From fardigola@univer.kharkov.ua Mon Jul 16 15:47:05 2001 Date: Tue, 22 May 2001 09:54:30 +0300 From: Larissa V. FARDIGOLA <fardigola@univer.kharkov.ua> To: congress@imath.kiev.ua Subject: ABSTRACT (13) \\documentclass[12pt]{article} \\usepackage[koi8-u]{inputenc} \\usepackage[english,ukrainian]{babel} \\begin{document} %%%%%%%%%%%%%%%%%%%%%%%%% FARDIGOLA %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% BEGIN %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\title{ On stabilizability of evolution systems of partial differential equations by time-delayed feedback control} \\author{Larissa~V.~Fardigola (Kharkov National University,\\\\ Institute for Low Temperature Physics of\\\\ the National Academy of Sciences of Ukraine)} \\date{} \\maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In the present paper we consider the following system $$ {\\partial w(x,t)}/{\\partial t}=A\\left( D_x \\right) w(x,t) +B\\left( D_x \\right) u(x,t),\\qquad x\\in{\\mathbf R}^n,\\ t> h, \\eqno{(1)} $$ where $D_x=\\left( -i\\partial/{\\partial x_1},\\ldots, -i\\partial/{\\partial x_n} \\right)$, $A(\\sigma)$ and $B(\\sigma)$ are polynomial matrices $(m\\times m)$, $\\det B(\\sigma)\\not\\equiv 0$ on ${\\mathbf R}^n$, $u$ is a control, $h>0$. We use the following Sobolev spaces ${{\\mathcal M}}=\\bigcap\\limits_{q\\in{\\mathbf N}_0} \\bigcup\\limits_{\\gamma\\in{\\mathbf R}}C_{\\gamma}^q$, $C_\\gamma^{-\\infty}=\\bigcup\\limits_{q\\in{\\mathbf N}_0}C_\\gamma^q$, $C_\\gamma^q=\\left\\{ g\\in C^q({\\mathbf R}^n )\\mid \\sup\\left\\{ | D^\\alpha_x g(x) | (1+|x|)^{-\\gamma}\\mid x\\in{\\mathbf R}^n\\wedge |\\alpha|\\leq q \\right\\}< +\\infty \\right\\}$. Further we assume that $\\gamma\\ge 0$. {\\bfseries Definition.} The system (1) is said to be stabilizable in $C_\\gamma^{-\\infty}$ if there exists such a matrix $(m\\times m)$ $q\\in{\\mathbf N}_0$ such that for every solution of this system with $u(x,t)\\equiv P\\left( D_x \\right) w(x,t-h)$ under the initial condition: $D_x^\\alpha w\\in C\\left( {\\mathbf R}^n\\times [0,h] \\right)$ $(|\\alpha|\\le q)$, the following assertions are true: $D^\\alpha_x w\\in C\\left({\\mathbf R}^n\\times[0,+\\infty) \\right)$ $(|\\alpha|\\le r)$ and $\\|w(\\cdot,t)\\|^r_\\gamma\\longrightarrow 0$ as $t\\longrightarrow +\\infty$. {\\bfseries Theorem.} {\\itshape If the system (1) satisfies the following two conditions $$ \\forall\\sigma\\in{\\mathbf R}^n\\qquad \\left[ \\det B(\\sigma)=0 \\Longrightarrow \\sup\\left\\{\\Re\\lambda\\mid \\det\\left\\{ \\lambda I-A(\\sigma)\\right\\}=0 \\right\\}<0 \\right], \\eqno{(2)} $$ $$ \\forall\\sigma\\in{\\mathbf R}^n\\qquad \\sup\\left\\{\\Re\\lambda\\mid \\det\\left\\{ \\lambda I-A(\\sigma)\\right\\}=0 \\right\\}<\\frac1 h \\eqno{(3)} $$ then this system is stabilizable in $C_\\gamma^{-\\infty}$. Moreover, (2) is necessary for stabilizability of (1) in $C_\\gamma^{-\\infty}$, and in the case $m=1$ (3) is also necessary for stabilizabilty of this system in $C_\\gamma^{-\\infty}$.} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FARDIGOLA %%%%%%%%%%%%%%%%%%%%%%%%%% \\end{document}

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