\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 80, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/80\hfil Controllability of matrix second order systems] {Controllability of matrix second order systems: A trigonometric matrix approach} \author[J. P. Sharma, R. K. George\hfil EJDE-2007/80\hfilneg] {Jaita Pankaj Sharma, Raju K. George} % in alphabetical order \address{Jaita Pankaj Sharma \newline Department of Applied Mathematics, Faculty of Tech. \& Eng., M.S. University of Baroda, Vadodara 390001, India} \email{jaita\_sharma@yahoo.co.uk} \address{Raju K. George \newline Department of Applied Mathematics, Faculty of Tech. \& Eng., M. S. University of Baroda, Vadodara 390001, India} \email{raju\_k\_george@yahoo.com} \thanks{Submitted February 15, 2007. Published May 29, 2007.} \subjclass[2000]{93B05, 93C10} \keywords{Controllability; matrix second order linear system; \hfill\break\indent cosine and sine matrices; Banach contraction principle} \begin{abstract} Many of the real life problems are modelled as Matrix Second Order Systems, (refer Wu and Duan \cite{Wu}, Hughes and Skelton \cite{Skel1}). Necessary and sufficient condition for controllability of Matrix Second Order Linear (MSOL) Systems has been established by Hughes and Skelton \cite{Skel1}. However, no scheme for computation of control was proposed. In this paper we first obtain another necessary and sufficient condition for the controllability of MSOL and provide a computational algorithm for the actual computation of steering control. We also consider a class of Matrix Second Order Nonlinear systems (MSON) and provide sufficient conditions for its controllability. In our analysis we make use of Sine and Cosine matrices and employ P\'ade approximation for the computation of matrix Sine and Cosine. We also invoke tools of nonlinear analysis like fixed point theorem to obtain controllability result for the nonlinear system. We provide numerical example to substantiate our results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} In this paper, we investigate the controllability property of the system governed by a Matrix Second Order Nonlinear (MSON) differential equation: $$\label{nld} \begin{gathered} \frac{d^{2}x(t)}{dt^{2}} + A^{2} x(t) = Bu(t)+f(t,x(t))\\ x(0) = x_{0}, \quad x'(0) = y_{0}. \end{gathered}$$ where, the state $x(t)$ is in ${R}^{n}$ and the control $u(t)$ is in ${R}^{m}$, $A^{2}$ is a constant matrix of order $n\times n$ and $B$ is a constant matrix of order $n\times m$ and $f:[0,T]\times R^{n} \to R^{n}$ is a nonlinear function satisfying Caratheodory conditions, that is, $f$ is measurable with respect to $t$ for all $x$ and continuous with respect to $x$ for almost all $t \in [0,T]$. The initial states $x_{0}$ and $y_{0}$ are in $R^{n}$. The corresponding Matrix Second Order Linear (MSOL) system is: $$\label{ld} \begin{gathered} \frac{d^{2}x(t)}{dt^{2}} + A^{2} x(t) = Bu(t)\\ x(0) = x_{0}, \quad x'(0) = y_{0}. \end{gathered}$$ The system \eqref{ld} has been studied by many researchers due to the fact that it can model the dynamics of many natural phenomenon to a significantly large extent(refer Hughes and Skelton \cite{Skel1,Skel2}, Balas \cite{Bala}, Diwakar and Yedavalli \cite{Diwa1,Diwa2}, Laub and Arnold \cite{Arno}, Fitzgibbon \cite{Fitz}). \begin{definition} \label{def1.1} \rm The system \eqref{nld} is said to be controllable on $[0,T]$ if for each pair $x_{0},x_{1}\in R^{n}$, there exists a control $u(t)\in L^{2}([t_{0},T];R^{m})$ such that the corresponding solution of \eqref{nld} together with $x(0)=x_{0}$ also satisfies $x(T) = x_{1}$. \end{definition} We note that in our controllability definition we are concerned only in steering the states but not the velocity vector $y_{0}$ in \eqref{nld}. A necessary and sufficient condition for the controllability of the MSOL system has been proved in (Hughes and Skelton \cite{Skel1}). They converted the second order system into first order system and obtained controllability