{"id":54518,"date":"2024-12-19T19:46:31","date_gmt":"2024-12-19T11:46:31","guid":{"rendered":"https:\/\/swarma.org\/?p=54518"},"modified":"2024-12-19T19:46:31","modified_gmt":"2024-12-19T11:46:31","slug":"koopman-%e7%ae%97%e7%ac%a6%e7%90%86%e8%ae%ba%e7%ae%80%e4%bb%8b%ef%bd%9c%e5%85%a5%e9%97%a8%e8%b7%af%e5%be%84","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=54518","title":{"rendered":"Koopman \u7b97\u7b26\u7406\u8bba\u7b80\u4ecb\uff5c\u5165\u95e8\u8def\u5f84"},"content":{"rendered":"
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\u672c\u6587\u9009\u6458\u7f16\u8bd1\u81ea2019\u5e74\u7684\u7efc\u8ff0\u6587\u7ae0 Notes on Koopman Operator Theory\uff0c\u65e8\u5728\u7b80\u8981\u4ecb\u7ecd Koopman \u7b97\u7b26\u7406\u8bba\uff0c\u5e2e\u52a9\u611f\u5174\u8da3\u7684\u8bfb\u8005\u5feb\u901f\u5165\u95e8\u3002\u5e0c\u671b\u6df1\u5165\u4e86\u89e3\u7684\u670b\u53cb\u53ef\u4ee5\u9605\u8bfb\u539f\u6587\u3002<\/span><\/strong><\/p>\n


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\u597d\u7684\u7406\u8bba\u548c\u5de5\u5177\u4f1a\u5927\u5927\u52a0\u5feb\u79d1\u7814\u8fdb\u5c55\uff0cKoopman \u7b97\u7b26\u6709\u671b\u6210\u4e3a\u5206\u6790\u548c\u9884\u6d4b\u52a8\u6001\u7cfb\u7edf\u7684\u6807\u51c6\u7406\u8bba\u3002\u96c6\u667a\u5b66\u56ed\u9080\u8bf7\u5317\u4eac\u90ae\u7535\u5927\u5b66\u5170\u5cb3\u6052\u6559\u6388\u5f00\u8bbe\u300cKoopman\u5206\u6790\u5728\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u4e2d\u7684\u5e94\u7528\u300d\u7cfb\u5217\u8bfe\u7a0b<\/a>\uff0c\u9762\u5411\u81f4\u529b\u4e8e\u5b66\u4e60\u548c\u53d1\u5c55 Koopman \u7b97\u7b26\u7406\u8bba\u7684\u793e\u533a\u670b\u53cb\uff0c\u57286\u8282\u8bfe\u7a0b\u4e2d\uff0c\u901a\u8fc7\u7ec6\u81f4\u7684\u8bb2\u89e3\uff0c\u6df1\u5165\u7684\u4e92\u52a8\uff0c\u5f62\u6210\u82cf\u683c\u62c9\u5e95\u5f0f\u8bfe\u5802\u6c1b\u56f4\uff0c\u5171\u540c\u8fdb\u6b65\u3002\u6b22\u8fce\u611f\u5174\u8da3\u7684\u670b\u53cb\u52a0\u5165\u8bfe\u7a0b\uff01<\/span><\/strong><\/p>\n<\/section>\n

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Notes on Koopman Operator Theory<\/span><\/p>\n

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https:\/\/fluids.ac.uk\/files\/meetings\/KoopmanNotes.1575558616.pdf<\/span><\/p>\n<\/blockquote>\n


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1931 \u5e74\uff0cBernard O. Koopman \u8bc1\u660e\uff0c\u4f5c\u7528\u4e8e\u7cfb\u7edf\u72b6\u6001\u7684\u6d4b\u91cf\u51fd\u6570\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u7684\u65e0\u9650\u7ef4\u7ebf\u6027\u7b97\u7b26<\/strong>\u53ef\u4ee5\u7528\u6765\u8868\u793a\u975e\u7ebf\u6027\u52a8\u529b\u7cfb\u7edf<\/strong>\u3002\u8fd9\u79cd\u6240\u8c13\u7684<\/strong> <\/strong>Koopman<\/strong> <\/strong>\u7b97\u7b26\u662f\u5b8c\u5168\u7ebf\u6027\u7684\uff0c\u4f46\u662f\u5176\u8c31\u5206\u89e3\u5b8c\u5168\u8868\u5f81\u4e86\u975e\u7ebf\u6027\u7cfb\u7edf\u7684\u884c\u4e3a<\/strong>\u3002\u5728\u7ebf\u6027\u6846\u67b6\u4e2d\u8868\u8fbe\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u7684\u601d\u60f3\u5f88\u6709\u5438\u5f15\u529b\uff0c\u56e0\u4e3a\u7ebf\u6027\u7cfb\u7edf\u6709\u7740\u4e30\u5bcc\u7684\u6700\u4f18\u4f30\u8ba1\u548c\u63a7\u5236\u6280\u672f\uff0c\u5e76\u4e14\u80fd\u591f\u5206\u6790\u9884\u6d4b\u7cfb\u7edf\u7684\u672a\u6765\u72b6\u6001\u3002\u7136\u800c\uff0c\u8fd9\u79cd\u7ebf\u6027\u7b97\u7b26\u4e5f\u662f\u65e0\u9650\u7ef4\u7684<\/strong>\uff0c\u56e0\u4e3a\u63cf\u8ff0\u72b6\u6001\u7684\u6240\u6709\u53ef\u80fd\u7684\u6d4b\u91cf\u51fd\u6570 g \u7684\u7a7a\u95f4\u9700\u8981\u65e0\u9650\u591a\u4e2a\u81ea\u7531\u5ea6\u3002\u5728 Koopman \u7684\u5e74\u4ee3\u8fdb\u884c\u8fd9\u4e2a\u91cf\u7ea7\u7684\u8ba1\u7b97\u5341\u5206\u56f0\u96be\uff0c\u800c\u4e14\u5f53\u65f6\u4e5f\u6ca1\u6709\u8feb\u5207\u7684\u9700\u6c42\uff0c\u6240\u4ee5 Koopman \u7b97\u7b26\u7406\u8bba\u8bde\u751f\u540e\u5e76\u672a\u5f97\u5230\u5b9e\u9645\u5e94\u7528\u3002\u4f46\u5728\u7406\u8bba\u65b9\u9762\uff0c\u51af\u8bfa\u4f9d\u66fc\u53d7\u5230\u542f\u53d1\uff0c\u5c06 Koopman \u7406\u8bba\u5e94\u7528\u4e8e\u904d\u5386\u95ee\u9898\uff0c\u5e76\u8bc1\u660e\u4e86\u5e73\u5747\u904d\u5386\u5b9a\u7406\u3002<\/span><\/p>\n


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\u52a8\u529b\u5b66\u5b8c\u5168\u7531\u77e9\u9635 A <\/strong>\u7684\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf\u8868\u5f81\uff0c\u7531 A <\/strong>\u7684\u8c31\u5206\u89e3<\/span>\uff08\u7279\u5f81\u5206\u89e3\uff09<\/span>\u7ed9\u51fa\uff1a<\/span><\/p>\n


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\u5f53 A <\/strong>\u6709 n \u4e2a\u4e0d\u540c\u7684\u7279\u5f81\u503c\u65f6\uff0c\u039b <\/strong>\u662f\u5305\u542b\u7279\u5f81\u503c<\/span> \u03bbj<\/sub> <\/span>\u7684\u5bf9\u89d2\u77e9\u9635\uff0cT <\/strong>\u662f\u4e00\u4e2a\u77e9\u9635\uff0c\u5176\u5217\u662f\u4e0e\u7279\u5f81\u503c \u03bb<\/span>j <\/sub>\u5173\u8054\u7684\u7ebf\u6027\u72ec\u7acb\u7279\u5f81\u5411\u91cf<\/span> \u03be<\/span>j<\/span><\/sub> \u3002\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u53ef\u4ee5\u5199\u6210 A <\/strong>= T\u039bT<\/strong>\u22121<\/sup>\uff0c\u5f0f(6)\u4e2d\u7684\u89e3\u53d8\u4e3a<\/span><\/p>\n


