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An equation in which the unknown is a matrix of functions appearing in the equation together with its derivative.
 
An equation in which the unknown is a matrix of functions appearing in the equation together with its derivative.
  
 
Consider a linear matrix differential equation of the form
 
Consider a linear matrix differential equation of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m0628001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$X'=A(t)X,\quad t\in\mathbf R,\label{1}\tag{1}$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m0628002.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m0628003.png" />-dimensional matrix function with locally Lebesgue-integrable entries, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m0628004.png" /> be an absolutely-continuous solution of equation (1) satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m0628005.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m0628006.png" /> is the identity matrix. Then the vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m0628007.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m0628008.png" />, is a solution of the linear system
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where $A(t)$ is an $(n\times n)$-dimensional matrix function with locally Lebesgue-integrable entries, and let $X(t)$ be an absolutely-continuous solution of equation \eqref{1} satisfying the condition $X(t_0)=I$, where $I$ is the identity matrix. Then the vector function $x(t)=X(t)h$, $h\in\mathbf R^n$, is a solution of the linear system
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m0628009.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$x'=A(t)x\label{2}\tag{2}$$
  
satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280010.png" />. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280012.png" /> is a solution of the system (2) satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280014.png" />, then the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280015.png" /> with as columns the solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280016.png" /> is a solution of the matrix differential equation (1). If, in addition, the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280017.png" /> are linearly independent, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280018.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280019.png" />.
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satisfying the condition $x(t_0)=h$. Conversely, if $h_1,\dots,h_n\in\mathbf R^n$ and $x_i(t)$ is a solution of the system \eqref{2} satisfying the condition $x_i(t_0)=h_i$, $i=1,\dots,n$, then the matrix $X(t)$ with as columns the solutions $x_i(t)$ is a solution of the matrix differential equation \eqref{1}. If, in addition, the vectors $h_1,\dots,h_n$ are linearly independent, then $\det X(t)\neq0$ for all $t\in\mathbf R$.
  
Equation (1) is a particular case of the following matrix differential equation (arising in the theory of stability)
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Equation \eqref{1} is a particular case of the following matrix differential equation (arising in the theory of stability)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
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$$X'=A(t)X-XB(t)+C(t).\label{3}\tag{3}$$
  
The solution of (3) with initial condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280021.png" /> is given by the formula
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The solution of \eqref{3} with initial condition $X(t_0)=X_0$ is given by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280022.png" /></td> </tr></table>
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$$X(t)=U(t,t_0)X_0V(t,t_0)+\int\limits_{t_0}^tU(t,s)C(s)V(s,t)ds,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280023.png" /> is the solution of (1) with the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280024.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280025.png" /> is the solution of the matrix differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280026.png" /> with the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280027.png" />.
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where $U(t,s)$ is the solution of \eqref{1} with the condition $X(s,s)=I$, and $V(t,s)$ is the solution of the matrix differential equation $X'=B(t)X$ with the condition $X(s,s)=I$.
  
 
In various applied problems (theories of stabilization, optimal control, filtration of control system, and others) an important role is played by the so-called matrix Riccati differential equation
 
In various applied problems (theories of stabilization, optimal control, filtration of control system, and others) an important role is played by the so-called matrix Riccati differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280028.png" /></td> </tr></table>
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$$X'=A(t)X-XB(t)+C(t)+XD(t)X.$$
  
 
Thus, if the matrix Riccati equation
 
Thus, if the matrix Riccati equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280029.png" /></td> </tr></table>
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$$X'=-(F(t)+\lambda I)^TX-X(F(t)+\lambda I)-I+XG(t)G^T(t)X,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280030.png" /> stands for transposition, has for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280031.png" /> a bounded solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280032.png" /> on the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280033.png" />, and if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280034.png" />, all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280035.png" />, and some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280036.png" />, the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280037.png" /> holds, then every solution of the controllable system
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where $^T$ stands for transposition, has for $\lambda\geq0$ a bounded solution $X(t)$ on the line $\mathbf R$, and if for all $h\in\mathbf R^n$, all $t\in\mathbf R$, and some $\epsilon>0$, the inequality $h^TX(t)h\geq\epsilon h^Th$ holds, then every solution of the controllable system
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280038.png" /></td> </tr></table>
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$$x'=F(t)x+G(t)u,\quad x\in\mathbf R^n,\quad u\in\mathbf R^m,$$
  
closed by the feedback law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280039.png" />, satisfies the inequality
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closed by the feedback law $u=-G^T(t)X(t)x/2$, satisfies the inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280040.png" /></td> </tr></table>
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$$|x(t)|\leq M|x(s)|e^{-\lambda(t-s)},\quad s\leq t,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280041.png" /> is the Euclidean norm and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280042.png" /> does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280043.png" />.
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where $|\cdot|$ is the Euclidean norm and $M$ does not depend on $s$.
  
