[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki9\/archives\/283258#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki9\/archives\/283258","headline":"\u30cf\u30df\u30eb\u30c8\u30f3-\u30e4\u30b3\u30d3-\u30d9\u30eb\u30de\u30f3\u65b9\u7a0b\u5f0f – Wikipedia","name":"\u30cf\u30df\u30eb\u30c8\u30f3-\u30e4\u30b3\u30d3-\u30d9\u30eb\u30de\u30f3\u65b9\u7a0b\u5f0f – Wikipedia","description":"\u30cf\u30df\u30eb\u30c8\u30f3-\u30e4\u30b3\u30d3-\u30d9\u30eb\u30de\u30f3(HJB)\u65b9\u7a0b\u5f0f\uff08\u30cf\u30df\u30eb\u30c8\u30f3\u2013\u30e4\u30b3\u30d3\u2013\u30d9\u30eb\u30de\u30f3\u307b\u3046\u3066\u3044\u3057\u304d\u3001\u82f1: Hamilton\u2013Jacobi\u2013Bellman equation\uff09\u306f\u3001\u6700\u9069\u5236\u5fa1\u7406\u8ad6\u306e\u6839\u5e79\u3092\u306a\u3059\u504f\u5fae\u5206\u65b9\u7a0b\u5f0f\u3067\u3042\u308b\u3002\u305d\u306e\u89e3\u3092\u300c\u4fa1\u5024\u95a2\u6570(value function)\u300d\u3068\u547c\u3073\u3001\u5bfe\u8c61\u306e\u52d5\u7684\u30b7\u30b9\u30c6\u30e0\u3068\u305d\u308c\u306b\u95a2\u3059\u308b\u30b3\u30b9\u30c8\u95a2\u6570(cost function)\u306e\u6700\u5c0f\u5024\u3092\u4e0e\u3048\u308b\u3002 HJB\u65b9\u7a0b\u5f0f\u306e\u5c40\u6240\u89e3\u306f\u6700\u9069\u6027\u306e\u5fc5\u8981\u6761\u4ef6\u3092\u4e0e\u3048\u308b\u304c\u3001\u5168\u72b6\u614b\u7a7a\u9593\u3067\u89e3\u3051\u3070\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u3092\u4e0e\u3048\u308b\u3002\u89e3\u306f\u958b\u30eb\u30fc\u30d7\u5236\u5fa1\u5247\u3068\u306a\u308b\u304c\u3001\u9589\u30eb\u30fc\u30d7\u89e3\u3082\u5c0e\u3051\u308b\u3002\u4ee5\u4e0a\u306e\u624b\u6cd5\u306f\u78ba\u7387\u30b7\u30b9\u30c6\u30e0\u3078\u3082\u62e1\u5f35\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u307b\u304b\u3001\u53e4\u5178\u7684\u5909\u5206\u554f\u984c\u3001\u4f8b\u3048\u3070\u6700\u901f\u964d\u4e0b\u7dda\u554f\u984c\u3082\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002 HJB\u65b9\u7a0b\u5f0f\u306f1950\u5e74\u4ee3\u306e\u30ea\u30c1\u30e3\u30fc\u30c9\u30fb\u30d9\u30eb\u30de\u30f3\u3068\u305d\u306e\u5171\u540c\u7814\u7a76\u8005\u3092\u5148\u99c6\u3068\u3059\u308b\u300c\u52d5\u7684\u8a08\u753b\u6cd5(Dynamic programming)\u300d\u7406\u8ad6\u306e\u6210\u679c\u3068\u3057\u3066\u5f97\u3089\u308c\u305f[1]\u3002\u305d\u306e\u96e2\u6563\u6642\u9593\u5f62\u5f0f\u306f\u901a\u5e38\u300c\u30d9\u30eb\u30de\u30f3\u65b9\u7a0b\u5f0f\u300d\u3068\u547c\u79f0\u3055\u308c\u308b\u3002 \u9023\u7d9a\u6642\u9593\u306b\u304a\u3044\u3066\u306f\u3001\u53e4\u5178\u7269\u7406\u5b66\u306b\u304a\u3051\u308b\u30cf\u30df\u30eb\u30c8\u30f3-\u30e4\u30b3\u30d3\u65b9\u7a0b\u5f0f (\u30a6\u30a3\u30ea\u30a2\u30e0\u30fb\u30ed\u30fc\u30ef\u30f3\u30fb\u30cf\u30df\u30eb\u30c8\u30f3 (William Rowan Hamilton) \u304a\u3088\u3073\u3001\u30ab\u30fc\u30eb\u30fb\u30b0\u30b9\u30bf\u30d5\u30fb\u30e4\u30b3\u30d6\u30fb\u30e4\u30b3\u30d3 (Carl Gustav Jacob Jacobi)\u306b\u3088\u308b) \u306e\u62e1\u5f35\u5f62\u3068\u307f\u306a\u305b\u308b\u3002 Table","datePublished":"2022-04-23","dateModified":"2022-04-23","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki9\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki9\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/cd810e53c1408c38cc766bc14e7ce26a?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/cd810e53c1408c38cc766bc14e7ce26a?