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\u66f4\u4e00\u822c\u5730\uff0c\u5728\u91cd\u590d\u7279\u5f81\u503c\u7684\u60c5\u51b5\u4e0b\uff0c\u77e9\u9635 \u039b<\/strong> <\/strong>\u5c06\u7531 Jordan \u5757\u7ec4\u6210 \u3002\u8bf7\u6ce8\u610f\uff0c\u8fde\u7eed\u65f6\u95f4\u7cfb\u7edf\u4f1a\u4ea7\u751f\u79bb\u6563\u65f6\u95f4\u52a8\u529b\u7cfb\u7edf\uff0c\u5176\u4e2d Ft <\/sub>\u7531 (6) \u4e2d\u7684\u6620\u5c04 eAt <\/sup>\u7ed9\u51fa\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u79bb\u6563\u65f6\u95f4\u7279\u5f81\u503c\u7531 e\u03bb<\/span>t <\/sup>\u7ed9\u51fa\u3002\u77e9\u9635 T-1 <\/sup>\u5b9a\u4e49\u4e86\u4e00\u4e2a\u53d8\u6362\uff0cz<\/strong> = T<\/strong>-1<\/sup>x<\/strong>\uff0c\u5230\u5185\u5728\u7279\u5f81\u5411\u91cf\u5750\u6807 z <\/strong>\uff0c\u5176\u4e2d\u52a8\u529b\u5b66\u53d8\u5f97\u89e3\u8026\uff1a<\/span><\/p>\n


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\u6362\u53e5\u8bdd\u8bf4\uff0c\u6bcf\u4e2a\u5750\u6807 z<\/strong>j <\/sub>\u4ec5\u53d6\u51b3\u4e8e\u5176\u81ea\u8eab\uff0c\u5177\u6709\u7b80\u5355\u7684\u52a8\u529b\u5b66<\/span><\/p>\n


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\u56e0\u6b64\uff0c\u975e\u5e38\u9700\u8981\u4f7f\u7528\u7ebf\u6027\u7cfb\u7edf\uff0c\u56e0\u4e3a\u53ef\u4ee5\u5c06\u7cfb\u7edf\u8f6c\u6362\u4e3a\u7279\u5f81\u5411\u91cf\u5750\u6807\uff0c\u5176\u4e2d\u52a8\u529b\u5b66\u53d8\u5f97\u89e3\u8026\u3002\u5bf9\u4e8e\u975e\u7ebf\u6027\u7cfb\u7edf\u6765\u8bf4\uff0c\u901a\u5e38\u4e0d\u5b58\u5728\u8fd9\u6837\u7684\u5c01\u95ed\u5f0f\u89e3\u6216\u7b80\u5355\u7684\u5750\u6807\u7ebf\u6027\u53d8\u5316\u3002<\/span><\/p>\n


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3. Koopman \u7406\u8bba\u7684\u6570\u5b66\u8868\u8ff0<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


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\u6211\u4eec\u8003\u8651\u5b9e\u503c\u6d4b\u91cf\u51fd\u6570 g : M \u2192 R\uff0c\u5b83\u4eec\u662f\u65e0\u9650\u7ef4\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u7684\u5143\u7d20\u3002\u51fd\u6570 g \u901a\u5e38\u4e5f\u79f0\u4e3a\u53ef\u89c2\u6d4b\u91cf\u3002\u901a\u5e38\uff0c\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u7531 M \u4e0a\u7684\u52d2\u8d1d\u683c\u5e73\u65b9\u53ef\u79ef\u51fd\u6570\u7ed9\u51fa\uff1b\u6d4b\u5ea6\u7a7a\u95f4\u7684\u5176\u4ed6\u9009\u62e9\u4e5f\u662f\u6709\u6548\u7684\u3002Koopman \u7b97\u7b26  <\/sub>\u662f\u65e0\u9650\u7ef4\u7ebf\u6027\u7b97\u7b26\uff0c\u4f5c\u7528\u4e8e\u6d4b\u91cf\u51fd\u6570 g \uff0c \u5982\u4e0b\uff1a<\/span><\/p>\n


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\u5176\u4e2d \u25e6 \u662f\u590d\u5408\u8fd0\u7b97\u7b26\u3002\u5bf9\u4e8e\u65f6\u95f4\u6b65\u957f\u4e3a \u0394t \u7684\u79bb\u6563\u65f6\u95f4\u7cfb\u7edf\uff0c\u5219\u53d8\u4e3a\uff1a<\/span><\/p>\n


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\u6362\u53e5\u8bdd\u8bf4\uff0cKoopman \u7b97\u7b26\u5b9a\u4e49\u4e86\u4e00\u4e2a\u65e0\u9650\u7ef4\u7ebf\u6027\u52a8\u529b\u7cfb\u7edf\uff0c\u5b83\u5c06\u72b6\u6001 gk<\/sub>=g(x<\/strong>k<\/sub>) \u7684\u89c2\u6d4b\u63a8\u8fdb\u5230\u4e0b\u4e00\u4e2a\u65f6\u95f4\u6b65\uff1a<\/span><\/p>\n


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\u8fd9\u5bf9\u4e8e\u4efb\u4f55\u53ef\u89c2\u5bdf\u51fd\u6570 gk<\/sub> \u548c\u4efb\u4f55\u72b6\u6001 x<\/strong>k <\/sub>\u90fd\u662f\u6210\u7acb\u7684\u3002Koopman \u7b97\u7b26\u662f\u7ebf\u6027\u7684\uff0c\u8be5\u5c5e\u6027\u7ee7\u627f\u81ea\u51fd\u6570\u7a7a\u95f4\u4e2d\u52a0\u6cd5\u8fd0\u7b97\u7684\u7ebf\u6027\uff1a<\/span><\/p>\n


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\u5bf9\u4e8e\u8db3\u591f\u5e73\u6ed1\u7684\u52a8\u529b\u7cfb\u7edf\uff0c\u8fd8\u53ef\u4ee5\u5b9a\u4e49 Koopman \u52a8\u529b\u7cfb\u7edf\u7684\u8fde\u7eed\u65f6\u95f4\u6a21\u62df\uff1a<\/span><\/p>\n


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\u7b97\u7b26 <\/span> <\/sub>\u662f\u5355\u53c2\u6570\u53d8\u6362\u65cf <\/span> <\/sub>\u7684\u65e0\u7a77\u5c0f\u751f\u6210\u5143\u3002\u5b83\u7531\u5176\u5bf9\u53ef\u89c2\u5bdf\u51fd\u6570 g \u7684\u4f5c\u7528\u6765\u5b9a\u4e49\uff1a<\/span><\/p>\n


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\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u539f\u59cb\u72b6\u6001 x<\/strong>k <\/sub>\u53ef\u80fd\u672c\u8eab\u662f\u53ef\u89c2\u6d4b\u91cf\uff0c\u65e0\u9650\u7ef4\u7b97\u7b26 <\/span> <\/sub>\u4f1a\u63a8\u8fdb\u8fd9\u4e2a\u51fd\u6570\u3002\u7136\u800c\uff0c\u4e00\u65e6\u901a\u8fc7\u52a8\u529b\u5b66\u8fed\u4ee3\uff0c\u5728\u9009\u5b9a\u7684\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u57fa\u7840\u4e0a\u53ef\u89c2\u6d4b\u91cf\u7684\u7b80\u5355\u8868\u793a g=x \u53ef\u80fd\u4f1a\u53d8\u5f97\u4efb\u610f\u590d\u6742\u3002<\/span><\/p>\n


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3.1 Koopman \u672c\u5f81\u51fd\u6570\u548c\u5185\u5728\u5750\u6807<\/span><\/strong><\/span><\/p>\n


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\u5bf9\u5e94\u4e8e\u7279\u5f81\u503c \u03bb \u7684\u79bb\u6563\u65f6\u95f4 Koopman \u7279\u5f81\u51fd\u6570 <\/span>\u03c6<\/span>(<\/span>x) \u6ee1\u8db3<\/span><\/p>\n


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\u5728\u8fde\u7eed\u65f6\u95f4\u4e2d\uff0cKoopman \u672c\u5f81\u51fd\u6570 \u03c6<\/span><\/span> <\/span><\/span>(<\/span>x) <\/span>\u6ee1\u8db3<\/span><\/p>\n


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\u4ece\u6570\u636e\u6216\u5206\u6790\u8868\u8fbe\u5f0f\u4e2d\u83b7\u53d6 Koopman \u672c\u5f81\u51fd\u6570\u662f\u73b0\u4ee3\u52a8\u529b\u7cfb\u7edf\u4e2d\u7684\u4e00\u4e2a\u6838\u5fc3\u5e94\u7528\u6311\u6218\u3002\u53d1\u73b0\u8fd9\u4e9b\u7279\u5f81\u51fd\u6570\u53ef\u4ee5\u5b9e\u73b0\u5f3a\u975e\u7ebf\u6027\u7cfb\u7edf\u7684\u5168\u5c40\u7ebf\u6027\u8868\u793a\u3002\u5c06\u94fe\u5f0f\u6cd5\u5219\u5e94\u7528\u4e8e Koopman \u672c\u5f81\u51fd\u6570 \u03c6<\/span><\/span><\/span>(x) \u7684\u65f6\u95f4\u5bfc\u6570\uff0c\u5f97\u51fa<\/span><\/p>\n


