppose that S 2 is controllable. We have to show that W 1 (t 0 , t N ) is non-singular. Since W 1 (t 0 , t N ) is symmetric, clearly it is positive semi definite. Now suppose that there exists some column\par
vector? 1 ?0 such that ? ? 1 1 0 1 * ( , ) W t t N = 0 ? ? ? ? 1 1 0 1 0 1 0 * * * ( , ( ))K( ) ( ) ( , ( )) ? ? ? ? t s s k s t s s t t N ? = 0 ? ? ? 1 0 1 0 0 * ( , ) ( , ) s t s t s t t N ? ? = 0, where ?1(s, t0) = K s t s * * ( ) ( , ( )) ? ? 1 0 1 ? ? ? 1 2 0 0 ?s t t N ? = . Hence ?1 ? 0 on [t0, tN].\par
By our assumption since S 2 is completely controllable, there exists a control V (t) (say) making X (t N ) = 0 if X (t 0 ) ? 1 g where g is any non zero 1xn matrix. i.e X 0 can not determined with the knowledge of Y(t) in this case. This contradicts our assumption that S 3 is completely observable. Therefore V(t 0 , t N ) is positive definite and hence V(t 0 , t N ) is non -singular.X (t N ) = 0 ? X (t0) + ? ? ? 1 0 2 0 0 ( , ( ))K( )V( ) ( , ) * t s s s t s s t t N ? ? = 0 ? ?1g = -? ? ? 1 0 2 0 0 ( , ( ))K( )V( ) ( , ) * t s s s t s s t t N ? ? Now ? 1 2 g = (?1g)* (?1g) = - ? ? ? 2 0 1 0 1 0 ( , )V ( )K ( ) ( ,\textbf{(}\par
Observation 4.1: From the theorem (4.2) it is observed that the controllability matrix is independent of ? 2 . The fundamental matrix ? 1 alone determines the controllability criterion of the dynamical system S 2 . We also observe that the controllability criterion can be determined using the fundamental matrix ? 2 alone. In this case the controllability matrix is given by W 2 (t 0 ,t N ) = ( ( , ( ))K( )K ( ) ( , ( )) that the observability matrix is independent of ? 2 . The fundamental matrix ? 1 alone determines the observability criterion of the dynamical system S 2 . We can also determine the observability criterion of S 2 using the fundamental matrix ? 2 alone.\begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.30142436149312374\textwidth}P{0.08934184675834969\textwidth}P{0.0008349705304518663\textwidth}P{0.0008349705304518663\textwidth}P{0.0008349705304518663\textwidth}P{0.0025049115913555993\textwidth}P{0.0008349705304518663\textwidth}P{0.0008349705304518663\textwidth}P{0.0008349705304518663\textwidth}P{0.012524557956777996\textwidth}P{0.02254420432220039\textwidth}P{0.0008349705304518663\textwidth}P{0.0008349705304518663\textwidth}P{0.0008349705304518663\textwidth}P{0.0008349705304518663\textwidth}P{0.0008349705304518663\textwidth}P{0.0008349705304518663\textwidth}P{0.36655206286836933\textwidth}P{0.01669941060903733\textwidth}P{0.008349705304518664\textwidth}P{0.0025049115913555993\textwidth}P{0.0025049115913555993\textwidth}P{0.004174852652259332\textwidth}P{0.005844793713163065\textwidth}P{0.004174852652259332\textwidth}P{0.0008349705304518663\textwidth}}
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep Proof : Please refer [6]\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep Theorem 3.2 : Suppose u -(t) is any other control taking\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep X (t 0 ) = x 0 to x(t N ) = xf.\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep Then\tabcellsep t ? N\tabcellsep ( ) u s\tabcellsep 2\tabcellsep s ?\tabcellsep t ? N\tabcellsep ( ) u s\tabcellsep s 2 ?\tabcellsep ,\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep t\tabcellsep o\tabcellsep t\tabcellsep o\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep where u(s) = -B*(s) ? * to, ? (s)) W -1 (t o ,t N ) [x o -? (t o ,t N )xf]\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep provided u(s) ? u(s) ?s ? [t o , t N ] .\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep Proof : Please refer [6]\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep Theorem 3.3 : The time varying dynamical system S 1 on\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep measure chain T=[ t 0 ,t N ] is completely observable if\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep and only if the symmetric observability matrix V 1 (t 0 , t N ) =\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep t\tabcellsep N\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep ?\tabcellsep ( ? * , s t\tabcellsep 0\tabcellsep )\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep t\tabcellsep o\\