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/35ccef2d3dc751e081375d51c111709d8a1d7ac6","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/35ccef2d3dc751e081375d51c111709d8a1d7ac6","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/jp\/wiki9\/archives\/283258","about":["wiki"],"wordCount":14208,"articleBody":"\u30cf\u30df\u30eb\u30c8\u30f3-\u30e4\u30b3\u30d3-\u30d9\u30eb\u30de\u30f3(HJB)\u65b9\u7a0b\u5f0f\uff08\u30cf\u30df\u30eb\u30c8\u30f3\u2013\u30e4\u30b3\u30d3\u2013\u30d9\u30eb\u30de\u30f3\u307b\u3046\u3066\u3044\u3057\u304d\u3001\u82f1: Hamilton\u2013Jacobi\u2013Bellman equation\uff09\u306f\u3001\u6700\u9069\u5236\u5fa1\u7406\u8ad6\u306e\u6839\u5e79\u3092\u306a\u3059\u504f\u5fae\u5206\u65b9\u7a0b\u5f0f\u3067\u3042\u308b\u3002\u305d\u306e\u89e3\u3092\u300c\u4fa1\u5024\u95a2\u6570(value function)\u300d\u3068\u547c\u3073\u3001\u5bfe\u8c61\u306e\u52d5\u7684\u30b7\u30b9\u30c6\u30e0\u3068\u305d\u308c\u306b\u95a2\u3059\u308b\u30b3\u30b9\u30c8\u95a2\u6570(cost function)\u306e\u6700\u5c0f\u5024\u3092\u4e0e\u3048\u308b\u3002  HJB\u65b9\u7a0b\u5f0f\u306e\u5c40\u6240\u89e3\u306f\u6700\u9069\u6027\u306e\u5fc5\u8981\u6761\u4ef6\u3092\u4e0e\u3048\u308b\u304c\u3001\u5168\u72b6\u614b\u7a7a\u9593\u3067\u89e3\u3051\u3070\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u3092\u4e0e\u3048\u308b\u3002\u89e3\u306f\u958b\u30eb\u30fc\u30d7\u5236\u5fa1\u5247\u3068\u306a\u308b\u304c\u3001\u9589\u30eb\u30fc\u30d7\u89e3\u3082\u5c0e\u3051\u308b\u3002\u4ee5\u4e0a\u306e\u624b\u6cd5\u306f\u78ba\u7387\u30b7\u30b9\u30c6\u30e0\u3078\u3082\u62e1\u5f35\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u307b\u304b\u3001\u53e4\u5178\u7684\u5909\u5206\u554f\u984c\u3001\u4f8b\u3048\u3070\u6700\u901f\u964d\u4e0b\u7dda\u554f\u984c\u3082\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002HJB\u65b9\u7a0b\u5f0f\u306f1950\u5e74\u4ee3\u306e\u30ea\u30c1\u30e3\u30fc\u30c9\u30fb\u30d9\u30eb\u30de\u30f3\u3068\u305d\u306e\u5171\u540c\u7814\u7a76\u8005\u3092\u5148\u99c6\u3068\u3059\u308b\u300c\u52d5\u7684\u8a08\u753b\u6cd5(Dynamic programming)\u300d\u7406\u8ad6\u306e\u6210\u679c\u3068\u3057\u3066\u5f97\u3089\u308c\u305f[1]\u3002\u305d\u306e\u96e2\u6563\u6642\u9593\u5f62\u5f0f\u306f\u901a\u5e38\u300c\u30d9\u30eb\u30de\u30f3\u65b9\u7a0b\u5f0f\u300d\u3068\u547c\u79f0\u3055\u308c\u308b\u3002\u9023\u7d9a\u6642\u9593\u306b\u304a\u3044\u3066\u306f\u3001\u53e4\u5178\u7269\u7406\u5b66\u306b\u304a\u3051\u308b\u30cf\u30df\u30eb\u30c8\u30f3-\u30e4\u30b3\u30d3\u65b9\u7a0b\u5f0f (\u30a6\u30a3\u30ea\u30a2\u30e0\u30fb\u30ed\u30fc\u30ef\u30f3\u30fb\u30cf\u30df\u30eb\u30c8\u30f3 (William Rowan Hamilton) \u304a\u3088\u3073\u3001\u30ab\u30fc\u30eb\u30fb\u30b0\u30b9\u30bf\u30d5\u30fb\u30e4\u30b3\u30d6\u30fb\u30e4\u30b3\u30d3 (Carl Gustav Jacob Jacobi)\u306b\u3088\u308b) \u306e\u62e1\u5f35\u5f62\u3068\u307f\u306a\u305b\u308b\u3002  Table of Contents\u6700\u9069\u5236\u5fa1\u554f\u984c[\u7de8\u96c6]HJB\u65b9\u7a0b\u5f0f[\u7de8\u96c6]HJB\u65b9\u7a0b\u5f0f\u306e\u5c0e\u51fa[\u7de8\u96c6]HJB\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5[\u7de8\u96c6]\u78ba\u7387\u30b7\u30b9\u30c6\u30e0\u3078\u306e\u62e1\u5f35[\u7de8\u96c6]\u30cf\u30df\u30eb\u30c8\u30f3\u2013\u30e4\u30b3\u30d3\u2013\u30d9\u30eb\u30de\u30f3\u2013\u30a2\u30a4\u30b6\u30c3\u30af\u30b9\u65b9\u7a0b\u5f0f[\u7de8\u96c6]\u6700\u9069\u505c\u6b62\u554f\u984c[\u7de8\u96c6]Linear