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\u7ed3\u5408 (18)\uff0c\u5f97\u51fa\u672c\u5f81\u51fd\u6570 <\/span>\u03a6<\/span>(x) \u7684\u504f\u5fae\u5206\u65b9\u7a0b (PDE)\uff1a<\/span><\/p>\n


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\u5229\u7528\u8fd9\u79cd\u975e\u7ebf\u6027\u504f\u5fae\u5206\u65b9\u7a0b\uff0c\u53ef\u4ee5\u901a\u8fc7\u6c42\u89e3\u6d1b\u6717\u7ea7\u6570\u6216\u901a\u8fc7\u4f7f\u7528\u6570\u636e\u56de\u5f52\u6765\u8fd1\u4f3c\u7279\u5f81\u51fd\u6570\uff0c\u8fd9\u4e24\u79cd\u65b9\u6cd5\u90fd\u5c06\u5728\u4e0b\u9762\u8fdb\u884c\u63a2\u8ba8\u3002\u4e0a\u9762\u7684\u516c\u5f0f\u5047\u8bbe\u52a8\u529b\u5b66\u65e2\u8fde\u7eed\u53c8\u53ef\u5fae\u3002(3)\u4e2d\u7684\u79bb\u6563\u65f6\u95f4\u52a8\u529b\u5b66\u66f4\u4e3a\u666e\u9002\uff0c\u5c3d\u7ba1\u5728\u8bb8\u591a\u793a\u4f8b\u4e2d\uff0c\u8fde\u7eed\u65f6\u95f4\u52a8\u529b\u5b66\u6bd4\u957f\u65f6\u95f4\u7684\u79bb\u6563\u65f6\u95f4\u6620\u5c04\u5177\u6709\u66f4\u7b80\u5355\u7684\u8868\u793a\u3002\u4f8b\u5982\uff0c\u7b80\u5355\u7684\u6d1b\u4f26\u5179\u7cfb\u7edf\u5177\u6709\u7b80\u5355\u7684\u8fde\u7eed\u65f6\u95f4\u8868\u793a\uff0c\u4f46\u5373\u4f7f\u5bf9\u4e8e\u4e2d\u7b49\u957f\u79bb\u6563\u65f6\u95f4\u6b65\u7684\u66f4\u65b0\u901a\u5e38\u4e5f\u65e0\u6cd5\u8868\u793a\u3002<\/span><\/p>\n


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(17) \u548c (18) \u7684\u5173\u952e\u8981\u70b9\u662f\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u5728\u672c\u5f81\u51fd\u6570\u5750\u6807\u4e2d\u5b8c\u5168\u7ebf\u6027\u5316\uff0c\u7531 \u03c6<\/span>(x) \u7ed9\u51fa\u3002\u4e3e\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\uff0c\u52a8\u529b\u7cfb\u7edf\u7684\u4efb\u4f55\u5b88\u6052\u91cf\u90fd\u5bf9\u5e94\u4e8e\u7279\u5f81\u503c \u03bb = 0 \u7684 Koopman \u7279\u5f81\u51fd\u6570\u3002\u8fd9\u5efa\u7acb\u4e86\u8457\u540d\u7684\u8bfa\u7279\u5b9a\u7406\u7684 Koopman \u6269\u5c55\uff0c\u8fd9\u610f\u5473\u7740\u63a7\u5236\u65b9\u7a0b\u4e2d\u7684\u4efb\u4f55\u5bf9\u79f0\u6027\u90fd\u4f1a\u4ea7\u751f\u4e00\u4e2a\u65b0\u7684\u7279\u5f81\u503c \u03bb = 0 \u7684Koopman\u7279\u5f81\u51fd\u6570\u3002\u4f8b\u5982\uff0c\u54c8\u5bc6\u987f\u80fd\u91cf\u51fd\u6570\u662f\u4fdd\u5b88\u7cfb\u7edf\u7684Koopman\u7279\u5f81\u51fd\u6570\u3002\u6b64\u5916\uff0c\u5bf9\u4e8e\u6bcf\u4e2a\u52a8\u529b\u7cfb\u7edf\uff0c\u5e38\u6570\u51fd\u6570 \u03c6<\/span>(x)=1 \u59cb\u7ec8\u662f\u5bf9\u5e94\u4e8e \u03bb = 0 \u7684\u5e73\u51e1\u672c\u5f81\u51fd\u6570\u3002<\/span><\/p>\n


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\u7279\u5f81\u503c\u70b9\u9635\uff1a<\/strong>\u6709\u8da3\u7684\u662f\uff0c\u4e00\u7ec4 Koopman \u672c\u5f81\u51fd\u6570\u53ef\u7528\u4e8e\u751f\u6210\u66f4\u591a\u672c\u5f81\u51fd\u6570\u3002\u5728\u79bb\u6563\u65f6\u95f4\u4e2d\uff0c\u6211\u4eec\u53d1\u73b0\u4e24\u4e2a\u672c\u5f81\u51fd\u6570 \u03c6<\/span>1<\/sub>(x) \u548c \u03c6<\/span>2<\/sub>(x) \u7684\u4e58\u79ef\u4e5f\u662f\u672c\u5f81\u51fd\u6570\u3002<\/span><\/p>\n


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\u5bf9\u5e94\u4e8e\u7531 \u03c6<\/span>1<\/sub>(x) \u548c\u03c6<\/span>2<\/sub>(x) \u4e24\u4e2a\u7279\u5f81\u503c\u7684\u4e58\u79ef\u7ed9\u51fa\u7684\u65b0\u7279\u5f81\u503c \u03bb<\/span>1<\/sub>\u03bb<\/span>2<\/sub>\u3002<\/span><\/p>\n


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\u5728\u8fde\u7eed\u65f6\u95f4\u5185\uff0c\u5173\u7cfb\u53d8\u4e3a\uff1a<\/span><\/p>\n


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\u6709\u8da3\u7684\u662f\uff0c\u8fd9\u610f\u5473\u7740Koopman\u672c\u5f81\u51fd\u6570\u96c6\u5728\u9010\u70b9\u4e58\u6cd5\u4e0b\u5efa\u7acb\u4e86\u4e00\u4e2a\u53ef\u4ea4\u6362\u5e7a\u534a\u7fa4\uff1b\u5e7a\u534a\u7fa4\u5177\u6709\u7fa4\u7684\u7ed3\u6784\uff0c\u53ea\u662f\u5143\u7d20\u4e0d\u9700\u8981\u6709\u9006\u5143\u3002\u56e0\u6b64\uff0c\u6839\u636e\u52a8\u529b\u7cfb\u7edf\uff0c\u53ef\u80fd\u5b58\u5728\u4e00\u7ec4\u6709\u9650\u7684\u672c\u5f81\u51fd\u6570\u5143\u7d20\u751f\u6210\u5143\uff0c\u53ef\u7528\u4e8e\u6784\u9020\u6240\u6709\u5176\u4ed6\u672c\u5f81\u51fd\u6570\u3002\u76f8\u5e94\u7684\u7279\u5f81\u503c\u7c7b\u4f3c\u5730\u6839\u636e\u4e58\u79ef \u03bb<\/span>1<\/sub>\u03bb<\/span>2<\/sub> \u6216\u603b\u548c \u03bb<\/span>1<\/sub>+\u03bb<\/span>2 <\/sub>\u5f62\u6210\u4e00\u4e2a\u70b9\u9635\uff0c\u5177\u4f53\u53d6\u51b3\u4e8e\u52a8\u529b\u5b66\u662f\u79bb\u6563\u65f6\u95f4\u8fd8\u662f\u8fde\u7eed\u65f6\u95f4\u7684\u3002\u4f8b\u5982\uff0c\u7ed9\u5b9a\u7ebf\u6027\u7cfb\u7edf<\/sub>\uff0c\u5219\u662f\u7279\u5f81\u503c\u4e3a \u03bb <\/span>\u7684\u7279\u5f81\u51fd\u6570\u03c6<\/span>(x)\u3002\u6b64\u5916\uff0c\u03c6<\/span><\/span>\u03b1<\/span><\/sub>(x)=x\u03b1<\/span><\/sup> \u4e5f\u662f\u4e00\u4e2a\u7279\u5f81\u51fd\u6570\uff0c\u5bf9\u4e8e\u4efb\u4f55 \u03b1 <\/span>\u5176\u7279\u5f81\u503c\u4e3a \u03b1<\/span>\u03bb <\/span>\u3002<\/span><\/p>\n


