Лемма вытекает из оценки
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.
Следствие. Пусть
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,
![](data:image/png;base64,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)
— нормированная при
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фундаментальная матрица уравнения (3). Любая фундаментальная матрица уравнения (3) ограничена, не ограничена или бесконечно мала по норме вместе с
![](data:image/png;base64,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)
.
Теорема 1. 1) Для того чтобы уравнение (3) было устойчивым по Ляпунову, необходимо и достаточно, чтобы его фундаментальные матрицы были ограничены при
![](data:image/png;base64,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)
. 2) Для того чтобы уравнение (3) было асимптотически устойчивым, необходимо и достаточно, чтобы его фундаментальные матрицы были бесконечно малыми при
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.
Доказательство. 1) Достаточность. Пусть
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ограничена на
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. Решение
![](data:image/png;base64,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)
задается формулой
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. (*)
Так как
![](data:image/png;base64,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)
, то
![](data:image/png;base64,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)
. Следовательно, уравнение (3) устойчиво по Ляпунову, так как устойчиво его тривиальное решение. Действительно, если
![](data:image/png;base64,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)
, то при всех
![](data:image/png;base64,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)
. (**)
Необходимость. Пусть уравнение (3) устойчиво по Ляпунову. Тогда устойчиво его тривиальное решение, и выполняется (**). Пусть
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAATCAMAAADPoV+/AAAAAXNSR0ICQMB9xQAAAG9QTFRFAQEBFBQvLxQULhMTPyY/IyNPIiJOJ0BATyMjTiIiXjZNQ1h4RFl5Xn6XX3+YeFhDeVlEvKuYvq2awc3Y2M3Bws7Z2c7C09fg3OPm2+Ll5uPc5eLb/v7+7/Ly8vLv+fn4+Pn59/j4+Pj38fHu////YiXZZgAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACvSURBVCjPvZJRE4IgEIQ3zNI00xQyPBWT//8bE1Ss8cF6kQeGmfvubncH6B8OdoGIaBvKgHwLKvy6CXIHkRtMC9PeuNYiUhNUADjVpiCu5DB5LKfLQOYhGbcVAYcJ09eEfISay3PpH7C4ozWkXykO5SJGena3haQ3Twr5VzoVOz9WmmyLs1exWI3u0sF+Fk/uJMupD5NZkXLqoq4P+JyTZIA/RpB8Lr4DfMdf8B/0Bjj+VhqZiXnOAAAAAElFTkSuQmCC)
фиксировано. Положим
![](data:image/png;base64,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)
. Если
![](data:image/png;base64,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)
, то
![](data:image/png;base64,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)
. Из (*) и (**) имеем
![](data:image/png;base64,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)
, т. е.
![](data:image/png;base64,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)
ограничена. Аналогично доказывается ограниченность
![](data:image/png;base64,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)
, а вместе с ними и матрицы
![](data:image/png;base64,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)
.
2) Достаточность. Пусть
![](data:image/png;base64,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)
при
![](data:image/png;base64,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)
. В силу (*)
![](data:image/png;base64,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)
при всех
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAYCAMAAAArvOYAAAAAAXNSR0ICQMB9xQAAAF1QTFRFAQEBFBQvLhMTRFl5Xn6XX3+YeFhDeVlEenpraWmGmH9fgo6kmay9mq2+pI6CvayZr7fGucK5xrev2c7C2M3Bws7Z2+Ll3OPm5uPc/v7++Pn5+Pj39/j4+fn4////t8IdFAAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAABtSURBVCjPxZBHDoAwDAQNhJ5gSCgxxf9/JogmEq5I7MGH0XpkGdgP/E+IrnESA0FPLUTjYwsFxq7HhtIzT0m3m+gmOhMbQJAHITLK5rNuSq6GnSDEvKSgsGMjnQv1wLZ2yLtjSzo9dwpQH/5nBYRAK8+pV4x0AAAAAElFTkSuQmCC)
, что и дает асимптотическую устойчивость.