Quadratic Gaussian (LQG)\u5236\u5fa1\u3078\u306e\u5fdc\u7528[\u7de8\u96c6]HJB\u65b9\u7a0b\u5f0f\u306e\u5fdc\u7528[\u7de8\u96c6]\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u6ce8\u91c8[\u7de8\u96c6]\u51fa\u5178[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u95a2\u9023\u6587\u732e[\u7de8\u96c6]\u6700\u9069\u5236\u5fa1\u554f\u984c[\u7de8\u96c6]\u6642\u9593\u7bc4\u56f2\u00a0[0,T]{displaystyle [0,T]} \u306b\u304a\u3051\u308b\u6b21\u5f0f\u306e\u6700\u9069\u5236\u5fa1\u554f\u984c\u306b\u3064\u3044\u3066\u8003\u3048\u308b\u3002  V(x(0),0)=minu{\u222b0TC[x(t),u(t)]dt+D[x(T)]}{displaystyle V(x(0),0)=min _{u}left{int _{0}^{T}!!!C[x(t),u(t)],dt;+;D[x(T)]right}}\u3053\u3053\u3067\u3001C[\u00a0]{displaystyle C[~]}\u306f\u3001\u30b9\u30ab\u30e9\u30fc\u306e\u5fae\u5206\u30b3\u30b9\u30c8\u95a2\u6570(cost rate function)\u3001D[\u00a0]{displaystyle D[~]}\u306f\u7d42\u7aef\u72b6\u614b\u306e\u671b\u307e\u3057\u3055\u3001\u306a\u3044\u3057\u7d4c\u6e08\u4fa1\u5024\u3092\u4e0e\u3048\u308b\u95a2\u6570\u3001x(t){displaystyle x(t)}\u306f\u30b7\u30b9\u30c6\u30e0\u306e\u72b6\u614b\u30d9\u30af\u30c8\u30eb\u3001x(0){displaystyle x(0)}\u306f\u305d\u306e\u521d\u671f\u5024\u3001u(t){displaystyle u(t)}\u306f\u6211\u3005\u304c\u6c42\u3081\u305f\u3044\u3068\u8003\u3048\u3066\u3044\u308b\u6642\u9593 0\u2264t\u2264T{displaystyle 0leq tleq T} \u306e\u5236\u5fa1\u5165\u529b\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u3002\u5bfe\u8c61\u3068\u3059\u308b\u30b7\u30b9\u30c6\u30e0\u306f\u4ee5\u4e0b\u306e\u30c0\u30a4\u30ca\u30df\u30af\u30b9\u306b\u5f93\u3046\u3068\u3059\u308b\u3002x\u02d9(t)=F[x(t),u(t)]{displaystyle {dot {x}}(t)=F[x(t),u(t)],}\u3053\u3053\u3067\u3001\u00a0F[\u00a0]{displaystyle F[~]}\u306f\u30b7\u30b9\u30c6\u30e0\u306e\u72b6\u614b\u306e\u6642\u9593\u767a\u5c55\u3092\u4e0e\u3048\u308b\u95a2\u6570\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u3002HJB\u65b9\u7a0b\u5f0f[\u7de8\u96c6]\u3053\u306e\u30b7\u30b9\u30c6\u30e0\u306b\u95a2\u3059\u308b\u30cf\u30df\u30eb\u30c8\u30f3-\u30e4\u30b3\u30d3-\u30d9\u30eb\u30de\u30f3(HJB)\u65b9\u7a0b\u5f0f\u306f\u6b21\u306e\u504f\u5fae\u5206\u65b9\u7a0b\u5f0f\u3067\u8868\u3055\u308c\u308b\u3002V\u02d9(x,t)+minu{\u2207V(x,t)\u22c5F(x,u)+C(x,u)}=0{displaystyle {dot {V}}(x,t)+min _{u}left{nabla V(x,t)cdot F(x,u)+C(x,u)right}=0}\u305d\u306e\u7d42\u7aef\u6761\u4ef6\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3002V(x,T)=D(x),{displaystyle V(x,T)=D(x),,}\u3053\u3053\u3067\u3001\u00a0a\u22c5b{displaystyle acdot b}\u00a0\u306f\u30d9\u30af\u30c8\u30eb a{displaystyle a} \u3068 b{displaystyle b} \u306e\u5185\u7a4d\u3001\u00a0\u2207{displaystyle nabla } \u306f\u00a0\u52fe\u914d \u30aa\u30da\u30ec\u30fc\u30bf\u30fc\u3002\u4e0a\u8ff0\u306e\u65b9\u7a0b\u5f0f\u306b\u73fe\u308c\u308b\u672a\u77e5\u306e\u30b9\u30ab\u30e9\u30fc\u95a2\u6570\u00a0V(x,t){displaystyle V(x,t)} \u3092\u30d9\u30eb\u30de\u30f3\u306e\u300c\u4fa1\u5024\u95a2\u6570\u300d\u3068\u547c\u3076\u3002V(x,t){displaystyle V(x,t)}\u306f\u521d\u671f\u72b6\u614b x{displaystyle x}\u3068\u6642\u523b t{displaystyle t}\u304b\u3089\u3001\u6642\u523bT{displaystyle