\multicolumn{18}{l}{Definition 2.7 (Observability) : The linear time varying}\\
\multicolumn{18}{l}{dynamical system defined by S 1 on measure chains is}\\
\multicolumn{18}{l}{completely observable if and only if the knowledge of}\\
\multicolumn{18}{l}{the control u(t) and the out put y(t) suffice to determine}\\
\multicolumn{15}{l}{x(t 0 ) = x 0 uniquely for a finite time t N ? t 0 .}\tabcellsep \tabcellsep \\
III.\tabcellsep \multicolumn{17}{l}{Controllability and Observability}\\
\tabcellsep \multicolumn{17}{l}{of Vector Dynamical Systems on}\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \multicolumn{6}{l}{Measure Chains}\tabcellsep \tabcellsep \\
\tabcellsep \multicolumn{17}{l}{Theorem 3.1: The system S 1 is completely}\\
\multicolumn{18}{l}{controllable if and only if the n × n symmetric}\\
\multicolumn{11}{l}{controllability matrix W (t o , t N ) =}\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
\tabcellsep t\tabcellsep \multicolumn{2}{l}{N}\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
\tabcellsep \multicolumn{3}{l}{?}\tabcellsep (\tabcellsep ( ?\tabcellsep t\tabcellsep o\tabcellsep ,\tabcellsep ?\tabcellsep ( ) ) ( ) ( ) * ? s B s B s\tabcellsep *\tabcellsep (\tabcellsep t\tabcellsep o\tabcellsep ,\tabcellsep ?\tabcellsep ( ) ) s\tabcellsep )\tabcellsep s\\
\tabcellsep \multicolumn{2}{l}{t}\tabcellsep o\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \end{longtable} \par
  {\small\itshape [Note: (Controllability) 2.6 : The linear time varying dynamical system S 1 on measure chain T=[t 0 ,t N ] is completely controllable if "for any initial time t0 and any initial state x (t o ) = x o and any given final state xf, there exists a finite time tN > to and a control u(t), to ? t ? t N such that x (t N ) = xf". C* (s) C(s) ? (s,t 0 ) ? s is non singular.Proof : Please refer\hyperref[b5]{[6]} ]} 
\caption{\label{tab_0}?}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.11226415094339622\textwidth}P{0.344811320754717\textwidth}P{0.004009433962264151\textwidth}P{0.004009433962264151\textwidth}P{0.004009433962264151\textwidth}P{0.004009433962264151\textwidth}P{0.004009433962264151\textwidth}P{0.004009433962264151\textwidth}P{0.004009433962264151\textwidth}P{0.12429245283018867\textwidth}P{0.004009433962264151\textwidth}P{0.004009433962264151\textwidth}P{0.004009433962264151\textwidth}P{0.004009433962264151\textwidth}P{0.008018867924528302\textwidth}P{0.008018867924528302\textwidth}P{0.06014150943396226\textwidth}P{0.004009433962264151\textwidth}P{0.016037735849056604\textwidth}P{0.008018867924528302\textwidth}P{0.016037735849056604\textwidth}P{0.06415094339622641\textwidth}P{0.020047169811320757\textwidth}P{0.020047169811320757\textwidth}}
\tabcellsep \multicolumn{2}{l}{t N ?}\tabcellsep ?\tabcellsep 2\tabcellsep \multicolumn{2}{l}{t}\tabcellsep \multicolumn{2}{l}{0}\tabcellsep ?\tabcellsep s\tabcellsep \multicolumn{2}{l}{s}\tabcellsep \tabcellsep *\tabcellsep \multicolumn{2}{l}{s}\tabcellsep ?\tabcellsep * 2\tabcellsep t\tabcellsep o\tabcellsep ?\tabcellsep s s ?\tabcellsep , and\\
\tabcellsep t\tabcellsep 0\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
\multicolumn{11}{l}{the control U(t) defined by}\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \multicolumn{8}{l}{U(t) = -K * (t)}\\
(\tabcellsep \multicolumn{5}{l}{) ( ( )K ( ) K t t * ? ? 1 1 1 ?}\tabcellsep \multicolumn{2}{l}{t}\tabcellsep 0\tabcellsep \multicolumn{3}{l}{) [ t X , ( ) ?}\tabcellsep 0\tabcellsep ?\tabcellsep ?\tabcellsep 1\tabcellsep \multicolumn{2}{l}{t t X N ( , ) 0}\tabcellsep f\tabcellsep ?\tabcellsep * 2\tabcellsep ] ( , ) t t N 0\\
\tabcellsep \multicolumn{17}{l}{W t t 2 1 0 ? ( , ) ( , )K( )K ( ) t t t t N 2 0}\end{longtable} \par
  {\small\itshape [Note: * ? Observation 4.2 : From the theorem (4.3) it is observed]} 
\caption{\label{tab_3}}\end{figure}
 			\footnote{© 2014 Global Journals Inc. (US)} 		 		\backmatter  			  				\begin{bibitemlist}{1}
\bibitem[P?tzsche et al. ()]{b10}\label{b10} 	 		‘A spectral characterization of exponential stability for linear time-invariant systems on time scales’.  		 			S P?tzsche 		,  		 			F Sigmund 		,  		 			Wirth 		.  	 	 		\textit{Discrete and Continuous Dynamical Systems}  		2003. 9  (5)  p. .  	 