result. However, no computational scheme for the steering control was proposed. In this paper we prove another controllability result and also provide a computational algorithm for the actual computation of controlled state and steering control. We do not reduce the system into first order and analyse the original second order form itself. We use matrix Sine and Cosine operators to find the solution of the systems \eqref{nld} and \eqref{ld}. We employ P\'ade approximation for the computation of matrix Sine and Cosine operators. Section 2 provides the necessary preliminaries on matrix Sine and matrix Cosine and Section 3 deals with the solution of MSOL and MSON. In section 4, we prove controllability results for MSOL, and controllability result of MSON is provided in Section 5. Section 6 concludes with the computational algorithm for Sine and Cosine matrices and steering control for linear and nonlinear systems. Examples are provided to illustrate the results. \section{Preliminaries} As we know the matrix exponential $y(t) = e^{At}y_{0}$ provides the solution to the first order differential system $\frac{dy}{dt} = Ay,\quad y(0) = y_{0}.$ Trigonometric matrix functions play a similar role in second order differential matrix system $\frac{d^{2}y}{dt^{2}} + Ay = 0,\quad y(0) = x_{0},\quad y'(0) = y_{0},$ That is, the solution of the above second order system, using Sine and Cosine matrices, is given by (refer Hargreaves and Higham \cite{Higm}) $y(t) = \cos(\sqrt{A}t)x_{0} + (\sqrt{A})^{-1}\sin(\sqrt{A}t)y_{0}.$ where $\cos(\sqrt{A}t)$ and $\sin(\sqrt{A}t)$ are matrix sine and cosine as defined below. The complex exponential of a matrix is defined as the series, (refer Chen \cite{Chen}) \begin{align*} e^{iAt} & = I + iAt + \frac{(iAt)^{2}}{2!} + \frac{(iAt)^{3}}{3!} + \frac{(iAt)^{4}}{4!} + \frac{(iAt)^{5}}{5!} + \frac{(iAt)^{6}}{6!} + \frac{(iAt)^{7}}{7!}+\dots \\ & = (I - \frac{A^{2}t^{2}}{2!} + \frac{A^{4}t^{4}}{4!} - \frac{A^{6}t^{6}}{6!} + \dots ) + i(At - \frac{A^{3}t^{3}}{3!} + \frac{A^{5}t^{5}}{5!} - \frac{A^{7}t^{7}}{7!} + \dots ). \end{align*} Convergence of the above series has been well established, (refer Brockett \cite{Brok}). We define Cosine and Sine matrix of $A$ as the real and imaginary part of the above series. That is, \begin{gather}\label{cos} \cos(At) = I - \frac{(At)^{2}}{2!}+ \frac{(At)^{4}}{4!} - \frac{(At)^{6}}{6!}+\dots \\ \label{sin} \sin(At) = At - \frac{(At)^{3}}{3!} + \frac{(At)^{5}}{5!} - \frac{(At)^{7}}{7!}\dots \end{gather} Since exponential matrix series converges, the subseries defined in \eqref{cos} and \eqref{sin} also converge. Further, \begin{gather*} e^{iAt}=\cos(At) + i\sin(At), \\ e^{-iAt} = \cos(At) - i\sin(At) \end{gather*} Using the above identities, we have the following representation of Cosine and Sine matrices in terms of matrix exponentials: $$\cos(At) = \frac{e^{iAt} + e^{-iAt}}{2} ,\quad \sin(At) = \frac{e^{iAt} - e^{-iAt}}{2i}$$ The Sine and Cosine matrices satisfy following properties: \begin{itemize} \item[(i)] $\cos(0)=I$. \item[(ii)] $\sin(0)=0$. \item[(iii)] $\frac{d}{dt}\cos(At) = -A\sin(At)$. \item[(iv)] $\frac{d}{dt}\sin(At) = A\cos(At)$. \item[(v)] $\cos(At)$ is non-singular matrix, if A is nonsingular. \item[(vi)] $\sin(A(t-s)) = \sin(At)\cos(As) - \cos(At)\sin(As)$ for all $t$. \item[(vii)] $A^{-1}\cos(At) = \cos(At)A^{-1}$. \end{itemize} \section{Solution of Second Order Systems Using Cosine and Sine Matrices} We use Sine and Cosine matrices to reduce the system \eqref{nld} to an integral equation. It can be shown easily that the matrices $X_{1}(t) = \cos(At)$ and $X_{2}(t) = A^{-1}\sin(At)$ satisfy the homogeneous linear matrix differential equation $$\label{mat} \frac{d^{2}X(t)}{dt^{2}} + A^{2}X(t) = 0$$ Here, if $A$ is a singular matrix, then $X_{2}$ is expanded as the power series, (refer Hargreaves and Higham \cite{Higm}) $$\label{inv-A} X_{2} = It - \frac{A^{2}t^{3}}{3!} + \frac{A^{4}t^{5}}{5!} - \frac{A^{6}t^{7}}{7!