T}\u307e\u3067\u30b7\u30b9\u30c6\u30e0\u3092\u6700\u9069\u306b\u5236\u5fa1\u3057\u305f\u5834\u5408\u306b\u5f97\u3089\u308c\u308b\u6700\u5c0f\u30b3\u30b9\u30c8\u3092\u8868\u3057\u3066\u3044\u308b\u3002HJB\u65b9\u7a0b\u5f0f\u306e\u5c0e\u51fa[\u7de8\u96c6]\u76f4\u611f\u7684\u306b\u306f\u3001HJB\u65b9\u7a0b\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5c0e\u51fa\u3067\u304d\u308b\u3002V(x(t),t){displaystyle V(x(t),t)}\u00a0\u304c\u4e0a\u8ff0\u306e\u4fa1\u5024\u95a2\u6570\uff08\u3059\u306a\u308f\u3061\u6700\u5c0f\u30b3\u30b9\u30c8\uff09\u3067\u3042\u3063\u305f\u3068\u3059\u308c\u3070\u3001Richard-Bellman\u306e\u300c\u6700\u9069\u6027\u306e\u539f\u7406\u300d\u304b\u3089\u3001\u00a0\u6642\u9593 t{displaystyle t}\u00a0\u304b\u3089\u00a0t+dt{displaystyle t+dt}\u307e\u3067\u306e\u5909\u5316\u306f\u6b21\u5f0f\u3067\u8868\u73fe\u3067\u304d\u308b\u3002V(x(t),t)=minu{\u222btt+dtC(x(s),u(s))ds+V(x(t+dt),t+dt)}.{displaystyle V(x(t),t)=min _{u}left{int _{t}^{t+dt}!!!!!!!!C(x(s),u(s)),ds;;+;;V(x(t!+!dt),t!+!dt)right}.}\u53f3\u8fba\u306e\u7b2c\u4e8c\u9805\u304c\u6b21\u306e\u3088\u3046\u306b\u00a0\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u00a0\u3067\u304d\u308b\u3053\u3068\u306b\u6ce8\u76ee\u3057\u3088\u3046\u3002V(x(t+dt),t+dt)=V(x(t),t)+V\u02d9(x(t),t)dt+\u2207V(x(t),t)\u22c5x\u02d9(t)dt+o(dt),{displaystyle V(x(t!+!dt),t!+!dt);=;V(x(t),t)+{dot {V}}(x(t),t),dt+nabla V(x(t),t)cdot {dot {x}}(t),dt;+;o(dt),}o(dt){displaystyle o(dt)} \u306f\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306e2\u6b21\u4ee5\u4e0a\u306e\u9ad8\u6b21\u9805\u3092\u30e9\u30f3\u30c0\u30a6\u8a18\u6cd5\u3067\u8868\u73fe\u3057\u305f\u3082\u306e\u306a\u306e\u3067\u7121\u8996\u3059\u308b\u3053\u3068\u306b\u3059\u308b\u3002\u4fa1\u5024\u95a2\u6570\u306e\u5f0f\u306b\u3053\u308c\u3092\u4ee3\u5165\u3057\u305f\u5f8c\u3001 \u4e21\u8fba\u306e\u00a0V(x(t),t){displaystyle V(x(t),t)} \u3092\u76f8\u6bba\u3057\u3001dt{displaystyle dt}\u3067\u5272\u3063\u3066\u30bc\u30ed\u306b\u6f38\u8fd1\u3055\u305b\u308c\u3070\u3001\u4e0a\u8ff0\u306eHJB\u65b9\u7a0b\u5f0f\u304c\u5c0e\u51fa\u3067\u304d\u308b\u3002HJB\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5[\u7de8\u96c6]HJB\u65b9\u7a0b\u5f0f\u306f\u901a\u5e38\u3001t=T{displaystyle t=T}\u00a0\u304b\u3089\u00a0 t=0{displaystyle t=0}\u3078\u5411\u304b\u3063\u3066\u6642\u9593\u3092\u9061\u308b\u65b9\u5411\u3067\u89e3\u304b\u308c\u308b\u3002\u5168\u72b6\u614b\u7a7a\u9593\u3067\u89e3\u304b\u308c\u305f\u5834\u5408\u3001HJB\u65b9\u7a0b\u5f0f\u306f\u6700\u9069\u6027\u306e\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u3092\u4e0e\u3048\u308b[2]\u3002 V{displaystyle V}\u306b\u95a2\u3057\u3066\u89e3\u3051\u308c\u3070\u3001\u305d\u3053\u304b\u3089\u30b3\u30b9\u30c8\u95a2\u6570\u3092\u6700\u5c0f\u5316\u3059\u308b\u5236\u5fa1\u5165\u529b\u00a0u{displaystyle u}\u304c\u5f97\u3089\u308c\u308b\u3002u(t)=arg\u2061minu{\u2207V(x,t)\u22c5F(x,u)+C(x,u)}{displaystyle u(t)=arg min _{u}left{nabla V(x,t)cdot F(x,u)+C(x,u)right}}\u4e00\u822c\u7684\u306bHJB\u65b9\u7a0b\u5f0f\u306f\u53e4\u5178\u7684\u306a\uff08\u306a\u3081\u3089\u304b\u306a\uff09\u89e3\u3092\u3082\u305f\u306a\u3044\u3002 \u305d\u306e\u3088\u3046\u306a\u5834\u5408\u306e\u89e3\u6cd5\u3068\u3057\u3066\u3001\u7c98\u6027\u89e3 (Pierre-Louis Lions \u3068\u3000Michael Crandall)\u3001\u30df\u30cb\u30de\u30c3\u30af\u30b9\u89e3 (Andrei