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\u8fde\u7eed\u65f6\u95f4\u548c\u79bb\u6563\u65f6\u95f4\u7684\u7279\u5f81\u503c\u70b9\u9635\u4ee5\u7b80\u5355\u7684\u65b9\u5f0f\u76f8\u5173\u8054\u3002\u5982\u679c\u8fde\u7eed\u65f6\u95f4\u7279\u5f81\u503c\u7531 \u03bb <\/span>\u7ed9\u51fa\uff0c\u5219\u76f8\u5e94\u7684\u79bb\u6563\u65f6\u95f4\u7279\u5f81\u503c\u7531 e\u03bb<\/span>t <\/sup>\u7ed9\u51fa\u3002\u56e0\u6b64\uff0c(21b)\u548c(22d)\u4e2d\u7684\u7279\u5f81\u503c\u8868\u8fbe\u5f0f\u7684\u5173\u8054\u4e3a\uff1a<\/span><\/p>\n


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<\/p>\n


<\/span><\/p>\n

\u4f5c\u4e3a\u8fde\u7eed\u65f6\u95f4\u548c\u79bb\u6563\u65f6\u95f4\u7279\u5f81\u503c\u4e4b\u95f4\u5173\u7cfb\u7684\u53e6\u4e00\u4e2a\u7b80\u5355\u6f14\u793a\uff0c\u8003\u8651\u5c06 (16) \u4e2d\u7684\u8fde\u7eed\u65f6\u95f4\u5b9a\u4e49\u5e94\u7528\u4e8e\u7279\u5f81\u51fd\u6570\uff1a<\/span><\/p>\n


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<\/span><\/h3>\n

3.2 Koopman \u6a21\u5f0f\u5206\u89e3\u548c\u6709\u9650\u8868\u793a<\/span><\/strong><\/span><\/h3>\n


<\/p>\n

\u5230\u76ee\u524d\u4e3a\u6b62\uff0c\u6211\u4eec\u5df2\u7ecf\u8003\u8651\u4e86\u7cfb\u7edf\u7684\u6807\u91cf\u6d4b\u91cf\uff0c\u5e76\u4e14\u6211\u4eec\u53d1\u73b0\u4e86\u968f\u65f6\u95f4\u7ebf\u6027\u6f14\u5316\u7684\u7279\u6b8a\u7279\u5f81\u6d4b\u91cf\u3002\u7136\u800c\uff0c\u6211\u4eec\u7ecf\u5e38\u5bf9\u4e00\u4e2a\u7cfb\u7edf\u8fdb\u884c\u591a\u6b21\u6d4b\u91cf\u3002\u5728\u6781\u7aef\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u6d4b\u91cf\u9ad8\u7ef4\u7a7a\u95f4\u7cfb\u7edf\u7684\u6574\u4e2a\u72b6\u6001\uff0c\u4f8b\u5982\u4e0d\u65ad\u53d8\u5316\u7684\u6d41\u4f53\u6d41\u52a8\u3002\u7136\u540e\u53ef\u4ee5\u5c06\u8fd9\u4e9b\u6d4b\u91cf\u7ed3\u679c\u6392\u5217\u5728\u5411\u91cf g \u4e2d\uff1a<\/span><\/p>\n


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<\/span><\/p>\n

\u6bcf\u4e2a\u5355\u72ec\u7684\u6d4b\u91cf\u90fd\u53ef\u4ee5\u6839\u636e\u672c\u5f81\u51fd\u6570 \u03c6<\/span>j<\/sub>(x) \u5c55\u5f00\uff0c\u8fd9\u4e3a\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u63d0\u4f9b\u4e86\u57fa\u7840\uff1a<\/span><\/p>\n


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<\/span><\/p>\n

\u56e0\u6b64\uff0c\u53ef\u89c2\u6d4b\u91cf\u5411\u91cf g <\/strong>\u53ef\u4ee5\u7c7b\u4f3c\u5730\u5c55\u5f00\uff1a<\/span><\/p>\n


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<\/p>\n


<\/p>\n

\u5176\u4e2d v<\/strong>j <\/sub>\u662f\u4e0e\u672c\u5f81\u51fd\u6570 \u03c6<\/span>j<\/sub> \u76f8\u5173\u7684\u7b2c j \u4e2a Koopman \u6a21\u5f0f\u3002\u5bf9\u4e8e\u4fdd\u5b88\u52a8\u529b\u7cfb\u7edf\uff0c\u4f8b\u5982\u53d7\u54c8\u5bc6\u987f\u52a8\u529b\u5b66\u63a7\u5236\u7684\u7cfb\u7edf\uff0cKoopman \u7b97\u7b26\u662f\u9149\u7684\u3002\u56e0\u6b64\uff0cKoopman \u7279\u5f81\u51fd\u6570\u5bf9\u4e8e\u4fdd\u5b88\u7cfb\u7edf\u662f\u6b63\u4ea4\u7684\uff0c\u5e76\u4e14\u53ef\u4ee5\u901a\u8fc7\u6295\u5f71\u76f4\u63a5\u8ba1\u7b97 Koopman \u6a21\u5f0f v<\/strong>j <\/sub>\uff1a<\/span><\/p>\n


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<\/span><\/p>\n

\u5176\u4e2d  <\/sub><\/span>\u662f\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u4e2d\u51fd\u6570\u7684\u6807\u51c6\u5185\u79ef\u3002\u5728\u7cfb\u7edf\u7684\u76f4\u63a5\u7a7a\u95f4\u6d4b\u91cf g(x) = x \u7684\u60c5\u51b5\u4e0b\uff0c\u8fd9\u4e9b\u6a21\u5f0f\u5177\u6709\u7269\u7406\u89e3\u91ca\uff0c\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u8fd9\u4e9b\u6a21\u5f0f\u662f\u76f8\u5e72\u7a7a\u95f4\u6a21\u5f0f\uff0c\u5176\u884c\u4e3a\u5177\u6709\u76f8\u540c\u7684\u65f6\u95f4\u52a8\u529b\u5b66<\/span>\uff08\u5373\u632f\u8361\uff0c\u53ef\u80fd\u5177\u6709\u7ebf\u6027\u589e\u957f\u6216\u8870\u51cf\uff09<\/span>\u3002\u8003\u8651\u5230\uff0827\uff09\u4e2d\u7684\u5206\u89e3\uff0c\u53ef\u4ee5\u5c06\u6d4b\u91cf g \u7684\u52a8\u529b\u5b66\u8868\u793a\u5982\u4e0b\uff1a<\/span><\/p>\n


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<\/p>\n

\u8fd9\u4e2a\u4e09\u5143\u7ec4\u5e8f\u5217 <\/sub> \u88ab\u79f0\u4e3a <\/strong>Koopman<\/strong> <\/strong>\u6a21\u5f0f\u5206\u89e3<\/strong>\uff0c\u7531 Mezic \u4e8e2005\u5e74\u5f15\u5165\u3002Koopman \u6a21\u5f0f\u5206\u89e3\u540e\u6765\u901a\u8fc7\u52a8\u6001\u6a21\u5f0f\u5206\u89e3\u8fde\u63a5\u5230\u6570\u636e\u9a71\u52a8\u56de\u5f52\u3002<\/span><\/p>\n


<\/span><\/p>\n

<\/p>\n

\u56fe1. \u975e\u7ebf\u6027\u52a8\u529b\u7cfb\u7edf\u7684Koopman\u7b97\u7b26\u793a\u610f\u56fe\u3002\u4ece yk<\/sub> \u2192 xk<\/sub> \u7684\u865a\u7ebf\u8868\u793a\u6211\u4eec\u5e0c\u671b\u80fd\u591f\u6062\u590d\u539f\u59cb\u72b6\u6001\u3002<\/span><\/p>\n


<\/span><\/strong><\/h4>\n

\u4e0d\u53d8\u7279\u5f81\u7a7a\u95f4\u548c\u6709\u9650\u7ef4\u6a21\u578b<\/span><\/strong><\/span><\/h4>\n


<\/span><\/p>\n

\u5e94\u7528 Koopman \u5206\u6790\u4e0d\u662f\u6355\u83b7\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u4e2d\u6240\u6709\u6d4b\u91cf\u51fd\u6570\u7684\u6f14\u5316\uff0c\u800c\u662f\u8fd1\u4f3c\u7531\u4e00\u7ec4\u6709\u9650\u6d4b\u91cf\u51fd\u6570\u7ec4\u6210\u7684\u4e0d\u53d8\u5b50\u7a7a\u95f4\u4e0a\u7684\u6f14\u5316\u3002Koopman \u4e0d\u53d8\u5b50\u7a7a\u95f4\u5b9a\u4e49\u4e3a\u4e00\u7ec4\u51fd\u6570 {g1<\/sub>, g2<\/sub>, \u00b7 \u00b7 \u00b7 , gp<\/sub>} \u7684\u8de8\u5ea6\uff0c\u5982\u679c\u5bf9\u4e8e\u8be5\u5b50\u7a7a\u95f4\u4e2d\u6240\u6709\u51fd\u6570g<\/span><\/p>\n