}\dots$$ General solution of the homogeneous system $\frac{d^{2}x(t)}{dt^{2}} + A^{2}x(t) = 0$ is given by \begin{gather*} x(t) = X_{1}(t)C_{1} + X_{2}(t)C_{2},\\ x(t) = \cos(At)C_{1} + A^{-1}\sin(At)C_{2} \end{gather*} where, $C_{1}$ and $C_{2}$ are arbitrary vectors in $R^{n}$. Now using the method of variation of parameter, a particular integral (P.I) for the nonhomogeneous system \eqref{ld} is given by \begin{equation*} P.I = - X_{1}(t) \int_{0}^{t} W^{-1}(s)X_{2}(s)Bu(s)ds + X_{2}(t) \int_{0}^{t} W^{-1}(s)X_{1}(s)Bu(s)ds \end{equation*} where, the Wronskian $W = \begin{vmatrix} X_{1} & X_{2} \\ X_{1}' & X_{2}' \end{vmatrix} = \begin{vmatrix} \cos(At) & A^{-1}\sin(At) \\ -A\sin(At) & A^{-1}A\cos(At) \end{vmatrix} = I\,,$ \begin{align*} P.I & = - \cos(At)\int_{0}^{t} A^{-1}\sin(As)Bu(s)ds + A^{-1}\sin(At)\int_{0}^{t}\cos(As)Bu(s)ds\\ & = \int_{0}^{t} A^{-1}(- \cos(At)\sin(As) + \sin(At)\cos(As))Bu(s)ds\\ & = \int_{0}^{t} A^{-1}\sin(A(t-s))Bu(s)ds, \end{align*} using property (vi). Hence the solution of \eqref{ld} is given by \begin{equation*} x(t) = \cos(At)C_{1} + A^{-1}\sin(At)C_{2} + \int_{0}^{t} A^{-1} \sin(A(t-s))Bu(s)ds. \end{equation*} Applying the initial conditions $x(0)= x_{0}$, $x'(0)=y_{0}$, the solution becomes $$\label{ldi} x(t) = \cos(At)x_{0} + A^{-1}\sin(At)y_{0} +\int_{0}^{t} A^{-1} \sin(A(t-s))Bu(s)ds.$$ Following the same approach the solution of the nonlinear system \eqref{nld} can be written as \label{nldi} \begin{aligned} x(t) &= \cos(At)x_{0} + A^{-1}\sin(At)y_{0} +\int_{0}^{t} A^{-1} \sin(A(t-s))Bu(s)ds\\ &\quad + \int_{0}^{t} A^{-1}\sin(A(t-s))f(s,x(s))ds \end{aligned} We remark that the above form of solution valid even if the matrix $A$ is singular, in that case $A^{-1}\sin(At)$ is to be taken as in \eqref{inv-A}. \section{Controllability Results For Linear Systems} In this section we obtain necessary and sufficient conditions for the controllability of the linear system \eqref{ld}. We make use of the following lemmas to prove the controllability result of \eqref{ld}. \begin{lemma}[Chen\cite{Chen}] \label{W-inv} Let $f_{i}$, for $i=1,2,\dots ,n,$ be $1\times p$ complex vector valued continuous functions defined on $[t_{1},t_{2}]$. Let $F$ be the $n\times p$ matrix with $f_{i}$ as its $i^{th}$ row. Define $W(t_{1},t_{2})= \int_{t_{1}}^{t_{2}}F(t){F}^{*}(t)dt$ Then $f_{1},f_{2},\dots ,f_{n}$ are linearly independent on $[t_{1},t_{2}]$ if and only if the $n\times n$ constant matrix $W(t_{1},t_{2})$ is nonsingular. \end{lemma} \begin{lemma}[Chen \cite{Chen}] \label{rank} Assume that for each $i$, $f_{i}$ is analytic on $[t_{1},t_{2}]$. Let $F$ be the $n\times p$ matrix with $f_{i}$ as its $i^{th}$ row, and let $F^{(k)}$ be the $k^{th}$ derivative of $F$. Let $t_{0}$ be any fixed point in $[t_{1},t_{2}]$. Then the $f_{i}$ are linearly independent on $[t_{1},t_{2}]$ if and only if $\mathop{\rm Rank}[F(t_{0}):{F}^{(1)}(t_{0}):\dots :{F}^{(n-1)}(t_{0}): \dots ]=n$ \end{lemma} The necessary and sufficient condition for the controllability of the linear system \eqref{ld} is given in the following theorem. \begin{theorem}\label{Lin_con} The following four statements regarding the linear system \eqref{ld} are equivalent: \begin{itemize} \item[(a)] The linear system \eqref{ld} is controllable on $[0,T]$. \item[(b)] The rows of $A^{-1}\sin(At)B$ are linearly independent. \item[(c)] The Controllability Grammian, $$\label{gram} W(0,T) = \int_{0}^{T} A^{-1}\sin(A(T-s))BB^{*}(A^{-1}\sin(A(T-s)))^{*}ds,$$ is nonsingular. \item[(d)] $$\label{ranklin} \mathop{\rm Rank}[B:A^{2}B:(A^{2})^{2}B:\dots :(A^{2})^{n-1}B]=n.$$ \end{itemize} \end{theorem} \begin{proof} First we shall prove the implication $(a)\Rightarrow (b)$, we prove this by contradiction. Suppose that the system \eqref{ld} is controllable but the rows of $A^{-1}\sin(At)B$ are linearly dependent functions on [0,T]. Then there exists a nonzero constant $1\times n$ row vector $\alpha$ such that $$\label{alpha} \alpha A^{-1}\sin(At)B = 0 \quad \forall t\in [0,T]$$ Let us choose $x(0)=x_{0}=0$, $x'(0)=y_{0}=0$. Therefore, the solution \eqref{ldi} becomes $x(t) = \int_{0}^{t} A^{-1}\sin(A(t-s))Bu(s)ds$ Since the system \eqref{ld} is controllable on $[0,T]$, taking $x(T)=\alpha^{*}$, where $\alpha^{*}$ is the conjugate transpose of $\alpha$. $x(T) = \alpha^{*} = \int_{0}^{T} A^{-1}\sin(A(T-s))Bu(s)ds\,.