Izmailovich Subbotin \u9732) \u306a\u3069\u304c\u5b58\u5728\u3059\u308b\u3002\u00a0\u78ba\u7387\u30b7\u30b9\u30c6\u30e0\u3078\u306e\u62e1\u5f35[\u7de8\u96c6]\u30b7\u30b9\u30c6\u30e0\u306e\u5236\u5fa1\u554f\u984c\u306b\u30d9\u30eb\u30de\u30f3\u306e\u6700\u9069\u6027\u539f\u7406\u3092\u9069\u7528\u3057\u3001\u6700\u9069\u5236\u5fa1\u6226\u7565\u3092\u6642\u9593\u3092\u9061\u308b\u5f62\u3067\u89e3\u304f\u624b\u6cd5\u306f\u3001\u78ba\u7387\u5fae\u5206\u65b9\u7a0b\u5f0f\u3067\u8868\u73fe\u3055\u308c\u308b\u30b7\u30b9\u30c6\u30e0\u306e\u5236\u5fa1\u554f\u984c\u3078\u62e1\u5f35\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u4e0a\u8ff0\u306e\u554f\u984c\u306b\u826f\u304f\u4f3c\u305f\u6b21\u306e\u554f\u984c\u3092\u8003\u3048\u3088\u3046\u3002minE\u2061{\u222b0TC(t,Xt,ut)dt+D(XT)}{displaystyle min operatorname {E} left{int _{0}^{T}C(t,X_{t},u_{t}),dt+D(X_{T})right}}\u3053\u3053\u3067\u306f\u3001\u6700\u9069\u5316\u3057\u305f\u3044\uff081\u6b21\u5143\uff09\u78ba\u7387\u904e\u7a0b (Xt)t\u2208[0,T]{displaystyle (X_{t})_{tin [0,T]},!} \u3068\u305d\u306e\u5165\u529b (ut)t\u2208[0,T]{displaystyle (u_{t})_{tin [0,T]},!} \u3092\u8003\u3048\u308b\u3002\u78ba\u7387\u904e\u7a0b (Xt)t\u2208[0,T]{displaystyle (X_{t})_{tin [0,T]},!} \u306f\u6b21\u306e\u78ba\u7387\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u5f93\u3046\u62e1\u6563\u904e\u7a0b\uff08\u82f1\u8a9e\u7248\uff09\u3067\u3042\u308b\u3068\u3059\u308b\u3002dXt=\u03bc(t,Xt,ut)dt+\u03c3(t,Xt,ut)dwt,{displaystyle dX_{t}=mu (t,X_{t},u_{t})dt+sigma (t,X_{t},u_{t})dw_{t},}\u305f\u3060\u3057\u3001(wt)t\u2208[0,T]{displaystyle (w_{t})_{tin [0,T]},!} \u306f\u6a19\u6e96\u30d6\u30e9\u30a6\u30f3\u904b\u52d5\uff08\u30a6\u30a3\u30fc\u30ca\u30fc\u904e\u7a0b\uff09\u3067\u3042\u308a\u3001\u03bc,\u03c3{displaystyle mu ,;sigma } \u306f\u6a19\u6e96\u7684\u306a\u4eee\u5b9a\u3092\u6e80\u305f\u3059\u53ef\u6e2c\u95a2\u6570\u3067\u3042\u308b\u3068\u3059\u308b\u3002\u76f4\u89b3\u7684\u306b\u89e3\u91c8\u3059\u308c\u3070\u3001\u72b6\u614b\u5909\u6570 X{displaystyle X} \u306f\u77ac\u9593\u7684\u306b \u03bcdt{displaystyle mu dt} \u3060\u3051\u5897\u6e1b\u3059\u308b\u304c\u3001\u540c\u6642\u306b\u6b63\u898f\u30ce\u30a4\u30ba \u03c3dwt{displaystyle sigma dw_{t}} \u306e\u5f71\u97ff\u3082\u53d7\u3051\u3066\u3044\u308b\u3002\u3053\u306e\u6642\u3001\u30d9\u30eb\u30de\u30f3\u306e\u6700\u9069\u6027\u539f\u7406\u3092\u7528\u3044\u3001\u6b21\u306b\u4fa1\u5024\u95a2\u6570 V(Xt,t){displaystyle V(X_{t},t)} \u3092\u4f0a\u85e4\u306e\u30eb\u30fc\u30eb\u3092\u4f7f\u3063\u3066\u5c55\u958b\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u4fa1\u5024\u95a2\u6570\u306b\u3064\u3044\u3066\u306eHJB\u65b9\u7a0b\u5f0f\u304c\u5f97\u3089\u308c\u308b\u3002\u2212\u2202V(x,t)\u2202t\u2212minu{AuV(x,t)+C(t,x,u)}=0,{displaystyle -{frac {partial V(x,t)}{partial t}}-min _{u}left{{mathcal {A}}^{u}V(x,t)+C(t,x,u)right}=0,}\u3053\u3053\u3067\u3001Au{displaystyle {mathcal {A}}^{u}} \u306f\u7121\u9650\u5c0f\u751f\u6210\u4f5c\u7528\u7d20\uff08\u82f1\u8a9e\u7248\uff09\u3068\u547c\u3070\u308c\u308b\u95a2\u6570\u4f5c\u7528\u7d20\u3067\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b\u3002AuV(x,t):=\u03bc(t,x,u)\u2202V(x,t)\u2202x+12(\u03c3(t,x,u))2\u22022V(x,t)\u2202x2{displaystyle {mathcal {A}}^{u}V(x,t):=mu (t,x,u){frac {partial V(x,t)}{partial x}}+{frac {1}{2}}{Big (}sigma (t,x,u){Big )}^{2}{frac {partial ^{2}V(x,t)}{partial x^{2}}}}\u975e\u78ba\u7387\u7684\u306a\u8a2d\u5b9a\u306e\u4e0b\u3067\u306f\u5b58\u5728\u3057\u306a\u304b\u3063\u305f \u03c32\/2{displaystyle sigma ^{2}\/2} \u306b\u4fa1\u5024\u95a2\u6570 V(x,t){displaystyle V(x,t)} \u306e x{displaystyle x} \u306b\u3064\u3044\u3066\u306e2\u56de\u5fae\u5206\u3092\u639b\u3051\u305f\u9805\u304c\u8db3\u3055\u308c\u3066\u3044\u308b\u304c\u3001\u3053\u306e\u9805\u306f\u4f0a\u85e4\u306e\u516c\u5f0f\u306b\u3088\u308a\u751f\u3058\u3066\u3044\u308b\u3002\u7d42\u7aef\u6761\u4ef6\u306f\u6b21\u5f0f\u3067\u3042\u308b\u3002V(x,T)=D(x).{displaystyle V(x,T)=D(x),!.}\u30e9\u30f3\u30c0\u30e0\u6027\u304c\u6d88\u3048\u305f\u3053\u3068\u306b\u6ce8\u610f\u3057\u3088\u3046\u3002 \u3053\u306e\u5834\u5408\u3001V{displaystyle V,!} \u306e\u89e3\u306f\u5143\u306e\u554f\u984c\u306e\u6700\u9069\u89e3\u306e\u5019\u88dc\u3067\u3042\u308b\u306b\u3059\u304e\u305a\u3001\u3055\u3089\u306a\u308b\u691c\u8a3c\u304c\u5fc5\u8981\u3067\u3042\u308b[\u6ce8\u91c8 1]\u3002 \u3053\u306e\u6280\u8853\u306f\u91d1\u878d\u5de5\u5b66\u306b\u304a\u3044\u3066\u3001\u5e02\u5834\u306b\u304a\u3051\u308b\u6700\u9069\u6295\u8cc7\u6226\u7565\u3092\u5b9a\u3081\u308b\u305f\u3081\u5e83\u304f\u7528\u3044\u3089\u308c\u3066\u3044\u308b \uff08\u4f8b\uff1a \u30de\u30fc\u30c8\u30f3\u306e\u30dd\u30fc\u30c8\u30d5\u30a9\u30ea\u30aa\u554f\u984c)\u3002\u30cf\u30df\u30eb\u30c8\u30f3\u2013\u30e4\u30b3\u30d3\u2013\u30d9\u30eb\u30de\u30f3\u2013\u30a2\u30a4\u30b6\u30c3\u30af\u30b9\u65b9\u7a0b\u5f0f[\u7de8\u96c6]\u30d7\u30ec\u30a4\u30e4\u30fc1\u30682\u306e\u4e8c\u4eba\u304b\u3089\u306a\u308b\u975e\u5354\u529b\u30bc\u30ed\u30b5\u30e0\u30b2\u30fc\u30e0\u3092\u8003\u3048\u308b[3]\u3002\u30df\u30cb\u30de\u30c3\u30af\u30b9\u539f\u7406\u306f\u3053\u306e\u8a2d\u5b9a\u3067\u3082\u6210\u7acb\u3057\u3001\u30d7\u30ec\u30a4\u30e4\u30fc1\u306e\u6700\u9069\u5236\u5fa1\u554f\u984c\u306f\u30d7\u30ec\u30a4\u30e4\u30fc1\u306e\u5236\u5fa1\u5909\u6570\u3092 u{displaystyle u} \u3068\u3057\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b\u3002maxuminvE\u2061{\u222b0TC(t,Xt,ut,vt)dt+D(XT)}{displaystyle max _{u}min _{v}operatorname {E} left{int _{0}^{T}C(t,X_{t},u_{t},v_{t}),dt+D(X_{T})right}}\u305f\u3060\u3057\u3001\u72b6\u614b\u5909\u6570 (Xt)t\u2208[0,T]{displaystyle (X_{t})_{tin [0,T]},!} \u306f\u6b21\u306e\u78ba\u7387\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u5f93\u3046\u3068\u3059\u308b\u3002dXt=\u03bc(t,Xt,ut,vt)dt+\u03c3(t,Xt,ut,vt)dwt{displaystyle dX_{t}=mu (t,X_{t},u_{t},v_{t})dt+sigma (t,X_{t},u_{t},v_{t})dw_{t}}\u3053\u306e\u554f\u984c\u306b\u304a\u3044\u3066\u306f\u30d7\u30ec\u30a4\u30e4\u30fc2\u306e\u5236\u5fa1\u5909\u6570 v{displaystyle v} \u304c\u554f\u984c\u306b\u5c0e\u5165\u3055\u308c\u3066\u3044\u308b\u3002\u30d7\u30ec\u30a4\u30e4\u30fc1\u306e\u554f\u984c\u306e\u4fa1\u5024\u95a2\u6570\u306f\u4ee5\u4e0b\u306e\u30cf\u30df\u30eb\u30c8\u30f3\u2013\u30e4\u30b3\u30d3\u2013\u30d9\u30eb\u30de\u30f3\u2013\u30a2\u30a4\u30b6\u30c3\u30af\u30b9\u65b9\u7a0b\u5f0f\uff08HJBI\u65b9\u7a0b\u5f0f\u3001\u82f1: Hamilton\u2013Jacobi\u2013Bellman\u2013Isaacs equation (HJBI equation)\uff09[\u6ce8\u91c8 