 
====References====
 
====References====

Latest revision as of 17:38, 14 February 2020

An equation in which the unknown is a matrix of functions appearing in the equation together with its derivative.

Consider a linear matrix differential equation of the form

$$X'=A(t)X,\quad t\in\mathbf R,\label{1}\tag{1}$$

where $A(t)$ is an $(n\times n)$-dimensional matrix function with locally Lebesgue-integrable entries, and let $X(t)$ be an absolutely-continuous solution of equation \eqref{1} satisfying the condition $X(t_0)=I$, where $I$ is the identity matrix. Then the vector function $x(t)=X(t)h$, $h\in\mathbf R^n$, is a solution of the linear system

$$x'=A(t)x\label{2}\tag{2}$$

satisfying the condition $x(t_0)=h$. Conversely, if $h_1,\dots,h_n\in\mathbf R^n$ and $x_i(t)$ is a solution of the system \eqref{2} satisfying the condition $x_i(t_0)=h_i$, $i=1,\dots,n$, then the matrix $X(t)$ with as columns the solutions $x_i(t)$ is a solution of the matrix differential equation \eqref{1}. If, in addition, the vectors $h_1,\dots,h_n$ are linearly independent, then $\det X(t)\neq0$ for all $t\in\mathbf R$.

Equation \eqref{1} is a particular case of the following matrix differential equation (arising in the theory of stability)

$$X'=A(t)X-XB(t)+C(t).\label{3}\tag{3}$$

The solution of \eqref{3} with initial condition $X(t_0)=X_0$ is given by the formula

$$X(t)=U(t,t_0)X_0V(t,t_0)+\int\limits_{t_0}^tU(t,s)C(s)V(s,t)ds,$$

where $U(t,s)$ is the solution of \eqref{1} with the condition $X(s,s)=I$, and $V(t,s)$ is the solution of the matrix differential equation $X'=B(t)X$ with the condition $X(s,s)=I$.

In various applied problems (theories of stabilization, optimal control, filtration of control system, and others) an important role is played by the so-called matrix Riccati differential equation

$$X'=A(t)X-XB(t)+C(t)+XD(t)X.$$

Thus, if the matrix Riccati equation

$$X'=-(F(t)+\lambda I)^TX-X(F(t)+\lambda I)-I+XG(t)G^T(t)X,$$

where $^T$ stands for transposition, has for $\lambda\geq0$ a bounded solution $X(t)$ on the line $\mathbf R$, and if for all $h\in\mathbf R^n$, all $t\in\mathbf R$, and some $\epsilon>0$, the inequality $h^TX(t)h\geq\epsilon h^Th$ holds, then every solution of the controllable system

$$x'=F(t)x+G(t)u,\quad x\in\mathbf R^n,\quad u\in\mathbf R^m,$$

closed by the feedback law $u=-G^T(t)X(t)x/2$, satisfies the inequality

$$|x(t)|\leq M|x(s)|e^{-\lambda(t-s)},\quad s\leq t,$$

where $|\cdot|$ is the Euclidean norm and $M$ does not depend on $s$.

References

[1] I.A. Lappo-Danilevsky, "Mémoire sur la théorie des systèmes des équations différentielles linéaires" , Chelsea, reprint (1953)
[2] Yu.L. Daletskii, M.G. Krein, "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc. (1974) (Translated from Russian)
[3] F.V. Atkinson, "Discrete and continuous boundary problems" , Acad. Press (1964)
[4] W.T. Reid, "Riccati differential equations" , Acad. Press (1972)
[5] M.Kh. Zakhar-Itkin, "The matrix Riccati differential equation and the semi-group of linear fractional transformations" Russian Math. Surveys , 28 : 3 (1973) pp. 89–131 Uspekhi Mat. Nauk , 28 : 3 (1973) pp. 83–120


Comments

References

[a1] J.K. Hale, "Ordinary differential equations" , Wiley (1969)
How to Cite This Entry:
Matrix differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matrix_differential_equation&oldid=19156
This article was adapted from an original article by E.L. Tonkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article