$ Now premultiplying both sides by $\alpha$, we have $\alpha \alpha^{*} = \int_{0}^{T} \alpha A^{-1}\sin(A(T-s))Bu(s)ds\,.$ From equation \eqref{alpha} $\alpha \alpha^{*} = 0$ and hence $\alpha = 0$. Hence it contradicts our assumption that $\alpha$ is non-zero. This implies that rows of $A^{-1}\sin(At)B$ are linearly independent on $[0,T]$. Now we prove the implication $(b)\Rightarrow (a)$. Suppose that the rows of $A^{-1}\sin(At)B$ are linearly independent on $[0,T]$. Therefore by Lemma \ref{W-inv}, the $n\times n$ constant matrix $W(0,T)=\int_{0}^{T} A^{-1}\sin(A(T-s))BB^{*}(A^{-1}\sin(A(T-s)))^{*}ds$ is nonsingular. Now we claim that the control $$\label{cont} u(t) = B^{*}(A^{-1}\sin(A(T-t)))^{*}W^{-1}(0,T)(x_{1} - \cos(AT)x_{0} - A^{-1}\sin(AT)y_{0})$$ transfers the initial state $x_{0}$ to the final state $x_{1}$ during $[0,T]$. Substituting \eqref{cont} for $u(t)$ in the solution \eqref{ldi}, we obtain \begin{align*} x(t) &= \cos(At)x_{0} + A^{-1}\sin(At)y_{0} + \int_{0}^{t} A^{-1}\sin(A(t-s))BB^{*}\\ &\quad\times (A^{-1}\sin(A(T-s)))^{*}W^{-1}(0,T)(x_{1} - \cos(AT)x_{0} - A^{-1}\sin(AT)y_{0})ds \end{align*} At $t=T$, we have \begin{align*} x(T) & = \cos(AT)x_{0}+A^{-1}\sin(AT)y_{0}+\int_{0}^{T} A^{-1}\sin(A(T-s))BB^{*}\\ &\quad (A^{-1}\sin(A(T-s)))^{*}W^{-1}(0,T)(x_{1} - \cos(AT)x_{0} - A^{-1}\sin(AT)y_{0})ds\\ & = \cos(AT)x_{0} + A^{-1}\sin(AT)y_{0} + W(0,T)W^{-1}(0,T)\\ &\quad (x_{1} - \cos(AT)x_{0} - A^{-1}\sin(AT)y_{0})\\ & = \cos(AT)x_{0}+A^{-1}\sin(AT)y_{0} + (x_{1} - \cos(AT)x_{0} - A^{-1}\sin(AT)y_{0})\\ & = x_{1} \end{align*} Hence the system is controllable. The implications $(b)\Rightarrow(c)$ and $(c)\Rightarrow(b))$ follow directly from Lemma \ref{W-inv}. Now we shall obtain the implication $(c)\Rightarrow(d)$. The controllability Grammian $W(0,T) = \int_{0}^{T} A^{-1}\sin(A(T-s))BB^{*}(s){(A^{-1}\sin(A(T-s)))}^{*}$ is nonsingular. Hence by Lemma \ref{gram}, the rows of $A^{-1}\sin(At)B$ are linearly independent on $[0,T]$. Since the entries of $A^{-1}\sin(At)B$ are analytic functions, applying the Lemma \ref{rank}, the rows of $A^{-1}\sin(At)B$ are linearly independent on [0,T] if and only if $\mathop{\rm Rank}[A^{-1}\sin(At)B:A^{-1}\cos(At)AB:-A^{-1}\sin(At)A^{2}B: -A^{-1}\cos(At)A^{3}B:$$A^{-1}\sin(At)A^{4}B:A^{-1}\cos(At)A^{5}B\dots ] = n.$ for any $t \in [0,T]$. Let $t = 0$, then this reduces to \begin{gather*} \mathop{\rm Rank}[0:B:0:A^{2}B:0:\dots :(A^{2})^{n-1}B:\dots ]=n,\\ \mathop{\rm Rank}[B:A^{2}B:{(A^{2})}^{2}B:\dots :{(A^{2})}^{n-1}B:\dots ]=n \end{gather*} Using Cayley-Hamilton theorem, $\mathop{\rm Rank}[B:A^{2}B:{(A^{2})}^{2}B:\dots :{(A^{2})}^{n-1}B]=n$ Now to prove the implication $(d)\Rightarrow(c)$, we assume that $\mathop{\rm Rank}[B:A^{2}B:{(A^{2})}^{2}B:\dots :{(A^{2})}^{n-1}B]=n$ Thus by Lemma \ref{rank}, the rows of $A^{-1}\sin(At)B$ are linearly independent. Hence Lemma \ref{W-inv} implies $W(0,T) =\int_{0}^{T} A^{-1}\sin(A(T-s))BB^{*}(s){(A^{-1}\sin(A(T-s)))}^{*}ds$ is nonsingular. Thus for the linear system\eqref{ld} , the control $u(t)$ defined by \eqref{cont}, steers the state from $x_{0}$ to $x_{1}$ during $[0,T]$. Since $x_{0}$ and $x_{1}$ are arbitrary, the system \eqref{ld} is controllable. \end{proof} \begin{remark} \label{rmk4.1} \rm Hughes and Skelton \cite{Skel1} obtained the condition \eqref{ranklin} by converting the system into first order system. However, our approach is different and the result obtained is directly from the second order system and also it provides a method to compute the steering control as we will see this in the next section. \end{remark} \section{Controllability of the Nonlinear Systems} We now investigate the