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<\/p>\n


<\/span><\/p>\n

\u7ecf Koopman \u7b97\u7b26<\/span> <\/span> <\/sub>\u4f5c\u7528\u540e\u4ecd\u4fdd\u7559\u5728\u8be5\u5b50\u7a7a\u95f4\u4e2d\uff1a<\/span><\/p>\n


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<\/p>\n


<\/span><\/p>\n

\u901a\u8fc7\u5c06 Koopman \u7b97\u7b26\u9650\u5236\u4e3a\u7531\u6709\u9650\u6570\u91cf\u7684\u51fd\u6570 <\/sub> \u5f20\u6210\u7684\u4e0d\u53d8\u5b50\u7a7a\u95f4\uff0c\u53ef\u4ee5\u83b7\u5f97 Koopman \u7b97\u7b26\u7684\u6709\u9650\u7ef4\u77e9\u9635\u8868\u793a\u3002\u77e9\u9635\u8868\u793a K \u4f5c\u7528\u4e8e\u5411\u91cf\u7a7a\u95f4 <\/sub>\uff0c\u5750\u6807\u7531 gj<\/sub>(x<\/strong>) \u7684\u503c\u7ed9\u51fa\u3002\u8fd9\u4ea7\u751f\u4e86\u4e00\u4e2a\u6709\u9650\u7ef4\u7ebf\u6027\u7cfb\u7edf\uff0c\u5982\uff0813\uff09\u548c\uff0815\uff09\u4e2d\u6240\u793a\u3002<\/span><\/p>\n


<\/span><\/p>\n

Koopman \u7b97\u7b26\u7684\u4efb\u4f55\u6709\u9650\u7279\u5f81\u51fd\u6570\u96c6\u90fd\u5c06\u5f20\u6210\u4e0d\u53d8\u5b50\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u53d1\u73b0\u8fd9\u4e9b\u7279\u5f81\u51fd\u6570\u5750\u6807\u662f\u4e00\u4e2a\u6838\u5fc3\u6311\u6218\uff0c\u56e0\u4e3a\u5b83\u4eec\u63d0\u4f9b\u4e86\u52a8\u529b\u5b66\u7684\u7ebf\u6027\u8868\u73b0\u7684\u5185\u5728\u5750\u6807\u3002\u5728\u5b9e\u8df5\u4e2d\uff0c\u6211\u4eec\u66f4\u6709\u53ef\u80fd\u786e\u5b9a\u4e00\u4e2a\u8fd1\u4f3c\u4e0d\u53d8\u7684\u5b50\u7a7a\u95f4\uff0c\u7531\u4e00\u7ec4\u51fd\u6570 <\/span><\/sub> \u7ed9\u51fa\uff0c\u5176\u4e2d\u6bcf\u4e2a\u51fd\u6570 gj<\/sub> \u90fd\u53ef\u4ee5\u901a\u8fc7\u7279\u5f81\u51fd\u6570\u7684\u6709\u9650\u548c\u6765\u5f88\u597d\u5730\u8fd1\u4f3c\uff1a<\/span><\/sub>\u3002<\/span><\/span><\/p>\n


<\/span><\/h3>\n

3.3 \u5e94\u7528<\/span><\/strong><\/span><\/h3>\n


<\/span><\/h4>\n

\u793a\u4f8b1\uff1a\u5177\u6709\u5355\u4e0d\u52a8\u70b9\u548c\u6162\u6d41\u5f62\u7684\u975e\u7ebf\u6027\u7cfb\u7edf<\/span><\/strong><\/span><\/h4>\n


<\/p>\n

\u5728\u8fd9\u91cc\uff0c\u6211\u4eec\u8003\u8651\u4e00\u4e2a\u5177\u6709\u5355\u4e2a\u4e0d\u52a8\u70b9\u7684\u793a\u4f8b\u7cfb\u7edf\uff0c\u7531\u4e0b\u5f0f\u7ed9\u51fa\uff1a<\/span><\/p>\n


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<\/p>\n


<\/span><\/p>\n

\u5bf9\u4e8e \u03bb < \u03bc < 0\uff0c\u7cfb\u7edf\u8868\u73b0\u51fa\u7531  <\/sub>\u7ed9\u51fa\u7684\u6162\u5438\u5f15\u6d41\u5f62\u3002\u53ef\u4ee5\u7528\u975e\u7ebf\u6027\u6d4b\u91cf  <\/sub>\u6765\u589e\u5f3a\u72b6\u6001 x\uff0c\u4ee5\u5b9a\u4e49\u4e09\u7ef4 Koopman \u4e0d\u53d8\u5b50\u7a7a\u95f4\u3002\u5728\u8fd9\u4e9b\u5750\u6807\u4e2d\uff0c\u52a8\u529b\u5b66\u53d8\u4e3a\u7ebf\u6027\uff1a<\/span><\/p>\n


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<\/p>\n


<\/span><\/p>\n

\u5b8c\u6574\u7684\u4e09\u7ef4 Koopman \u53ef\u89c2\u6d4b\u5411\u91cf\u7a7a\u95f4\u5982\u56fe 2 \u6240\u793a\u3002\u4ece\u4e0d\u53d8\u6d41\u5f62 <\/sub> <\/span>\uff08\u84dd\u8272\u8868\u9762\uff09<\/span>\u5f00\u59cb\u7684\u8f68\u8ff9\u88ab\u9650\u5236\u7559\u5728\u8be5\u6d41\u5f62\u4e0a\u3002\u5b58\u5728\u4e00\u4e2a\u6162\u5b50\u7a7a\u95f4\uff0c\u7531\u5bf9\u5e94\u4e8e\u6162\u7279\u5f81\u503c \u03bc \u548c 2\u03bc \u7684\u7279\u5f81\u5411\u91cf\u5f20\u6210\uff1b\u8be5\u5b50\u7a7a\u95f4\u7531\u7eff\u8272\u8868\u9762\u53ef\u89c6\u5316\u3002\u6700\u540e\uff0c\u5b58\u5728\u539f\u59cb\u7cfb\u7edf\u7684\u539f\u59cb\u6e10\u8fd1\u5438\u5f15\u6d41\u5f62\uff0c<\/sub><\/span> <\/sub>(\u7ea2\u8272\u8868\u9762)<\/span>\u3002\u84dd\u8272\u548c\u7ea2\u8272\u629b\u7269\u9762\u59cb\u7ec8\u76f8\u4ea4\u4e8e\u6cbf y2<\/sub>-y3<\/sub> \u65b9\u5411\u503e\u659c 45\u00b0 \u89d2\u7684\u629b\u7269\u7ebf\u3002\u968f\u7740\u5feb\u52a8\u529b\u5b66\u4e0e\u6162\u52a8\u529b\u5b66\u7684\u6bd4\u7387\u53d8\u5f97\u8d8a\u6765\u8d8a\u5927\uff0c\u7eff\u8272\u8868\u9762\u63a5\u8fd1 45\u00b0 \u503e\u89d2\u3002\u5728\u5b8c\u6574\u7684\u4e09\u7ef4 Koopman \u53ef\u89c2\u6d4b\u7a7a\u95f4\u4e2d\uff0c\u52a8\u529b\u5b66\u4ea7\u751f\u5355\u4e2a\u7a33\u5b9a\u8282\u70b9\uff0c\u5176\u8f68\u8ff9\u8fc5\u901f\u88ab\u5438\u5f15\u5230\u7eff\u8272\u5b50\u7a7a\u95f4\u4e0a\uff0c\u7136\u540e\u7f13\u6162\u63a5\u8fd1\u56fa\u5b9a\u70b9\u3002<\/span><\/p>\n


<\/span><\/p>\n

<\/p>\n

\u56fe2.  \u65b9\u7a0b(33a) \u8868\u793a\u7684\u4e09\u7ef4\u7ebf\u6027Koopman\u7cfb\u7edf\u7684\u53ef\u89c6\u5316\u4ee5\u53ca x1<\/sub>-x2<\/sub> \u5e73\u9762\u4e0a\u7684\u52a8\u6001\u6295\u5f71\u3002\u8fd9\u91cc\uff0c\u03bc = \u22120.05 \u4e14 \u03bb = 1\u3002<\/span><\/p>\n


<\/strong><\/span><\/p>\n

\u7531<\/strong> <\/strong>Koopman<\/strong> <\/strong>\u7b97\u7b26\u7684\u7279\u5f81\u51fd\u6570\u5b9a\u4e49\u7684\u5185\u5728\u5750\u6807\u3002<\/strong>Koopman \u7b97\u7b26\u7684\u5de6\u7279\u5f81\u5411\u91cf\u4ea7\u751f Koopman \u7279\u5f81\u51fd\u6570<\/span>\uff08\u5373\u7279\u5f81\u53ef\u89c2\u6d4b\u91cf\uff09<\/span>\u3002(33a) \u5bf9\u5e94\u4e8e\u7279\u5f81\u503c \u03bc \u548c \u03bb \u7684 Koopman \u7279\u5f81\u51fd\u6570\u4e3a\uff1a<\/span><\/p>\n