Необходимость. Пусть для любых
![](data:image/png;base64,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)
при
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACwAAAAQCAMAAABa4m3lAAAAAXNSR0ICQMB9xQAAAJZQTFRFAAAAAQEBFBQvExMuLhMTPyY/IiJOXTVMTDVdTTZeXUw1RFl5Q1h4Xn6XeFhDeWp5mH9fl35eh2pqkXiFhoaGgo6kmay9mq2+pI6CpY+DvayZvq2ar7fGxrevws7Zwc3Y2c7CzsHBwcHO2M3B2+Ll3OPm5eLb5uPc/v7+8vLv+Pn5+fn4+Pj37/Ly7vHx8fHu5uvp////5uPxVQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC4SURBVCjPvZHrDoIwDIU7vN8QvCAiKipj5abu/V/OAYUFTUyM0fNjSU++tSctyA8E38P4Do78tuuGzTdEegjGqBe2YdEYHjPCqMvYoYYtoEJr16losZbxfDxJ0gWvY9hOM5OUmXAsnL1iAlCdhE9wPOAlGzAtgHLYKVf2NFHMmWCvqNrKzCpY0dnuX6W8EZyazsuqPINTZryw1QzRpcxieF/mT9sY1YaKtpGW3kbAqI3Wlv/q3H+HH0MZeJdcBSF3AAAAAElFTkSuQmCC)
. Положим
![](data:image/png;base64,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)
. В силу (*)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGMAAAAdCAMAAACJzSIWAAAAAXNSR0ICQMB9xQAAALRQTFRFAAAAAQEBExMuFBQvLhMTPyY/IiJOIyNPTyMjTiIiQCdATDVdXTVMTTZeXjZNQ1h4RFl5Xn6XX3+YeVlEeFhDakVqaURpmH9fl35ekHeEmJismay9mKu8pI6CrZmZpY+DvayZvKuYvq2av7i4r7fGucbNwcHOwc3Y2c7Cws7ZzsHB2M3B3+bf2+Ll3OPm5uPc5eLb/v7++fn47/Ly8vLv8fHu7vHx+Pn5+Pj39/j45uvp////TqH5SgAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAIHSURBVEjH7ZWJUsIwEIYTECiiclgErGhUlEaEkjZc5v3fy5xN0FBwxHHGMcO0pdnst/tvsgXs5wf4ZQaRv+JB9psUMsDWNCF+G3A8RnpZzX6awdLBcRjkEAZxTYsYhLgM/Y9A9ZI+FzAWXXGn3eU+BoJwbBlGa2OOmqudjLimUMnp1MewSnBDGoytW7XepJGGAFsGz5GIPNcDEfniSiLmI5Y0lh7GcGpCCrmPWAbrMAyCxU9hO2esw9prHTY3rAdbfEE/EsHMK9wVansYSWVqHmb64jKAFQWV9BxIw9YqjlhfqcfomZjoAQgxTyTz1AOVFCQW8tM63mLkaSQjPhdphnCEIpacYLtSZyMdaAbJxzoEYy+Da50j2C2f7FdVYkAoyhlpGKkoJUNlIwNRjBjaAQDEhqFCUwyuNTAbmKehJyWjmikGdhhq3zkMZ6ShRHyuhyg4DaCMdTiTltyNYND6WDDm0lhHx5MiH7Ry6lGe6d6DTTDAINhLRsW+pNdS1gmvutxXqDy9v1kEEbM1TxpvYo94a56cm44TNzeLADvHTt1ECsjo2tHnQ2qNjY9eJN7IWFHHs3fvZvnjg1kG3CbyuPL2KxTZd/TCHDJ1Gg/uiRoRY7aXYXoIy3U4kKHMJm1yAIN3wyy/foEBdZuEuKC371p/zG/UP6PIAupKsUIb+C3GscZfYbwDPhZPEiMcQnIAAAAASUVORK5CYII=)
, следовательно,
![](data:image/png;base64,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)
. Аналогично доказывается, что
![](data:image/png;base64,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)
,
![](data:image/png;base64,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)
, что означает
![](data:image/png;base64,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)
при
![](data:image/png;base64,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)
. Теорема доказана.
