{"id":1289,"date":"2023-05-08T21:10:02","date_gmt":"2023-05-08T13:10:02","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=1289"},"modified":"2023-07-10T23:02:22","modified_gmt":"2023-07-10T15:02:22","slug":"gronwallbudengshijiqizaichangweifenfangchengjieguanyucanshudeguanghuayilaixingzhongdeyingyong","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=1289","title":{"rendered":"Gr\u00f6nwall\u4e0d\u7b49\u5f0f\u53ca\u5176\u5728\u5e38\u5fae\u5206\u65b9\u7a0b\u89e3\u5173\u4e8e\u53c2\u6570\u7684\u5149\u6ed1\u4f9d\u8d56\u6027\u4e2d\u7684\u5e94\u7528"},"content":{"rendered":"

1. Gr\u00f6nwall\u4e0d\u7b49\u5f0f<\/a><\/span>\n\n

\u5b9a\u7406 1<\/span> (Gr\u00f6nwall \u4e0d\u7b49\u5f0f<\/span>).<\/span> \u5047\u8bbe $u,v:[a,b]\\to\\mathbb{R}$ \u662f\u8fde\u7eed\u51fd\u6570, \u4e14 $u\\geq0$. \u5982\u679c
\n\\[
\nv(t)\\leq C+\\int_a^t v(s)u(s)\\rd s,\\quad t\\in [a,b],
\n\\]
\n\u8fd9\u91cc $C$ \u662f\u4e00\u4e2a\u5e38\u6570, \u90a3\u4e48
\n\\[
\nv(t)\\leq C\\exp\\left(
\n\\int_a^t u(s)\\rd s
\n\\right).
\n\\]
\n<\/div>
\n
\u8bc1\u660e<\/span>.<\/span> \u4ee4 $\\alpha(t)=C+\\int_a^t v(s)u(s)\\rd s$, $\\beta(t)=C\\exp\\left(\\int_a^t u(s)\\rd s\\right)$, \u5219 $\\alpha(a)=C=\\beta(a)$, \u4e14$$\\left(\\frac{\\alpha(t)}{\\beta(t)}\\right)^\\prime=\\frac{1}{\\beta^2(t)}(\\alpha^\\prime\\beta-\\alpha\\beta^\\prime)=\\frac{1}{\\beta^2(t)}(u v \\beta-\\alpha\\beta u )=\\frac{u}{\\beta}(v-\\alpha),$$ \u6ce8\u610f\u5230 $v-\\alpha\\leq 0$ \u4ee5\u53ca $\\alpha(a)\/\\beta(a)=1$, \u8fd9\u6837, \u82e5$C>0$ \u5219 $\\beta>0$, \u4ece\u800c $\\alpha\/\\beta\\leq 1$, \u5373 $\\alpha\\leq\\beta$; \u82e5 $C<0$, \u5219 $\\beta<0$, \u4ece\u800c $\\alpha\/\\beta\\geq 1$, \u6545\u4ecd\u6709 $\\alpha\\leq \\beta$. \u82e5 $C=0$, \u5219\u53ef\u4ee4 $C_n=1\/n$ \u5229\u7528\u524d\u9762\u7ed3\u679c\u5e76\u4ee4 $n\\to\\infty$ \u5f97\u5230 $v(t)\\leq 0$.
\n<\/div><\/p>\n

\u7279\u522b\u5730\uff0c\u82e5$u$\u662f\u4e00\u4e2a\u5927\u4e8e\u96f6\u7684\u5e38\u6570$K$, \u5219\u6211\u4eec\u5f97\u5230\u5982\u4e0b
\n

\u63a8\u8bba 2<\/span>.<\/span> \u5047\u8bbe$X(t)$\u662f$[t_0,T]$\u4e0a\u7684\u4e00\u4e2a\u975e\u8d1f\u8fde\u7eed\u5b9e\u51fd\u6570\uff0c \u82e5\u5b58\u5728\u5e38\u6570$C$\u4ee5\u53ca\u5e38\u6570$K>0$\u4f7f\u5f97
\n \\[
\n X(t)\\leq C+K\\int_{t_0}^tX(s)ds,
\n \\]
\n \u5728$t\\in[t_0,T]$\u4e0a\u6052\u6210\u7acb\uff0c\u5219
\n \\[
\n X(t)\\leq Ce^{K(t-t_0)},
\n \\]
\n \u5bf9\u4efb\u610f\u7684$t\\in[t_0,T]$\u90fd\u6210\u7acb\u3002
\n<\/div>
\n2. \u5e38\u5fae\u5206\u65b9\u7a0b\u4e2d\u89e3\u5173\u4e8e\u53c2\u6570\u7684\u8fde\u7eed\u4f9d\u8d56\u6027<\/a><\/span>\n\n2.1. \u5e38\u5fae\u5206\u65b9\u7a0b\u7684\u521d\u503c\u95ee\u9898<\/a><\/span>\n\n\u8003\u5bdf\u5982\u4e0b\u7684\u5e38\u5fae\u5206\u65b9\u7a0b\u7684\u521d\u503c\u95ee\u9898:
\n\\begin{equation}
\n \\begin{cases}
\n \\begin{aligned}
\n \\dot x&=f(t,x),\\\\
\n x(t_0)&=a,
\n \\end{aligned}
\n \\end{cases}\\label{eq:ODE}
\n\\end{equation}
\n\u4ee5\u53ca\u5e26\u6709\u53c2\u6570\u7684\u5e38\u5fae\u5206\u65b9\u7a0b\u7684\u521d\u503c\u95ee\u9898\uff1a
\n\\begin{equation}
\n \\begin{cases}
\n \\begin{aligned}
\n \\dot x&=f(t,x,\\mu),\\\\
\n x(t_0)&=a,
\n \\end{aligned}
\n \\end{cases}\\label{eq:ODE-para}
\n\\end{equation}
\n\u8fd9\u91cc$\\mu\\in \\mathbb{R}^k$, $a\\in \\mathbb{R}^n$, $f(t,x),f(t,x,\\mu)$\u90fd\u662f\u4ece$[t_0,T]\\to\\mathbb{R}^n$\u7684\u8fde\u7eed\u51fd\u6570\u3002 <\/p>\n