controllability of the nonlinear system \eqref{nld}. We assume that the corresponding linear system \eqref{ld} is controllable and the control function $u$ belongs to $L^{2}([0,T],R^{m})$. We use the following definition. \begin{definition} \label{def5.1} \rm An $m\times n$ matrix function $P(t)$ with entries in $L^{2}([0,T])$ is said to be a steering function for \eqref{ld} on $[0,T]$ if $\int_{0}^{T} A^{-1} \sin(A(T-s))BP(s)ds = I,$ where $I$ is the identity matrix on $R^{n}$. \end{definition} The linear system \eqref{ld} is controllable if and only if there exists a steering function $P(t)$ for the system \eqref{ld} (refer Russel \cite{Russ}). \begin{remark} \label{rmk5.1} \rm If the controllability Grammian \eqref{gram} is nonsingular then $$\label{p(t)} P(t) = B^{*}(A^{-1}\sin(A(T-t))^{*}W^{-1}(0,T)$$ defines a steering function for the linear system \eqref{ld}. \end{remark} Now the nonlinear system \eqref{nld} is controllable on $[0,T]$ if and only if for every given $x_{1}$ and $x_{0}$ in $R^{n}$ there exists a control $u$, such that \begin{align*} x_{1}&=x(T)\\ &= \cos(AT)x_{0} + A^{-1}\sin(AT)y_{0} +\int_{0}^{T} A^{-1}\sin(A(T-s))f(s,x(s))ds\\ &\quad + \int_{0}^{T} A^{-1}\sin(A(T-s))Bu(s)ds \end{align*} Consider the control $u(t)$ defined by $$\label{concal} u(t) = P(t)\{x_{1} - \cos(AT)x_{0} - A^{-1}\sin(AT)y_{0} - \int_{0}^{T} A^{-1}\sin(A(T-s))f(s,x(s))ds\}$$ where, $P(t)$ is the steering function for the linear system \eqref{ld}. Now substituting this control $u(t)$ into equation \eqref{nldi}, we have \label{subs} \begin{aligned} x(t) & = \cos(At)x_{0}+A^{-1}\sin(At)y_{0}+\int_{0}^{t} A^{-1}\sin(A(T-s))f(s,x(s))ds \\ &\quad +\int_{0}^{t} A^{-1}\sin(A(T-s))BP(s)\{x_{1} - \cos(AT)x_{0} - A^{-1}\sin(AT)y_{0} \\ &\quad - \int_{0}^{T} A^{-1}sinA(T-\tau)f(\tau,x(\tau))d\tau\}ds \end{aligned} If this equation is solvable then $x(t)$ satisfies $x(0)=x_0$ and $x(T)=x_{1}$. This implies that the system \eqref{nld} is controllable with control $u(t)$ given by \eqref{concal}. Hence, controllability of the system \eqref{nld} is equivalent to the solvability of the equation \eqref{subs}. Now applying Banach contraction principle, we will prove the solvability of the equation \eqref{subs}. \begin{theorem}[Banach contraction Principle, Limaye \cite{Lim}] Let $X$ be a Banach space and $T:X \to X$ be a contraction on $X$. Then $T$ has precisely one fixed point, and the fixed point can be computed by the iterative scheme $x_{n+1} = Tx_{n}$, $x_{0}$ being any arbitrary initial guess. \end{theorem} We define a mapping $F:C([0,T];R^{n})\to C([0,T];R^{n})$ by \label{Fx} \begin{aligned} (Fx)(t) & = \cos(At)x_{0}+A^{-1}\sin(At)y_{0}+\int_{0}^{t} A^{-1}\sin(A(t-s))f(s,x(s)) ds\\ &\quad + \int_{0}^{t}A^{-1}\sin(A(t-s)) BP(s)\{x_{1} - \cos(AT)x_{0} - A^{-1}\sin(AT)y_{0}\\ & \quad - \int_{0}^{T} A^{-1}sinA(T-\tau)f(\tau,x(\tau))d\tau\}ds\,. \end{aligned} The following lemma proves that $F$ is a contraction under some assumptions on the system components. \begin{lemma}\label{contrac} Under the following assumptions the nonlinear operator $F$ is a contraction: \begin{itemize} \item[(i)] $a = \sup_{t\in [0,T]}\|A^{-1}\sin(At)\|$. \item[(ii)] $b= \|B\|$. \item[(iii)] $p=\sup_{t\in[0,T]}\|P(t)\|$. \item[(iv)] The nonlinear function $f:[0,T]\times R^{n} \to R^{n}$ is Lipschitz continuous. That is, there exists $\alpha>0$ such that $\|f(t,x) - f(t,y)\|\leq \alpha \|x -y\|\quad \forall x,y \in R^{n},\quad t\in[0,T].