\bibitem[Bartosiewicz and Paw?uszewicz ()]{b1}\label{b1} 	 		 			Z Bartosiewicz 		,  		 			E Paw?uszewicz 		.  		\textit{Realizations of linear control systems on time scales, Control \& Cybernetics},  				2006. 35 p. .  	 
\bibitem[Lakshmikantham et al. ()]{b5}\label{b5} 	 		\textit{Dynamic systems on measure chains},  		 			V Lakshmikantham 		,  		 			S Sivasundaram 		,  		 			B Kaymakclan 		.  		1996. Kluwer academic Publishers.  	 
\bibitem[Da Cunha ()]{b4}\label{b4} 	 		‘Instability results for slowly time varying linear dynamic systems on time scales’.  		 			J J Da Cunha 		.  	 	 		\textit{J. Math. Anal. Appl}  		2007. 328 p. .  	 
\bibitem[Barnet ()]{b11}\label{b11} 	 		\textit{Introduction to mathematical control},  		 			Stephen Barnet 		.  		1975. Clarendon Press-Oxford.  	 
\bibitem[Bartosiewicz and Paw?uszewicz ()]{b2}\label{b2} 	 		‘Linear control systems on time scale: unification of continuous and discrete’.  		 			J Z Bartosiewicz 		,  		 			E Paw?uszewicz 		.  	 	 		\textit{Proceedings of the 10th IEEE International Conference On Methods and Models in Automation and Robotics MMAR},  				 (the 10th IEEE International Conference On Methods and Models in Automation and Robotics MMARMiedzyzdroje, Poland)  		2004. 2004.  	 
\bibitem[Murty and Deekshitulu (1998)]{b8}\label{b8} 	 		\textit{Non linear Lyapunov type matrix system on time scales-Existence, Uniqueness and Sensitivity analysis. Dynamic systems and applications},  		 			K N Murty 		,  		 			G V S R Deekshitulu 		.  		Dec 1998. 7 p. .  	 
\bibitem[Marks et al. ()]{b6}\label{b6} 	 		‘Nonregressivity in switched linear circuits and mechanical systems’.  		 			R J Marks 		,  		 			II 		,  		 			I A Gravagne 		,  		 			J M Davis 		,  		 			J J Da Cunha 		.  	 	 		\textit{Mathematical and Computer Modelling}  		2006. 43 p. .  	 
\bibitem[Nguyen et al. ()]{b9}\label{b9} 	 		‘On the exponential stability of dynamic equations on time scales’.  		 			Huu Nguyen 		,  		 			Le Huy Du 		,  		 			Tien 		.  	 	 		\textit{J. Math. Anal. Appl}  		2007. 331 p. .  	 
\bibitem[Murty and Rao ()]{b7}\label{b7} 	 		‘S Two point boundary value problems inhomogeneous time scale linear dynamic process’.  		 			K N Murty 		,  		 			Y Rao 		.  	 	 		\textit{JMAA}  		1994. 184 p. .  	 
\bibitem[Da Cunha ()]{b3}\label{b3} 	 		‘Stability for time varying linear dynamic systems on time scales’.  		 			J J Da Cunha 		.  	 	 		\textit{J. Comput. Appl. Math}  		2005. 176 p. .  	 
\bibitem[Hamza and Oraby ()]{b0}\label{b0} 	 		‘Stability of abstract dynamic equations on time scales’.  		 			Alaa E Hamza 		,  		 			Karima M Oraby 		.  	 	 		\textit{Advances in Difference Equations},  				2012. 2012. p. 143.  	 
\end{bibitemlist}
 			 		 	 
\end{document}