\u9996\u5148\uff0c\u6211\u4eec\u8bf4\u660e\u521d\u503c\u95ee\u9898\u4e2d\u7684\u521d\u503c$a$, \u53ef\u4ee5\u8f6c\u5316\u4e3a\u5e26\u53c2\u6570\u521d\u503c\u95ee\u9898\u4e2d\u7684\u53c2\u6570\uff08\u521d\u503c\u4e3a\u96f6\uff09\uff1a\u4e8b\u5b9e\u4e0a\uff0c\u7ed9\u5b9a\\eqref{eq:ODE}\u7684\u4e00\u4e2a\u89e3$x=x(t)$, \u6784\u9020\u65b0\u7684\u51fd\u6570$\\bar x(t):=x(t)-a$, \u4ee5\u53ca$\\bar f(t,\\bar x, a):=f(t, x)=f(t,\\bar x+a)$, \u5219
\n\\[
\n \\begin{cases}
\n \\dot{\\bar x}=\\dot x=f(t,x)=\\bar f(t,\\bar x, a)\\\\
\n \\bar x(t_0)=x(t_0)-a=0,
\n \\end{cases}
\n\\]
\n\u56e0\u6b64\u5b83\u53ef\u4ee5\u8f6c\u5316\u4e3a\\eqref{eq:ODE-para}\u7684\u4e00\u4e2a\u89e3, \u4e14$a$\u4e3a\u53c2\u6570\u3002<\/p>\n

\u53cd\u8fc7\u6765\uff0c\u82e5\u6709\\eqref{eq:ODE-para}\u7684\u4e00\u4e2a\u89e3$x=x(t)$, \u5219\u6211\u4eec\u53ef\u4ee5\u6784\u9020\u65b0\u7684\u51fd\u6570$\\tilde{x}(t):=(x(t), \\mu)\\in \\mathbb{R}^{n+k}$\u4ee5\u53ca$\\tilde{f}(t,\\tilde{x}):=(f(t,x,\\mu),0)$, \u4f7f\u5f97
\n\\[
\n \\begin{cases}
\n \\dot{\\tilde{x}}=(\\dot x,0)=(f(t,x,\\mu),0)=\\tilde{f}(t,\\tilde{x})\\\\
\n \\dot{\\tilde{x}}(t_0)=(x(t_0),\\mu)=(a,\\mu).
\n \\end{cases}
\n\\]
\n\u53ef\u89c1$\\tilde x$\u662f\u521d\u503c\u95ee\u9898\\eqref{eq:ODE}\u7684\u4e00\u4e2a\u89e3\uff0c\u800c\u4e14\u53c2\u6570$\\mu$\u662f\u521d\u503c\u7684\u4e00\u90e8\u5206\u3002
\n2.2. \u89e3\u7684\u8fde\u7eed\u4f9d\u8d56\u6027<\/a><\/span>\n\n\u6b63\u662f\u7531\u4e8e\u4e0a\u8ff0\u5173\u7cfb\uff0c\u6211\u4eec\u53ea\u9700\u8981\u8ba8\u8bba\u5e38\u5fae\u5206\u65b9\u7a0b\u521d\u503c\u95ee\u9898\\eqref{eq:ODE-para}\u5173\u4e8e\u53c2\u6570$\\mu$\u7684\u8fde\u7eed\u4f9d\u8d56\u6027\u6216\u8005\u5149\u6ed1\u4f9d\u8d56\u6027\u3002<\/p>\n

\u56de\u5fc6\uff0c\u5bf9\u4e00\u4e2a\u5e26\u53c2\u6570\u7684\u51fd\u6570$f(t,t):[a,b]\\times\\Omega\\to \\mathbb{R}^n$, \u79f0$f(\\cdot,t)$\u662f\u5177\u6709Lipschtiz\u5e38\u6570$L$\u7684\u51fd\u6570\uff0c\u5982\u679c\u5bf9\u4efb\u610f\u7684$t\\in[a,b]$\u6052\u6709
\n\\[
\n \\lvert f(x,t)-f(y,t) \\rvert\\leq L \\lvert x-y \\rvert,\\quad \\forall x,y\\in\\Omega.
\n\\]<\/p>\n

\u51fd\u6570$x=x(\\mu)$\u7684\u8fde\u7eed\u6a21<\/span>\u5b9a\u4e49\u4e3a\u8fde\u7eed\u9012\u589e\u51fd\u6570\u51fd\u6570$\\omega:[0,+\\infty]\\to[0,+\\infty]$, \u4f7f\u5f97$\\lim_{\\tau\\to0}\\omega(\\tau)=\\omega(0)=0$, \u800c\u4e14$\\lvert x(\\mu_1)-x(\\mu_2) \\rvert\\leq \\omega( \\lvert \\mu_1-\\mu_2 \\rvert)$. <\/p>\n

\u6ce8\u8bb0 1<\/span>.<\/span> \u660e\u663e\u5730:
\n