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\u03c6<\/span>\u03bb<\/span><\/sub> <\/span>\u4e2d\u7684\u5e38\u6570 b \u53cd\u6620\u4e86\u8fd9\u6837\u4e00\u4e2a\u4e8b\u5b9e\uff1a\u5bf9\u4e8e\u6709\u9650\u7684\u6bd4\u7387 \u03bb\/\u03bc\uff0c\u52a8\u529b\u5b66\u4ec5\u5f71\u54cd\u6e10\u8fd1\u5438\u5f15\u6162\u6d41\u5f62 <\/span> <\/sub>\uff0c\u4f46\u5b9e\u9645\u4e0a\u9075\u5faa\u76f8\u90bb\u7684\u629b\u7269\u7ebf\u8f68\u8ff9\u3002\u56fe 2 \u4e2d\u9488\u5bf9\u4e0d\u540c\u6bd4\u7387 \u03bb\/\u03bc \u7684\u5404\u4e2a\u8868\u9762\u66f4\u6e05\u695a\u5730\u8bf4\u660e\u4e86\u8fd9\u4e00\u70b9\u3002\u8fd9\u6837\uff0c\u53ef\u4ee5\u6839\u636e\u7531 Koopman \u7b97\u7b26\u7684\u5de6\u7279\u5f81\u5411\u91cf\u5728\u4e0d\u53d8\u5b50\u7a7a\u95f4\u4e0a\u5b9a\u4e49\u7684\u53ef\u89c2\u6d4b\u51fd\u6570\u6765\u786e\u5b9a\u4e00\u7ec4\u5185\u5728\u5750\u6807\u3002\u5177\u4f53\u5730\u8bf4\uff0c<\/span><\/p>\n


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<\/p>\n

\u8fd9\u4e9b\u7279\u5f81\u53ef\u89c2\u6d4b\u91cf\u5b9a\u4e49\u4e86\u5728 Koopman \u7b97\u7b26\u4f5c\u7528\u4e0b\u4fdd\u6301\u4e0d\u53d8\u7684\u53ef\u89c2\u5bdf\u5b50\u7a7a\u95f4\uff0c\u5373\u4f7f\u5728\u5750\u6807\u53d8\u6362\u4e4b\u540e\u3002\u56e0\u6b64\uff0c\u5b83\u4eec\u53ef\u4ee5\u88ab\u89c6\u4e3a Koopman \u4e0d\u53d8\u5b50\u7a7a\u95f4\u4e0a\u7684\u5185\u5728\u5750\u6807\u3002<\/span><\/p>\n


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\u793a\u4f8b2\uff1a\u903b\u8f91\u65af\u8482\u6620\u5c04<\/span><\/strong><\/span><\/h4>\n


<\/span><\/p>\n

\u8003\u8651\u7531\u4e0b\u5f0f\u7ed9\u51fa\u7684\u903b\u8f91\u65af\u8482\u6620\u5c04\uff1a<\/span><\/p>\n


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\u4ee4\u53ef\u89c2\u5bdf\u5b50\u7a7a\u95f4\u5305\u62ec x \u548c x2<\/sup> \uff1a<\/span><\/p>\n


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\u5199\u51fa Koopman \u7b97\u7b26\uff0c\u7b2c\u4e00\u884c\u5f88\u5bb9\u6613\u5c31\u80fd\u5199\u51fa\uff1a<\/span><\/p>\n


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\u4f46\u7b2c\u4e8c\u6392\u5c31\u4e0d\u660e\u663e\u4e86\u3002\u8981\u627e\u5230\u6b64\u8868\u8fbe\u5f0f\uff0c\u9700\u8981\u5c55\u5f00  <\/sub>\uff1a<\/span><\/p>\n


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\u56e0\u6b64\uff0c\u9700\u8981\u4e09\u6b21\u548c\u56db\u6b21\u591a\u9879\u5f0f\u9879\u6765\u63a8\u8fdb x2 <\/sup>\u3002\u7c7b\u4f3c\u5730\uff0c\u8fd9\u4e9b\u9879\u5206\u522b\u9700\u8981\u516d\u9636\u548c\u516b\u9636\u591a\u9879\u5f0f\uff0c\u4f9d\u6b64\u7c7b\u63a8\uff0c\u65e0\u7a77\u65e0\u5c3d\uff1a<\/span><\/p>\n


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\u4e0a\u8ff0 Koopman \u7b97\u7b26\u5728\u591a\u9879\u5f0f\u57fa\u7840\u4e0a\u7684\u8868\u793a\u6709\u4e9b\u4ee4\u4eba\u4e0d\u5b89\u3002\u4e0d\u4ec5\u6ca1\u6709\u5c01\u95ed\u6027\uff0c\u800c\u4e14\u5f53 \u03b2 > 1 \u65f6\uff0c\u4efb\u4f55\u6709\u9650\u79e9\u622a\u65ad\u7684\u884c\u5217\u5f0f\u90fd\u975e\u5e38\u5927\u3002\u8fd9\u8bf4\u660e\u4e86\u4e0e\u7b80\u5355\u6df7\u6c8c\u7cfb\u7edf\u7684\u65e0\u9650\u7ef4 Koopman \u7b97\u7b26\u7684\u6734\u7d20\u8868\u793a\u76f8\u5173\u7684\u4e00\u4e2a\u7f3a\u9677\u3002\u622a\u65ad\u7cfb\u7edf\uff0c\u6216\u5bf9\u589e\u5e7f\u7684\u53ef\u89c2\u5bdf\u5411\u91cf\u8fdb\u884c\u6700\u5c0f\u4e8c\u4e58\u62df\u5408\u4f1a\u4ea7\u751f\u8f83\u5dee\u7684\u7ed3\u679c\uff0c\u622a\u65ad\u7684\u7cfb\u7edf\u4ec5\u4e0e\u5c11\u6570\u8fed\u4ee3\u7684\u771f\u5b9e\u52a8\u529b\u5b66\u4e00\u81f4\uff0c\u56e0\u4e3a\u8868\u793a\u7684\u590d\u6742\u6027\u589e\u957f\u5730\u975e\u5e38\u5feb\uff1a<\/span><\/p>\n


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3.4 \u7279\u5f81\u51fd\u6570\u7684\u89e3\u6790\u7ea7\u6570\u5c55\u5f00<\/span><\/strong><\/span><\/p>\n


<\/span><\/p>\n

\u7ed9\u5b9a\u65b9\u7a0b(1) \u4e2d\u7684\u52a8\u529b\u5b66\uff0c\u53ef\u4ee5\u4f7f\u7528\u6807\u51c6\u6280\u672f\u6c42\u89e3 (20) \u4e2d\u7684\u504f\u5fae\u5206\u65b9\u7a0b\uff0c\u4f8b\u5982\u9012\u5f52\u6c42\u89e3\u6cf0\u52d2\u7ea7\u6570\u6216\u6d1b\u6717\u7ea7\u6570\u4e2d\u7684\u9879\u3002\u4e0b\u9762\u63a2\u8ba8\u4e86\u4e00\u4e9b\u7b80\u5355\u7684\u4f8b\u5b50\u3002<\/span><\/p>\n


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\u7ebf\u6027\u52a8\u529b\u5b66<\/span><\/strong><\/span><\/h4>\n


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\u8003\u8651\u7b80\u5355\u7684\u7ebf\u6027\u52a8\u529b\u5b66<\/span><\/p>\n


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\u5047\u8bbe \u03c6<\/span>(x) <\/span>\u7684\u6cf0\u52d2\u7ea7\u6570\u5c55\u5f00\uff1a<\/span><\/p>\n


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<\/span><\/p>\n

\u90a3\u4e48\u68af\u5ea6\u548c\u65b9\u5411\u5bfc\u6570\u7531\u4e0b\u5f0f\u7ed9\u51fa\uff1a<\/span><\/p>\n


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<\/p>\n

\u6c42\u89e3 Koopman \u672c\u5f81\u51fd\u6570 PDE (20) \u4e2d\u7684\u9879\uff0c\u6211\u4eec\u53d1\u73b0 c0<\/sub>=0 \u5fc5\u987b\u6210\u7acb\u3002\u5bf9\u4e8e(20)\u4e2d\u7684\u4efb\u4f55\u6b63\u6574\u6570 \u03bb <\/span>\uff0c\u53ea\u6709\u4e00\u4e2a\u7cfb\u6570\u53ef\u4ee5\u662f\u975e\u96f6\u7684\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5bf9\u4e8e \u03bb = k \u2208 Z+<\/sup>\uff0c\u5219 \u03c6<\/span>(x)=cxk<\/sup> \u662f\u4efb\u4f55\u5e38\u6570 c \u7684\u7279\u5f81\u51fd\u6570\u3002\u4f8b\u5982\uff0c\u5982\u679c \u03bb = 1\uff0c\u5219 \u03c6<\/span>(x)=x \u3002<\/span><\/p>\n