$ \item[(v)] $\alpha aT(1 + abpT) < 1$. \end{itemize} \end{lemma} \begin{proof} From the definition of $F$, we have \begin{align*} &\|Fx - Fy\|\\ & = \sup_{t\in [0,T]}\|(Fx)(t) - (Fy)(t)\|\\ & = \sup_{t\in [0,T]}\|\int_{0}^{t} A^{-1}\sin(A(T-s))(f(s,x(s)) -f(s,y(s))ds+\int_{0}^{t}A^{-1}\\ &\quad\times \sin(A(T-s))B P(s)\int_{0}^{T} A^{-1}sinA(T-\tau)(f(\tau,x(\tau)) -f(\tau,y(\tau)))d\tau ds\|\\ & \leq \sup_{t\in [0,T]}\|\int_{0}^{t} A^{-1}\sin(A(T-s))(f(s,x(s)) -f(s,y(s))ds\|+\sup_{t\in [0,T]}\|\int_{0}^{t} A^{-1}\\ &\quad \times \sin(A(T-s))BP(s)\int_{0}^{T} A^{-1}sinA(T-\tau) (f(\tau,x(\tau)) -f(\tau,y(\tau)))d\tau ds\|\\ & \leq \sup_{t\in [0,T]}\int_{0}^{t} \|A^{-1}\sin(A(T-s))\|\; \|(f(s,x(s)) -f(s,y(s))\|ds\\ & \quad +\sup_{t\in [0,T]}\int_{0}^{t} \|A^{-1}\sin(A(T-s))\|\|B\| \|P(s)\|\\ & \quad \times \int_{0}^{T}\|A^{-1}sinA(T-\tau)\|\;\|(f(\tau,x(\tau)) -f(\tau,y(\tau)))\|d\tau ds\\ & \leq \sup_{t\in [0,T]}a\int_{0}^{t} \alpha\|x(s)-y(s)\|ds +\sup_{t\in [0,T]}a^{2}bpt \int_{0}^{T} \alpha\|y(\tau)) -x(\tau)\|d\tau\\ & \leq a\alpha\sup_{t\in [0,T]}\int_{0}^{t} \|x(s)-y(s)\|ds +a^{2}bpT\alpha \int_{0}^{T} \sup_{t\in [0,T]}\|y(\tau)) -x(\tau)\|d\tau\\ & \leq a\alpha T\|x-y\|+a^{2}bpT\alpha T \|x-y\|\\ & \leq a\alpha T(1 + abpT)\|x-y\| \end{align*} Since $a\alpha T(1 + abpT) < 1$, we have $F$ is a contraction. \end{proof} Now we have the following computational result for the controllability of the nonlinear system \eqref{nld}. \begin{theorem}\label{main} Under the assumptions of Lemma \ref{contrac}, the system \eqref{nld} is controllable and the steering control and the controlled solution can be computed by the following iterative scheme: $$\label{contit} u^{n}(t) = P(t)\{x_{1} - \cos(AT)x_{0} - A^{-1}\sin(AT)y_{0} - \int_{0}^{T} A^{-1}\sin(A(T-s))f(s,x^{n}(s))ds$$ \label{iter} \begin{aligned} x^{n+1}(t) &= \cos(At)x_{0} + A^{-1}\sin(At)y_{0} + \int_{0}^{t} A^{-1}\sin(A(t-s))f(s,x^{n}(s))ds \\ &\quad + \int_{0}^{t}A^{-1}\sin(A(t-s))Bu^{n}(s)ds \end{aligned} where $x^{0}(t) = x_{0}$ and $n = 1,2,3,\dots$. \end{theorem} \begin{proof} In Lemma \ref{contrac} we have proved that $F$, as defined in the equation \eqref{Fx}, is a contraction. Hence, from the Banach contraction principle, $F$ has a fixed point. Thus the equation \eqref{subs} is solvable, subsequently the system \eqref{nld} is controllable. Further, Theorem \ref{contrac} implies the convergence of the iterative scheme for the computation of control and controlled trajectory. \end{proof} \section{Computational Algorithm for the controlled state and steering control} Here we compute Cosine and Sine of a matrix $A\in R^{n \times n}$, using the algorithm proposed by Higham and Hargreaves \cite{Higm}. The algorithm makes use of P\'ade approximations of $\cos(A)$ and $\sin(A)$. We define $C_{i} = \cos(2^{i-m}A)$ and $S_{i} = \sin(2^{i-m}A)$. The value of m is chosen in such a way that $\|2^{-m}A\|$ is small enough, ensuring a good approximation of $C_{0} = \cos(2^{-m}A)$ and $S_{0} = \sin(2^{-m}A)$ by P\'ade approximation. By applying the cosine and sine double angle formulae $\cos(2A) = 2cos^{2}(A) - I$ and $\sin(2A) = 2\sin(A)\cos(A)$, we can compute $C_{m} = \cos(A)$ and $S_{m} = \sin(A)$, from $C_{0}$ and $S_{0}$ using the recurrence relation $C_{i+1} = 2C_{i}^{2}-I$ and $S_{i+1} = 2C_{i}S_{i}$, $i = 0,1,\dots m-1$. The algorithm for the computation of Sine and Cosine matrices is summarized as follows: \subsection*{Algorithm} Given a matrix $A\in R^{n \times n}$:\\ Choose $m$ such that $2^{-m}\|A\|$ is very small.\\ $C_{0}$ = pade approximation to $\cos(2^{-m}A)$. \\ $S_{0}$= pade approximation to $\sin(2^{-m}A)$. \\ for $i = 0\dots m-1$.\\ $C_{i+1} = 2C_{i}^{2}-I$.\\ $S_{i+1} = 2C_{i}S_{i}$.\\ end. \subsection*{Steering Control For The Linear System} The control which steers the initial state $x_{0}$ of the MSOL system \eqref{ld} to a desired state $x_{1}$ during $[0,T]$ is given by \label{ut} \begin{aligned} u(t) &= B^{*}(A^{-1}\sin(A(T-t)))^{*}W^{-1}(0,T)\{x_{1} - \cos(AT)x_{0} - A^{-1}\sin(AT)y_{0} \\ &\quad -\int_{0}^{T} A^{-1}\sin(A(T-s))f(s,x(s))ds\} \end{aligned} where, $\sin(At)$ and $\cos(At)$ are computed by the P\'ade approximation algorithm given in (Hargreaves and Higham \cite{Higm}), and $W^{-1}(0,T)$ is computed by using \eqref{gram}. \section*{Numerical Experiment For Matrix Second Order Linear System} \begin{example} \label{exa6.1} \rm Consider the Matrix Second Order Linear (MSOL) System $\frac{d^{2}x(t)}{dt^{2}} + A^{2}x(t) = Bu(t), \quad x(t)\in \mathbb{R}^{3}$ with initial conditions $x(0) = \begin{pmatrix} -1\\ 1\\ 0\end{pmatrix}$, $x'(0) = \begin{pmatrix} 1\\ 1\\ -1\end{pmatrix}$, where $A^{2}=\begin{pmatrix} 5 & -4 & 2\\ -4 & 7 & -2\\ 4 & -4 & 3\end{pmatrix}, \quad B = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}, \quad \text{and hence}\quad A= \begin{pmatrix} 1 & -2 & 0\\ -2 & 1 & -1\\ 0 & -2 & 1 \end{pmatrix}.