Применим теорему 1 к исследованию устойчивости уравнения (3) с постоянной матрицей коэффициентов P. Уравнение (3) в этом случае имеет фундаментальную матрицу
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACAAAAAVCAMAAAAQExzYAAAAAXNSR0ICQMB9xQAAAJ9QTFRFAAAAAQEBFBQvExMuLhMTLxQUIiJOIyNPTyMjXTVMXjZNQ1h4X3+YXn6XeFhDeVlEaURpaWmGl35emH9fhmlphHeQkHeEgo6kmpqumay9mKu8mq2+pI6CvayZt7e+rrbFwLm52c7Cws7Zz9XKwcHOwc3Y2M3B3OPm2+Ll5uPc5eLb/v7++Pj38vLv8fHu+Pn5+fn47/Ly9/j47vHx////eRnxSwAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADOSURBVCjPrZLZEoIwDEWJghub4r6gti6ApYDQ//8228qiDIMzah46eTi5yb1ThX0o5XuAkHYAKR2vXcHB4b4CSC5YFR2fVMAFEFoAODq/AsEk3nmlgmMwNhV8eSFz13QeFwAd+owli3s1voxGXnAgBZDYJu+PcX4NYYHG5VwwyhUIzPzI0AbYNAQVaHoqtWY+uam4IUlOxDIe4LWtAUhMuF1hCkmsvsK98OfaEyZQL5Mm3wGnn5HQMp5+9JSg2gq6ogOQlkQkVfvzf/gb8AD6vHpAGxbhqgAAAABJRU5ErkJggg==)
,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAD0AAAATCAMAAAAUGvWdAAAAAXNSR0ICQMB9xQAAAJNQTFRFAAAAAQEBExMuLhMTLxQUPyY/IyNPIiJOTyMjTiIiXTVMQ1h4RFl5X3+YXn6XeFhDeVlEakVqaWmGl35emH9fhmlpkHeEmpqumKu8mq2+vKuYvayZvq2awLm5ws7ZwcHOwc3Y2c7C2M3B2+Ll3OPm5eLb5uPc/v7++Pj38fHu7/Ly8vLv+fn4+Pn59/j47vHx////jy8WHgAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAE5SURBVDjLzZPbVoMwEEUzrQKVpqC0WkEjKdgA4ZL//zpzL0rf+mBZC0jI7MmZMwGJWy50DzS7ssgYuwr5747+XpNFUJMAkPZrSecA5ZymgMhCRYaF2MMy6ynoeFTO9z77vfvEjvhTJWevnfiTVgfQeHC0rEMrZ6qgdlOasD7dycHn4As51Gajx9o+NM0yACSVNynAkUcIgs6Ws2PeJsbyozGLhnKZb4ils+0g9oj0h0pJuNQgzg/xqAcFyPzyxkvaKFkTqtbhfUYr3AvP3GdNmyhJF3ZmjPD+UfUqVrUz5mW0bf5dN11Vhg4nFejoQvXkFHZWuRaGjZtyJd9az3kUTrLgsk/jkVGp3HieBRNrEux6jC/9pvHYRsT1m0cqLxD1xjK19py/mant/XM3Oy4fYM/QXfxj/0D/AGeFu/r/aGlIAAAAAElFTkSuQmCC)
, где
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAATCAMAAAB4HKeYAAAAAXNSR0ICQMB9xQAAAFdQTFRFAAAAAQEBExMuLhMTLxQUPyY/IiJOTyMjRFl5Q1h4Xn6XX3+YeFhDl35emay9mq2+vayZ2c7C2M3B3OPm5uPc5eLb/v7++Pn57/Ly+fn49/j48vLv////ZCYkfwAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAABVSURBVCjPY5BBBQzE8sWAAIkvyMjAyCGOJC/CzIeiXpBNAoXPy4ViniQPHwpfhFUYhS/AKQ7niyGUg/ggpUJMwnC+AKeUKAs3wr3S7IyM/CT7DwcfAPvXHJcJiY4EAAAAAElFTkSuQmCC)
— жорданова форма матрицы P. По теореме 1, лемме 1 и следствию к ней устойчивость по Ляпунову, асимптотическая устойчивость и неустойчивость уравнения (3) эквивалентны соответственно ограниченности, бесконечной малости и неограниченности матрицы
![](data:image/png;base64,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)
при
![](data:image/png;base64,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)
. Отсюда получаем следующую теорему:
Теорема 2. Линейная однородная система с постоянным коэффициентами: 1) устойчива по Ляпунову тогда и только тогда, когда среди собственных чисел матрицы коэффициентов нет таких, вещественные части которых положительны, а число мнимые и нулевые собственные числа либо простые, либо имеют только простые элементарные делители; 2) асимптотически устойчива тогда и только тогда, когда все собственные числа матрицы коэффициентов имеют отрицательные вещественные части.
Ниже рассматриваются необходимые и достаточные условия отрицательности корней характеристического уравнения линейной однородной системы с постоянными коэффициентами — критерий Гурвица (Рауса-Гурвица), а также частотный критерий Михайлова, являющийся геометрическим признаком, эквивалентным критерию Гурвица.
Определение. Полином
![](data:image/png;base64,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)
, где
![](data:image/png;base64,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)
,
![](data:image/png;base64,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)
,
![](data:image/png;base64,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)
называется полиномом Гурвица, если все его корни имеют отрицательные вещественные части.
Если полином
![](data:image/png;base64,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)
является полиномом Гурвица, то все
![](data:image/png;base64,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)
.