<\/span><\/p>\n

\u4e8c\u6b21\u975e\u7ebf\u6027\u52a8\u529b\u5b66<\/span><\/strong><\/h4>\n


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\u8003\u8651\u975e\u7ebf\u6027\u52a8\u529b\u7cfb\u7edf<\/span><\/p>\n


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\u9664\u4e86 \u03bb = 0 \u65f6\u7684\u5e73\u51e1\u89e3 \u03c6<\/span>(x)=0 \u5916\uff0c\u6ca1\u6709\u6ee1\u8db3 (20) \u7684\u6cf0\u52d2\u7ea7\u6570\u3002\u76f8\u53cd\uff0c\u6211\u4eec\u5047\u8bbe\u4e00\u4e2a\u6d1b\u6717\u7ea7\u6570\uff1a<\/span><\/p>\n


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\u68af\u5ea6\u548c\u65b9\u5411\u5bfc\u6570\u7531\u4e0b\u5f0f\u7ed9\u51fa\uff1a<\/span><\/p>\n


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\u6c42\u89e3\u6ee1\u8db3 (20) \u7684\u6d1b\u6717\u7ea7\u6570\u7cfb\u6570\uff0c\u6211\u4eec\u53d1\u73b0\u6240\u6709\u5177\u6709\u6b63\u7d22\u5f15\u7684\u7cfb\u6570\u90fd\u4e3a\u96f6\uff0c\u5373\u5bf9\u4e8e\u6240\u6709 k \u2265 1\uff0cck<\/sub> = 0\u3002\u7136\u800c\uff0c\u5bf9\u4e8e k \u2264 \u22121 \uff0c\u975e\u6b63\u7d22\u5f15\u7cfb\u6570\u7531\u9012\u5f52 \u03bbck+1<\/sub>= kck<\/sub>\u7ed9\u51fa\u3002\u56e0\u6b64\uff0c\u6d1b\u6717\u7ea7\u6570\u4e3a<\/span><\/p>\n


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\u8fd9\u9002\u7528\u4e8e <\/sub> <\/sub>\u7684\u6240\u6709\u503c\u3002\u8fd8\u6709\u5176\u4ed6Koopman\u672c\u5f81\u51fd\u6570\u53ef\u4ee5\u4ece\u6d1b\u6717\u7ea7\u6570\u4e2d\u8bc6\u522b\u51fa\u6765\u3002<\/span><\/p>\n


<\/span><\/p>\n

\u591a\u9879\u5f0f\u975e\u7ebf\u6027\u52a8\u529b\u5b66<\/span><\/strong><\/h4>\n


<\/span><\/p>\n

\u5bf9\u4e8e\u66f4\u4e00\u822c\u7684\u975e\u7ebf\u6027\u52a8\u529b\u7cfb\u7edf<\/span><\/p>\n


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\u662f\u6240\u6709 <\/span> <\/sub>\u7684\u7279\u5f81\u51fd\u6570\u3002\u5982\u4e0a\u6240\u8ff0\uff0c\u8fd8\u53ef\u4ee5\u901a\u8fc7\u83b7\u53d6\u8fd9\u4e9b\u539f\u59cb\u672c\u5f81\u51fd\u6570\u7684\u5e42\u6765\u751f\u6210\u65b0\u7684\u672c\u5f81\u51fd\u6570\uff1b\u6240\u5f97\u7279\u5f81\u503c\u5728\u590d\u5e73\u9762\u4e0a\u751f\u6210\u4e00\u4e2a\u70b9\u9635\u3002<\/span><\/p>\n


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4. \u5386\u53f2\u548c\u8fd1\u671f\u53d1\u5c55<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


<\/p>\n

Koopman \u4e8e1931\u5e74\u63d0\u51fa\u7684\u5206\u6790\u65b9\u6cd5\u6700\u521d\u662f\u4e3a\u4e86\u63cf\u8ff0\u54c8\u5bc6\u987f\u7cfb\u7edf\u6d4b\u91cf\u7684\u6f14\u5316\uff0c\u5e76\u57281932\u5e74\u4e0e\u51af\u00b7\u8bfa\u4f9d\u66fc\u4e00\u8d77\u5c06\u8fd9\u4e00\u7406\u8bba\u63a8\u5e7f\u5230\u5177\u6709\u8fde\u7eed\u7279\u5f81\u503c\u8c31\u7684\u7cfb\u7edf\u3002\u5728\u54c8\u5bc6\u987f\u6d41\u7684\u60c5\u5f62\u4e0b\uff0cKoopman \u7b97\u7b26 <\/span><\/sub> \u662f\u9149\u7684\uff0c\u5e76\u5728\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u4e2d\u5f62\u6210\u4e86\u4e00\u65cf\u5355\u53c2\u6570\u7684\u9149\u53d8\u6362\uff0c\u8fd9\u4e0e\u54c8\u5bc6\u987f\u7cfb\u7edf\u7684\u76f8\u7a7a\u95f4\u4f53\u79ef\u5b88\u6052\u6027\u8d28\u5bc6\u5207\u76f8\u5173\u3002Koopman \u5728\u5176\u539f\u59cb\u8bba\u6587\u4e2d\u63a2\u8ba8\u4e86\u8be5\u7b97\u7b26\u7684\u7279\u5f81\u503c\u8c31\u4e0e\u5b88\u6052\u91cf\u3001\u53ef\u79ef\u6027\u53ca\u904d\u5386\u6027\u4e4b\u95f4\u7684\u8054\u7cfb\uff0c\u8fd9\u4e00\u5de5\u4f5c\u76f4\u63a5\u63a8\u52a8\u4e86 Birkhoff \u548c\u51af\u00b7\u8bfa\u4f9d\u66fc\u5bf9\u904d\u5386\u5b9a\u7406\u7684\u8457\u540d\u8bc1\u660e\u3002\u6b64\u5916\uff0cKoopman \u7b97\u7b26\u4e5f\u88ab\u79f0\u4e3a\u7ec4\u5408\u7b97\u7b26\uff0c\u4e0e\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u7a7a\u95f4\u4e2d\u7684 Perron-Frobenius \u7b97\u7b26<\/span>\uff08\u63a8\u524d\u7b97\u7b26\uff09<\/span>\u4e92\u4e3a\u5bf9\u5076\uff0c\u5373\u5de6\u4f34\u968f\u7b97\u7b26\u3002<\/span><\/p>\n


<\/span><\/p>\n

\u8fd1\u5e74\u6765\uff0c\u968f\u7740 Mezi\u0107 \u53ca\u5176\u5408\u4f5c\u8005\u7684\u5f00\u521b\u6027\u7814\u7a76\uff0cKoopman\u5206\u6790\u91cd\u65b0\u5f15\u8d77\u4e86\u5e7f\u6cdb\u5173\u6ce8\u3002\u5728\u5e94\u7528\u65b9\u9762\uff0c\u5f53\u9009\u62e9\u591a\u9879\u5f0f\u57fa\u8868\u793a Koopman \u7b97\u7b26\u65f6\uff0c\u5176\u4e0e\u975e\u7ebf\u6027\u63a7\u5236\u4e2d\u7684 Carleman \u7ebf\u6027\u5316\u65b9\u6cd5\u5bc6\u5207\u76f8\u5173\uff0c\u5e76\u5728\u6d41\u4f53\u52a8\u529b\u5b66\u4e2d\u4e0e\u89e3\u7b97\u7b26\u7406\u8bba\u6709\u8054\u7cfb\u3002Koopman \u5206\u6790\u63d0\u4f9b\u4e86\u4e00\u79cd\u8865\u5145\u51e0\u4f55\u548c\u6982\u7387\u65b9\u6cd5\u7684\u65b0\u6846\u67b6\uff0c\u4f8b\u5982\uff0cKoopman \u7279\u5f81\u51fd\u6570\u7684\u6c34\u5e73\u96c6\u53ef\u5f62\u6210\u52a8\u529b\u7cfb\u7edf\u7684\u72b6\u6001\u7a7a\u95f4\u4e0d\u53d8\u5206\u533a\uff0c\u7528\u4e8e\u904d\u5386\u5206\u533a\u7684\u5206\u6790\u3002\u6b64\u5916\uff0c\u7814\u7a76\u8868\u660e Koopman \u5206\u6790\u80fd\u591f\u63a8\u5e7f Hartman-Grobman \u5b9a\u7406\u81f3\u7a33\u5b9a\u6216\u4e0d\u7a33\u5b9a\u5e73\u8861\u70b9\u53ca\u5468\u671f\u8f68\u9053\u7684\u5438\u5f15\u57df\u3002\u5f53\u524d\u7684\u4e3b\u8981\u6311\u6218\u5728\u4e8e\u5982\u4f55\u6709\u6548\u8868\u793a\u4e00\u822c\u52a8\u529b\u7cfb\u7edf\u7684 Koopman \u7279\u5f81\u51fd\u6570\uff0c\u7279\u522b\u662f\u901a\u8fc7\u6570\u636e\u9a71\u52a8\u65b9\u6cd5\u6765\u8bc6\u522b\u8fd9\u4e9b\u7279\u5f81\u51fd\u6570\u5e76\u5e94\u7528\u4e8e\u63a7\u5236\uff0c\u5176\u4e2d\u5305\u62ec\u5229\u7528\u6df1\u5ea6\u5b66\u4e60\u6280\u672f\u6316\u6398\u6570\u636e\u4e2d\u7684\u7279\u5f81\u51fd\u6570\u7ed3\u6784\u3002<\/span><\/p>\n