$ The controllability matrix is $Q = [B A^{2}B (A^{2})^{2}B] = \begin{pmatrix} 0 & 2 & 24\\ 0 & -2 & -28\\ 1 & 3 & 25 \end{pmatrix}$ and the $\mathop{\rm Rank}(Q) = 3$. Hence the system is controllable. The matrices $\sin(At)$ and $\cos(At)$ for $t = 1$ are \begin{gather*} \sin(A) = \begin{pmatrix} -0.1512 & -0.2810 & -0.4965\\ -0.2810 & -0.6478 & -0.1405\\ -0.9931 & -0.2810 & 0.3453 \end{pmatrix}, \\ \cos(A) = \begin{pmatrix} -0.0972 & 0.4385 & -0.3188\\ 0.4385 & -0.4160 & 0.2192\\ -0.6375 & 0.4385 & 0.2215 \end{pmatrix}. \end{gather*} The controllability Grammian matrix, $W(0,T)$ is $W = \begin{pmatrix} 0.0733 & -0.0406 & -0.2130\\ -0.0406 & 0.0272 & 0.1255\\ -0.2130 & 0.1255 & 0.6915 \end{pmatrix},$ taking $T = 2$. Now using the algorithm given in \eqref{ut} along with P\'ade approximation to the Sine and Cosine matrix, we compute the steering control $u(t)$, steering the state from $x_{0} = \begin{pmatrix} -1 \\1 \\0 \end{pmatrix}$ to $x_{1} = \begin{pmatrix} 1 \\-1 \\2 \end{pmatrix}$ during the time interval $[0,2]$. Furthermore, the controlled trajectory and steering control $u$ are computed and are depicted in Figure \ref{fig1}. \begin{figure}[ht] \begin{center} \includegraphics[width=4.0in]{figure1} \end{center} \caption{}\label{fig1} \end{figure} \end{example} \subsection*{Steering Control For The Nonlinear System} The steering control and controlled trajectories of the MSON system steering from $x_{0}$ to $x_{1}$ during $[0,T]$ can be approximated from the following algorithm: $u^{n}(t) = P(t)\{x_{1} - \cos(AT)x_{0} - A^{-1}\sin(AT)y_{0} - \int_{0}^{T} A^{-1}\sin(A(T-s))f(s,x^{n}(s))ds$ \label{xt} \begin{aligned} x^{n+1}(t) &= \cos(At)x_{0} + A^{-1}\sin(At)y_{0} + \int_{0}^{t} A^{-1}\sin(A(t-s))f(s,x^{n}(s))ds \\ &\quad + \int_{0}^{t}A^{-1}\sin(A(t-s))Bu^{n}(s)ds \end{aligned} with $x^{0}(t) = x_{0}$, $n = 1,2,3,\dots$, and $P(t)$ being the steering function given in equation \eqref{p(t)}. \section*{Numerical Experiment For Matrix Second Order Nonlinear System} \begin{figure}[ht] \begin{center} \includegraphics[width=4.0in]{figure2} \end{center} \caption{}\label{fig2} \end{figure} \begin{example} \label{exa2} \rm Consider the Matrix Second Order Nonlinear(MSON) system described by: $\frac{d^{2}x(t)}{dt^{2}} + A^{2}x(t) = B u(t) + f(t,x(t)),$ where $x(t)\in \mathbb{R}^{3}$ and $f(t,x(t)) = \begin{pmatrix} f_{1}(x_{1},x_{2},x_{3}) \\ f_{2}(x_{1},x_{2},x_{3}) \\ f_{3}(x_{1},x_{2},x_{3}) \end{pmatrix}$ with the initial conditions $x(0) = \begin{pmatrix} -1\\ 1\\ 0\end{pmatrix}$, $x'(0) = \begin{pmatrix} 1\\ 1\\ -1\end{pmatrix}$ and $A^{2} = \begin{pmatrix} 14 & -2 & 12\\ 10 & 14 & 30\\ 0 & -12 & 16 \end{pmatrix}, \quad B = \begin{pmatrix} 1 \\ 1 \\ 0\end{pmatrix}\quad\text{and hence}\quad A =\begin{pmatrix} -2 & 2 & 3\\ 2 & 4 & 3\\ 2 & -2 & 4\end{pmatrix}.$ The controllability matrix is $Q = [B A^{2}B(A^{2})^{2}B]= \begin{pmatrix} 0 & 10 & 100\\ 1 & 44 & 836\\ 1 & 4 & -464 \end{pmatrix}$ and $\mathop{\rm Rank}(Q) = 3$. Hence the corresponding linear system is controllable. We have the following numerical estimate, for the parameters given in Lemma \ref{contrac}, taking $T = 1$, \begin{gather*} a = \sup_{t\in [0,T]}\|A^{-1}\sin(At)\| = 1.0316,\quad b = \|B\| = 1.4142,\\ p = \sup_{t\in[0,T]}\|P(t)\| = 52.1831 \end{gather*} Let us take $f_{1}(x_{1},x_{2},x_{3}) = \frac{\sin(x_{1}(t))}{82},\quad f_{2}(x_{1},x_{2},x_{3})=\frac{\cos(x_{2}(t))}{81},\quad f_{3}(x_{1},x_{2},x_{3})=\frac{x_{3}(t)}{80}.