Составим
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACEAAAAPCAMAAABdnFYbAAAAAXNSR0ICQMB9xQAAAHtQTFRFAAAAAQEBLxQUTiIiXjZNXTVMUFBQRFl5Xn6XeVlEakVqaURpamqHaWmGl35ehmlphHeQmay9mq2+rZmZvayZvq2auL+4r7fGws7Zz9XK2c7C3+bf3OPm5eLb5uPc/v7+8vLv7vHx+Pn5+Pj37/Ly+fn48fHu9/j4////L1sq7wAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACZSURBVCjPrZDZDsIgEEXRutS6VFuXqhWwIp3//0LnMknBJ6ORhEk4HGZuUPRpqS8Nk9TUMAabiy3ufLytuhSyoceqeSxGOxJFhAjRwy6vk3YPg5VzEBII45LX7lm24cJvexk/QBibWUN6HZ567iHKANmwWUUkQ3zhSAclQiRl006PJ6RwCIkkEbJxqBB93r39QoTq77/+m/EChWxDklCr6KIAAAAASUVORK5CYII=)
-матрицу Гурвица вида
![](data:image/png;base64,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)
Теорема Гурвица (критерий Гурвица). Для того чтобы полином
![](data:image/png;base64,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)
являлся полиномом Гурвица, необходимо и достаточно, чтобы были положительны все главные диагональные миноры его матрицы Гурвица
![](data:image/png;base64,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)
:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHMAAAB7CAMAAABwzk+5AAAAAXNSR0ICQMB9xQAAAPBQTFRFAAAAAQEBAgICExMuFBQvLhMTLxQUPyY/IyNPIiJOJ0BATyMjTiIiTDVdTTZeQ1h4RFl5Xn6XX3+YXX2WeVlEeFhDaURpmH9fl35ehmlpkHeEj3aDhXiRm5CQmJisgo6kmZmtmKu8may9mq2+kaaxrZmZpI6CrJiYrpqavKuYvayZvq2apKSarr6+v7i4r6+vr7fGucbNxrevwMDNwc3Y2M3Bws7Z2c7C39bSz8LC0tbf09fg39/m3+bf3uXe3OPm2+Ll5uPc5eLb/v7+8fHu+Pn57/Ly8vLv+fn4+Pj39/j47vHx/f399vf38/Hx////a8ewGgAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAPnSURBVGje7ZoLV9MwFMeXITityjaGQ3ygwxegiAoNPnB22bqBSL//tzFt0zZpb9MmS3eOnNxzxmG9SX/LTZrc/pNWsHprWWbjTKJ9B0KIHnP6VZt5gNCpFvNwc6aJPOv4c+dUg7kYtFyVcGb/LrZpRaz2i2Mm/rzdq+gzrtdwL/vi3R2zP4rMxa5/tDYuQFIQOUR3PnBtwSil4nt+EMwfuMpM74RWe8VfHqLY+uGXy+0+9Y94P0bdK7IU8w2tNwwrw7bTZ/3Gm7ceVYiY3royc3IS3eM0DCkQ2+i2k4e+8BySycb9H0v059txNHQ3Z4tB0lg+tju9MLy9zBkRuzM24ulPPeiqjtvpi6hFZ2u/w8EEhHaLTJ3Wx5+Zk/ZmCsGbV1PH5UJUi3mE0kaBTIxQ58Lp+KkT93k3rU6R3kbHV51v2RO+K6kodQbkfLxq5tzpB3rMy6d/yotKnepzXzYVSEag1KnPXI1ZpmXeWqZiMklMMFtm22GZlrlCppjM5x4P3meOmU/mBaTgM8YEkvnUcj5jTDCZZ5bzmWKCyTyzxJe8cptiAsl81szYR9+c0cgos5DMc0zm8/bi15C6TFIV23wyz8eW+Q7dAI9gpvCU1dIwuJvIEusv4/BNFmQKQkctDaMms7ydotBRS8PgbiJL5r09wvVnudBRS8PgmBWZ/n5aHPevy4SOEg1DFDG05njc7l3DQkeJhiGKGJrrCm7DQkcdDUN/LYOFDlHDyJ6j5WMblAodgoZB31zRyFhsS4UOUcP45s8f+4ZiS3uTEzr+ckKHqGEkjTbAzAkdbU7oKNzueGZwDBWm3XNoDcBu0ByTCR2in5xtkQaZoD/sWnfFTNNzwrJMWXrbUH4rS28bym9l6W1T+a0svW0ov82nsOIznTiZ1xAzSWG9DVSMb5rfsiXCFJOlsDez+bPCOEqcyRJhKrYshaUL1PdibBNnEC8RpueEIZJuOEVLhPF5SK6QR0uEYSYdmMfluSZbIgwzh+AakBhbIowwUaBkbRNMw2aZlmmZlmmZlmmZ/w0zzPfzn4aZi0F3lv/Y/rRMy7RMy7xNTNW1hCzPVD3bvMRZaO2zzfpnoQPNs83K5SFm1dnmwunmyvKVTGBfUHq6GSqvygT2BaWnm6HyqsyqfcG8KFxZvprJ7QvCsc1E4chKyisxs33BktimonBsUHnwsEQ5k9sXpNML9HtTUThyw+Xlemeeye0Leo9+OcDeQiIKx264vBqTs5tPzy9elhcuuKMLca9rM+dPXLxfXrjgDi+8HtAm0/Rfl0nv+E5ygLng5i4onHwWme/9uSN5Wym4uQsKJ59tbnK7mP8AG+dFx2IgqigAAAAASUVORK5CYII=)
Если степень полинома
![](data:image/png;base64,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)
сравнительно большая, то применение критерия Гурвица становится затруднительным. В этом случае для определения расположения корней полинома
![](data:image/png;base64,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)
на комплексной плоскости иногда оказывается более удобным использование частотного критерия Михайлова.