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Koopman\u5206\u6790\u5728\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u4e2d\u7684<\/strong><\/p>\n

\u5e94\u7528\u7cfb\u5217\u8bfe\u7a0b<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u662f\u590d\u6742\u7cfb\u7edf\u7814\u7a76\u4e2d\u91cd\u8981\u7684\u7406\u8bba\u57fa\u7840\u548c\u7814\u7a76\u5de5\u5177\uff0c\u5b83\u6709\u52a9\u4e8e\u7406\u89e3\u7cfb\u7edf\u7684\u672c\u8d28\u7279\u6027\uff0c\u901a\u8fc7\u6784\u5efa\u548c\u5206\u6790\u7cfb\u7edf\u7684\u7c97\u7c92\u5316\u7a0b\u5ea6\u4e0d\u540c\u7684\u6570\u5b66\u6a21\u578b\u6765\u89e3\u51b3\u5b9e\u9645\u95ee\u9898\u3002\u5c3d\u7ba1\u4f4e\u7ef4\u975e\u7ebf\u6027\u7cfb\u7edf\u7684\u7406\u8bba\u5df2\u7ecf\u76f8\u5f53\u5b8c\u5907\uff0c\u4f46\u5982\u4f55\u5c06\u5176\u7528\u5230\u9ad8\u7ef4\u975e\u7ebf\u6027\u4f53\u7cfb\u4e2d\uff0c\u83b7\u53d6\u7cfb\u7edf\u7684\u5173\u952e\u52a8\u529b\u5b66\u4fe1\u606f\u662f\u4e00\u4e2a\u957f\u671f\u56f0\u6270\u7814\u7a76\u8005\u7684\u6311\u6218\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u597d\u7684\u7406\u8bba\u548c\u5de5\u5177\u4f1a\u5927\u5927\u52a0\u5feb\u79d1\u7814\u8fdb\u5c55\uff0cKoopman\u7b97\u7b26\u6709\u671b\u6210\u4e3a\u5206\u6790\u548c\u9884\u6d4b\u52a8\u6001\u7cfb\u7edf\u7684\u6807\u51c6\u7406\u8bba\u3002\u96c6\u667a\u5b66\u56ed\u9080\u8bf7\u5317\u4eac\u90ae\u7535\u5927\u5b66\u5170\u5cb3\u6052\u6559\u6388\u5f00\u8bbe\u300cKoopman\u5206\u6790\u5728\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u4e2d\u7684\u5e94\u7528\u300d\u7cfb\u5217\u8bfe\u7a0b<\/a>\uff0c\u9762\u5411\u81f4\u529b\u4e8e\u5b66\u4e60\u548c\u53d1\u5c55Koopman\u7b97\u7b26\u7406\u8bba\u7684\u793e\u533a\u670b\u53cb\uff0c\u57286\u8282\u8bfe\u7a0b\u65f6\u95f4\u4e2d\uff0c\u901a\u8fc7\u7ec6\u81f4\u7684\u8bb2\u89e3\uff0c\u6df1\u5165\u7684\u4e92\u52a8\uff0c\u5f62\u6210\u82cf\u683c\u62c9\u5e95\u5f0f\u8bfe\u5802\u6c1b\u56f4\uff0c\u5171\u540c\u8fdb\u6b65\u3002\u6b22\u8fce\u611f\u5174\u8da3\u7684\u670b\u53cb\u52a0\u5165\u8bfe\u7a0b\uff01<\/span><\/section>\n

<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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<\/p>\n<\/section>\n

\u63a8\u8350\u9605\u8bfb<\/span><\/strong><\/strong><\/strong><\/p>\n

<\/strong><\/section>\n
1. <\/strong>\u524d\u6cbf\u8fdb\u5c55\uff1aKoopman\u795e\u7ecf\u7b97\u5b50\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b<\/a><\/strong><\/span><\/section>\n
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2. <\/strong>\u5085\u7acb\u53f6\u795e\u7ecf\u7b97\u5b50\uff1a\u5085\u7acb\u53f6\u53d8\u6362\u5e94\u7528\u4e8e\u6df1\u5ea6\u5b66\u4e60\uff0c\u987d\u5f3a\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b<\/a><\/strong><\/span><\/strong><\/strong><\/section>\n
3. <\/strong>Nat. Rev. Phys.\u901f\u9012\uff1a\u795e\u7ecf\u7b97\u5b50\u52a0\u901f\u79d1\u5b66\u4eff\u771f\u4e0e\u8bbe\u8ba1<\/a> <\/strong><\/span><\/section>\n
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4. <\/strong><\/span>\u5f20\u6c5f\uff1a\u7b2c\u4e09\u4ee3\u4eba\u5de5\u667a\u80fd\u6280\u672f\u57fa\u7840\u2014\u2014\u4ece\u53ef\u5fae\u5206\u7f16\u7a0b\u5230\u56e0\u679c\u63a8\u7406 | \u96c6\u667a\u5b66\u56ed\u5168\u65b0\u8bfe\u7a0b<\/a><\/strong><\/strong><\/span><\/section>\n
<\/strong>5.<\/strong> \u9f99\u5e74\u5927\u8fd0\u8d77\uff0c\u5b66\u4e60\u6b63\u5f53\u65f6\uff01\u89e3\u9501\u96c6\u667a\u5168\u7ad9\u5185\u5bb9\uff0c\u5f00\u542f\u65b0\u5e74\u5b66\u4e60\u8ba1\u5212<\/a><\/strong><\/strong><\/strong><\/strong><\/span><\/strong><\/strong><\/section>\n

6. <\/strong>\u52a0\u5165\u96c6\u667a\uff0c\u4e00\u8d77\u590d\u6742\uff01<\/strong><\/span><\/a><\/p>\n


<\/p>\n


<\/p>\n

\u70b9\u51fb\u201c\u9605\u8bfb\u539f\u6587\u201d\uff0c\u62a5\u540d\u8bfe\u7a0b<\/span><\/strong><\/span><\/strong><\/span><\/strong><\/p>\n

<\/mp-style-type><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

\u5bfc\u8bed \u672c\u6587\u9009\u6458\u7f16\u8bd1\u81ea2019\u5e74\u7684\u7efc\u8ff0\u6587\u7ae0 Notes on Koopman Operator Theory\uff0c\u65e8\u5728\u7b80\u8981\u4ecb\u7ecd Koopman \u7b97\u7b26\u7406\u8bba\uff0c\u5e2e\u52a9\u611f\u5174\u8da3\u7684\u8bfb\u8005\u5feb\u901f\u5165\u95e8\u3002\u5e0c\u671b\u6df1\u5165\u4e86\u89e3\u7684\u670b\u53cb\u53ef\u4ee5\u9605\u8bfb\u539f\u6587\u3002 \u597d\u7684\u7406\u8bba\u548c\u5de5\u5177\u4f1a\u5927\u5927\u52a0\u5feb\u79d1\u7814\u8fdb\u5c55\uff0cKoopman \u7b97\u7b26\u6709\u671b\u6210\u4e3a\u5206\u6790\u548c\u9884\u6d4b\u52a8\u6001\u7cfb\u7edf\u7684\u6807\u51c6\u7406\u8bba\u3002…<\/p>\n","protected":false},"author":0,"featured_media":54446,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"special":[],"_links":{"self":[{"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/posts\/54518"}],"collection":[{"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"replies":[{"embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=54518"}],"version-history":[{"count":0,"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/posts\/54518\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/media\/54446"}],"wp:attachment":[{"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=54518"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=54518"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=54518"},{"taxonomy":"special","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fspecial&post=54518"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}