$ The nonlinear function $f(t,x(t))$ is Lipschitz continuous with Lipschitz constant $\alpha = 1/80$ and $\alpha aT(1 + abpT) < 1$. Hence, it satisfies all the assumption of the Theorem \ref{main}. So the MSON system is controllable. Now using the algorithm given in \eqref{xt} with P\'ade approximation to Sine and Cosine matrices, the controllability Grammian matrix, $W(0,T)$ is $W = \begin{pmatrix} 0.0682 & 0.1128 & 0.0241\\ 0.1128 & 0.1998 & 0.0525\\ 0.0241 & 0.0525 & 0.0994 \end{pmatrix}.$ We compute the steering control $u(t)$, steering the state from $x_{0} = \begin{pmatrix} -1 \\1 \\0 \end{pmatrix}$ to $x_{1} = \begin{pmatrix} 0 \\-1 \\1 \end{pmatrix}$ during the time interval $[0,1]$. Furthermore the controlled trajectory and the steering control $u(t)$ are computed and is shown in Figure \ref{fig2}. \end{example} \begin{thebibliography}{00} \bibitem{Bala} Balas, M. J.; \textit{Trends in Large Space Structure Control Theory: Fondest Hopes}, IEEE trans. on Automatic Control, vol. 27, no.3, pp.522-535, June 1982. \bibitem{Brok} Brockett, R. W.; \textit{Finite-Dimensional Linear Systems}, John Wiley and Sons, New York, 1970. \bibitem{Chen} Chen, C. T.; \textit{Linear System Theory and Design}, Saunder college publishing, Harcourt Brace college Publishers, New York, 1970. \bibitem{Deme} Demetriou, M. A.; \textit{UIO for Fault Detection in Vector Second Order Systems}, Proc. of the American Control Conference, Arlington, VA, pp. 1121-1126, June 25-27,2001. 2315, 2000. \bibitem{Diwa1} Diwakar, A. M.; Yedavalli, R. K.; \textit{Stability of Matrix Second-Order systems: New Conditions and Perspectives}, IEEE Trans. on Automatic Control,vol.44, no. 9, pp. 1773-1777, 1999. \bibitem{Diwa2} Diwakar, A. M.; Yedavalli, R. K.; \textit{Smart Structure Control in Matrix Second Order Form}, in proc. North American Conf. Smart Structures and Materials, pp. 24-34,1995. \bibitem{Fitz} Fitzgibbon, W. E.; \textit{Global Existence and Boundedness of Solutions to the Extensible Beam Equation}, SIAM J.Math. Anal. 13, 739-745, 1982. \bibitem{Gold} Goldstein, J. A.; \textit{Semigroups of Linear Operators and Applications}, Oxford Math. Monogr., Oxford Univ. Press, New-York, 1985. \bibitem{Higm} Hargreaves, G. I.; Higham, N. J.; \textit{Efficient Algorithms for The Matrix Cosine and Sine}, Numerical Algorithms 40: 383400. DOI 10.1007/s11075- 005-8141-0, 2005. \bibitem{Skel1} Hughes, P. C.; Skelton, R. E.; \textit{Controllability and Observability of Linear Matrix Second Order Systems}, ASME J.Appl. Mech., vol. 47, pp 415-420, 1980. \bibitem{Joshi} Joshi, M. C.; George, R. K.; \textit{Controllability of Nonlinear Systems}, Journal of Numer. Funct. Anal. and optimiz, pp.139-166,1989. \bibitem{Bose} Joshi, M. C.; Bose, R. K.; \textit{Some Topics in Nonlinear Functional Analysis}, Wiley Eastern Limited, New Delhi, 1985. \bibitem{Arno} Laub, A. J.; Arnold, W. F.; \textit{Controllability and Observability Criteria for Multivariable Linear Second-Order Model}, IEEE Trans. on Automatic Control, vol.29, no.2, pp.163-165, February 1984. \bibitem{Lim} Limaye, B. V.; \textit{Functional Analysis}, Wiley Eastern Limited, New Delhi,1980. \bibitem{Russ} Russel, D. L.; \textit{Mathematics of Finite-Dimensional Control System}, Marcel Dekker Inc., New York and Basel ,1979. \bibitem{Skel2} Skelton R. E., \textit{Adaptive Orthogonal Filters for Compensation of Model Errors in Matrix Second Order Systems.} J. Guidance, Contr. Dynam., pp 214-221, Mar. 1979. \bibitem{Trav1} Travis, C. C.; Webb, G. F.; \textit{Cosine Families and Abstract Nonlinear Second Order Differential Equaitons}, Acta Math. Acad. Sci. Hungar. 32,77-96, 1978. \bibitem{Trav2} Travis, C. C.; Webb, G. F.; \textit{Second Order Differential Equations in Banach Spaces}, in Nonlinear equations in Abstract spaces, Proc. Internat Sympos.(Univ. Texas, Arlington TX, 1977), Academic press, New York, 331-361,1978. \bibitem{Wu} Wu, Y. L.; Duan, G. R.; \textit{Unified Parameter Approaches for Observer Design in Matrix Second Order Linear Systems}, International Journal of Control, Automation, and Systems, vol.3, no.2, pp. 159-165, 2005. \end{thebibliography} \end{document}