Определение. Пусть
![](data:image/png;base64,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)
, где
![](data:image/png;base64,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)
,
![](data:image/png;base64,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)
,
![](data:image/png;base64,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)
. Кривая
![](data:image/png;base64,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)
,
![](data:image/png;base64,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)
называется годографом Михайлова функции
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACMAAAAVCAMAAAD7JKfbAAAAAXNSR0ICQMB9xQAAAIdQTFRFAAAAAQEBFBQvPyY/IyNPIiJOTyMjTiIiXTVMQCdATDVdXjZNTTZeRFl5Q1h4Xn6XX3+YeVlEeFhDakVqaURpl35emH9fmay9vayZvq2a2c7C2M3Bws7Zwc3Y3OPm2+Ll5d7e5uPc5eLb/v7+8fHu7/Ly8vLv7vHx+Pn5+fn47Orn5+rs////ajHj7gAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADWSURBVCjPtZJbE4IgEIWhtOxumWlqmq14Kf7/70vYEMKHpmbiBc/O5+6BA+GfF/mSAQC5DXrMVDMaiS1gqOugsZnKy5NNy1OHqUq5KCwmjCR5ZLpzuWxMBlK30SQAeot3BtPtCRFDai+XilIyyUUjZs4K5S+JK4r3B4cwE7bnmcGg4rGLdhDpq5HBYFfFvBCLibetnqWQ91ndIcKi8AzhBeDKuOW5nOb4cerZM+2XL7v76uz9VaAQjVaFkQ7GQuRN3dajAGSA2ZBFQh0jgFpnyn57P/9mnux7cRVCu2T6AAAAAElFTkSuQmCC)
.
Критерий Михайлова непосредственно следует из леммы:
Лемма 2. Угол поворота в положительном направлении ненулевого вектора
![](data:image/png;base64,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)
при
![](data:image/png;base64,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)
равен
![](data:image/png;base64,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)
, где
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAPCAMAAAA1b9QjAAAAAXNSR0ICQMB9xQAAAHJQTFRFAAAAAQEBFBQvLxQUIiJOIyNPNUxdTyMjTiIiXTVMRFl5RGlpXn6XeFhDaURpamqHaWmGl35emH9fhXiRkJCbmq2+vq2auL+4xrevws7Zz9XK2c7C3OPm5eLb5uPc/v7+8vLv7/Ly6uzn+Pn5+fn4////d3uEDwAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAABoSURBVCjPlY5JDoAgDABx3xdQVMQFxf7/i4JiTLyoTeYy6TRF8Bz00czzzWFEgNjouIvG7IioLda8V1yG+7XkSSV5LI3B9gDYWwAzc2fLGIhQcUTaTKUEHfC0G07TUACiAmLRPz+/mR35tCB/WfPvlAAAAABJRU5ErkJggg==)
— число корней полинома
![](data:image/png;base64,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)
с положительной вещественной частью с учетом их кратностей.
Критерий Михайлова. Для того чтобы полином
![](data:image/png;base64,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)
, не имеющий чисто мнимых корней, являлся полиномом Гурвица, необходимо и достаточно, чтобы угол поворота в положительном направлении вектора
![](data:image/png;base64,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)
при
![](data:image/png;base64,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)
был бы равен
![](data:image/png;base64,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)
.