2]\u306e\u7c98\u6027\u89e3\u3068\u306a\u308b\u3002\u2212\u2202V(x,t)\u2202t\u2212maxuminu{Au,vV(x,t)+C(t,x,u,v)}=0,{displaystyle -{frac {partial V(x,t)}{partial t}}-max _{u}min _{u}left{{mathcal {A}}^{u,v}V(x,t)+C(t,x,u,v)right}=0,}\u3053\u3053\u3067\u3001Au,v{displaystyle {mathcal {A}}^{u,v}} \u306f\u7121\u9650\u5c0f\u751f\u6210\u4f5c\u7528\u7d20\u3067\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b\u3002Au,vV(x,t):=\u03bc(t,x,u,v)\u2202V(x,t)\u2202x+12(\u03c3(t,x,u,v))2\u22022V(x,t)\u2202x2{displaystyle {mathcal {A}}^{u,v}V(x,t):=mu (t,x,u,v){frac {partial V(x,t)}{partial x}}+{frac {1}{2}}{Big (}sigma (t,x,u,v){Big )}^{2}{frac {partial ^{2}V(x,t)}{partial x^{2}}}}\u7d42\u7aef\u6761\u4ef6\u306f\u6b21\u5f0f\u3067\u3042\u308b\u3002V(x,T)=D(x).{displaystyle V(x,T)=D(x),!.}HJBI\u65b9\u7a0b\u5f0f\u306b\u542b\u307e\u308c\u308b u,v{displaystyle u,v} \u306b\u3064\u3044\u3066\u306e\u6700\u5927\u5316\u554f\u984c\u3068\u6700\u5c0f\u5316\u554f\u984c\u306e\u89e3\u304c\u3053\u306e\u30b2\u30fc\u30e0\u306e(\u30de\u30eb\u30b3\u30d5)\u30ca\u30c3\u30b7\u30e5\u5747\u8861\u3068\u306a\u308b\u3002\u6700\u9069\u505c\u6b62\u554f\u984c[\u7de8\u96c6]\u6b21\u306e\u6700\u9069\u505c\u6b62\u554f\u984c\u3092\u8003\u3048\u308b[4]\u3002max\u03c4E\u2061{\u222b0\u03c4C(t,Xt)dt+D(XT)1{\u03c4=T}+F(\u03c4,X\u03c4)1{\u03c4}{displaystyle {;cdot ;}} \u5185\u306e\u4e8b\u8c61\u304c\u8d77\u304d\u308c\u30701\u3001\u305d\u3046\u3067\u306a\u3051\u308c\u30700\u3092\u8fd4\u3059\u95a2\u6570\u3067\u3042\u308b\u3002\u72b6\u614b\u5909\u6570 (Xt)t\u2208[0,T]{displaystyle (X_{t})_{tin [0,T]},!} \u306f\u6b21\u306e\u78ba\u7387\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u5f93\u3046\u3068\u3059\u308b\u3002dXt=\u03bc(t,Xt)dt+\u03c3(t,Xt)dwt{displaystyle dX_{t}=mu (t,X_{t})dt+sigma (t,X_{t})dw_{t}}\u3059\u308b\u3068\u3001\u4fa1\u5024\u95a2\u6570 V(x,t){displaystyle V(x,t)} \u306f\u6b21\u306eHJB\u65b9\u7a0b\u5f0f\u306e\u7c98\u6027\u89e3\u3068\u306a\u308b\u3002min{\u2212\u2202V(x,t)\u2202t\u2212AV(x,t)\u2212C(t,x),V(x,t)\u2212F(t,x)}=0,{displaystyle min left{-{frac {partial V(x,t)}{partial t}}-{mathcal {A}}V(x,t)-C(t,x),quad V(x,t)-F(t,x)right}=0,}\u305f\u3060\u3057\u3001\u7121\u9650\u5c0f\u751f\u6210\u4f5c\u7528\u7d20 A{displaystyle {mathcal {A}}} \u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b\u3002AV(x,t):=\u03bc(t,x)\u2202V(x,t)\u2202x+12(\u03c3(t,x))2\u22022V(x,t)\u2202x2{displaystyle {mathcal {A}}V(x,t):=mu (t,x){frac {partial V(x,t)}{partial x}}+{frac {1}{2}}{Big (}sigma (t,x){Big )}^{2}{frac {partial ^{2}V(x,t)}{partial x^{2}}}}\u7d42\u7aef\u6761\u4ef6\u306f\u6b21\u5f0f\u3067\u3042\u308b\u3002V(x,T)=D(x).{displaystyle V(x,T)=D(x),!.}\u6700\u9069\u5236\u5fa1\u3068\u306a\u308b\u505c\u6b62\u6642\u523b\uff08\u82f1\u8a9e\u7248\uff09\u306f\u6b21\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u03c4\u2217:=min{inf{t\u2208[0,T]:V(Xt,t)=F(t,Xt)},T}{displaystyle tau ^{*}:=min{inf{tin [0,T];:;V(X_{t},t)=F(t,X_{t})},;T}}\u6700\u9069\u505c\u6b62\u554f\u984c\u306f\u30a2\u30e1\u30ea\u30ab\u30f3\u30aa\u30d7\u30b7\u30e7\u30f3\u306e\u4fa1\u683c\u4ed8\u3051\u554f\u984c\u306a\u3069\u3067\u73fe\u308c\u308b\u3002Linear Quadratic Gaussian (LQG)\u5236\u5fa1\u3078\u306e\u5fdc\u7528[\u7de8\u96c6]\u4e00\u4f8b\u3068\u3057\u3066\u3001\u4e8c\u6b21\u5f62\u5f0f\u306e\u30b3\u30b9\u30c8\u95a2\u6570\u3092\u6301\u3064\u7dda\u5f62\u78ba\u7387\u30b7\u30b9\u30c6\u30e0\u306e\u554f\u984c\u3092\u6271\u3063\u3066\u307f\u3088\u3046\u3002 \u4ee5\u4e0b\u306e\u30c0\u30a4\u30ca\u30df\u30af\u30b9\u3092\u6301\u3064\u30b7\u30b9\u30c6\u30e0\u3092\u8003\u3048\u308b\u3002dxt=(axt+but)dt+\u03c3dwt,{displaystyle dx_{t}=(ax_{t}+bu_{t})dt+sigma dw_{t},}\u5fae\u5206\u30b3\u30b9\u30c8\u95a2\u6570\u304c\u3001C(xt,ut)=r(t)ut2\/2+q(t)xt2\/2{displaystyle C(x_{t},u_{t})=r(t)u_{t}^{2}\/2+q(t)x_{t}^{2}\/2} \u3067\u4e0e\u3048\u3089\u308c\u308b\u3068\u3059\u308c\u3070\u3001HJB\u65b9\u7a0b\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u4e0e\u3048\u3089\u308c\u308b\u3002\u2212\u2202V(x,t)\u2202t=12q(t)x2+\u2202V(x,t)\u2202xax\u2212b22r(t)(\u2202V(x,t)\u2202x)2+12\u03c32\u22022V(x,t)\u2202x2.{displaystyle -{frac {partial V(x,t)}{partial t}}={frac {1}{2}}q(t)x^{2}+{frac {partial V(x,t)}{partial x}}ax-{frac {b^{2}}{2r(t)}}left({frac {partial V(x,t)}{partial x}}right)^{2}+{frac {1}{2}}sigma ^{2}{frac {partial ^{2}V(x,t)}{partial x^{2}}}.}\u4e8c\u6b21\u5f62\u5f0f\u306e\u4fa1\u5024\u95a2\u6570\u3092\u4eee\u5b9a\u3059\u308b\u4e8b\u306b\u3088\u308a\u3001\u901a\u5e38\u306eLQG\u5236\u5fa1\u3068\u540c\u69d8\u306b\u3001\u4fa1\u5024\u95a2\u6570\u306e\u30d8\u30b7\u30a2\u30f3\u306b\u95a2\u3059\u308b\u4e00\u822c\u7684\u306a\u00a0\u30ea\u30ab\u30c3\u30c1\u65b9\u7a0b\u5f0f\u3092\u5f97\u308b\u3053\u3068\u304c\u51fa\u6765\u308b\u3002HJB\u65b9\u7a0b\u5f0f\u306e\u5fdc\u7528[\u7de8\u96c6]HJB\u65b9\u7a0b\u5f0f\u306f\u9023\u7d9a\u6642\u9593\u306e\u6700\u9069\u5236\u5fa1\u306b\u304a\u3044\u3066\u57fa\u672c\u3068\u306a\u308b\u65b9\u7a0b\u5f0f\u3067\u3042\u308a\u3001\u69d8\u3005\u306a\u5206\u91ce\u3067\u5fdc\u7528\u3055\u308c\u3066\u3044\u308b\u3002\u4f8b\u3048\u3070\u3001\u306a\u3069\u304c\u6319\u3052\u3089\u308c\u308b\u3002\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u6ce8\u91c8[\u7de8\u96c6]\u51fa\u5178[\u7de8\u96c6]^ Bellman, R. E. (1957). Dynamic Programming. Princeton, NJ\u00a0^ Bertsekas, Dimitri P. (2005). Dynamic Programming and Optimal Control. Athena Scientific\u00a0^ Fleming, W.; Souganidis, P. (1989), \u201cOn the Existence of Value Functions of Two-Player, Zero-Sum Stochastic Differential Games\u201d, Indiana Univ. Math. J. 38 (2):  293\u2013314, http:\/\/www.iumj.indiana.edu\/docs\/38015\/38015.asp 2016\u5e749\u670824\u65e5\u95b2\u89a7\u3002\u00a0^ Pham, Huy\u00ean (2009), Continuous-Time Stochastic Control and Optimization with Financial Applications, Springer, ISBN\u00a03540894993\u00a0\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u51fa\u5178\u306f\u5217\u6319\u3059\u308b\u3060\u3051\u3067\u306a\u304f\u3001\u811a\u6ce8\u306a\u3069\u3092\u7528\u3044\u3066\u3069\u306e\u8a18\u8ff0\u306e\u60c5\u5831\u6e90\u3067\u3042\u308b\u304b\u3092\u660e\u8a18\u3057\u3066\u304f\u3060\u3055\u3044\u3002\u8a18\u4e8b\u306e\u4fe1\u983c\u6027\u5411\u4e0a\u306b\u3054\u5354\u529b\u3092\u304a\u9858\u3044\u3044\u305f\u3057\u307e\u3059\u3002\uff082016\u5e7410\u6708\uff09\u95a2\u9023\u6587\u732e[\u7de8\u96c6]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki9\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki9\/archives\/283258#breadcrumbitem","name":"\u30cf\u30df\u30eb\u30c8\u30f3-\u30e4\u30b3\u30d3-\u30d9\u30eb\u30de\u30f3\u65b9\u7a0b\u5f0f – Wikipedia"}}]}]