??
|
|
|
- えりか のしろ
- 5 years ago
- Views:
Transcription
1 ( ) /119
2 = (ISS) ISS ISS ISS iss-clf iss-clf ISS = (ISS) FB /119
3 = (ISS) ISS ISS ISS iss-clf iss-clf ISS R + : 0 K: γ: R + R + K γ γ(0) = 0 K : γ: R + R + K γ K γ(r) (r ) FB K K K K R + R /119
4 ( ) = (ISS) ISS ISS ISS iss-clf iss-clf ISS KL: β: R + R + R + KL s β(,s) K r β(r, ) β(r, s) 0, (s ) r FB s /119
5 = (ISS) = (ISS) ISS ISS ISS iss-clf iss-clf ISS FB =(Input-to-State Stability,ISS): ẋ = f(x, t)+g 1 (x, t)d x =0 t = KL β K χ t 0 (> 0) x(t 0 ) [0, ) d( ) ( ) x(t) β( x(t 0 ),t t 0 )+χ sup t 0 τ t d(τ), t>t 0 β χ d /119
6 = (ISS) ISS ISS ISS iss-clf iss-clf ISS FB ẋ = Ax + Bd x(t) = exp(a(t t 0 ))x(t 0 )+ t t 0 exp(a(t τ))bd(τ)dτ x(t) exp{(max Re(λ i (A)))(t t 0 )} x(t 0 ) + i t 0 t 0 τ t exp(a(τ))b dτ sup d(τ) A ISS /119
7 ISS = (ISS) ISS ISS ISS iss-clf iss-clf ISS FB ISS (Sontag and Wang 1995): ẋ = f(x, t)+g 1 (x, t)d f(0,t)=0 V : R n R + R + γ 1 ( x ) V (x, t) γ 2 ( x ) V = V t + V x f(x, t)+ V x g 1(x, t)d γ 3 ( x ), for x ρ( d ) γ 1, γ 2, ρ K γ 3 K ISS χ = γ 1 1 γ 2 ρ /119
8 ISS = (ISS) ISS ISS ISS iss-clf iss-clf ISS FB γ 1 ( x ) V (x) γ 2 ( x ) V ISS (Sontag and Wang 1996): ẋ = f(x)+g 1 (x)d f(0) = 0 V : R n R + V a( x )+b( d ) a( ), b( ) K ISS d a( ) K x V < /119
9 - = (ISS) ISS ISS ISS iss-clf iss-clf ISS FB - (ISS) x(t) β( x(t 0 ),t t 0 )+χ( d ) - (IISS) α( x(t) ) β( x(t 0 ),t t 0 )+ - t t t 0 α( x(τ) )dτ β( x(t 0 ),t t 0 )+ t 0 χ( d(τ) )dτ t t 0 χ( d(τ) )dτ - (ISS) - ISS IISS /119
10 ISS = (ISS) ISS ISS ISS iss-clf iss-clf ISS FB : ẋ = f(x)+g 1 (x)d + g 2 (x)u d u : u = α(x) ẋ = f(x)+g 1 (x)d = {f(x)+g 2 (x)α(x)} + g 1 (x)d ISS α(x) / 119
11 iss-clf = (ISS) ISS ISS ISS iss-clf iss-clf ISS FB ISS-control Lyapunov function (iss-clf): V iss-clf K ρ d x ρ( d ) inf u R m{l f V + L g1 Vd+ L g2 Vu} < 0, x 0 ISS u V (x) / 119
12 iss-clf = (ISS) ISS ISS ISS iss-clf iss-clf ISS FB Iss-clf : V iss-clf K ρ L g2 V (x) =0 L f V (x)+ L g1 V (x) ρ 1 ( x ) < 0, x 0 : d = {ρ 1 ( x )/ L g1 V }(L g1 V ) T : x ρ( d ) inf {L f V + L g1 Vd+ L g2 Vu} inf {L f V + L g1 V d + L g2 Vu} u u inf {L f V + L g1 V ρ 1 ( x )+L g2 Vu} < 0, x 0 u / 119
13 iss = (ISS) ISS ISS ISS iss-clf iss-clf ISS FB u g 2 (x) =0 ISS-clf 3 ISS : V K ρ L f V (x)+ L g1 V (x) ρ 1 ( x ) < 0, x 0 ISS / 119
14 ISS = (ISS) ISS ISS ISS iss-clf iss-clf ISS FB ẋ = f(x)+g 1 (x)d + g 2 (x)u ISS u = α(x) scp iss-clf : Sontag and Wang : Sontag-type () ISS Sontag-type u = α(x) ω + ω 2 +(L g2 V (L g2 V ) T ) 2 = L g2 V (L g2 V ) T (L g2 V ) T, L g2 V 0 0, L g2 V =0 ω = L f V + L g1 V ρ 1 ( x ) / 119
15 ISS ( ) = (ISS) ISS ISS ISS iss-clf iss-clf ISS Sontag-type V L g2 V 0 : x ρ( d ) V L g2 V ( d ρ 1 ( x )) ω 2 +(L g2 V (L g2 V ) T ) 2 < 0 FB L g2 V =0 : Iss-clf x ρ( d ) V L f V + L g2 V ρ 1 ( x ) < 0 ISS scp Sontag-type / 119
16 = (ISS) FB / 119
17 (1) = (ISS) FB : J = F (x) min : x R n : g(x) =0, g(x) R m : h i (x) 0, i =1,...,l (h(x) 0 ) Lagrange : J ext (x, λ, μ, γ) =F (x)+λ T g(x)+μ T (h(x) (γ1,...,γ 2 l 2 ) T ) λ R m, μ R l, γ R l J ext = J / 119
18 (2) = (ISS) FB J ext (x, λ, μ, γ) =0 x : J ext x = F x + λt g x λ : g(x) =0 μ : h(x) (γ1,...,γ 2 l 2)T =0 γ : μ i γ i =0 + μt h x =0 n + m +2l n + m +2l KKT : () h(x) 0, μ 0, diag.(μ)h(x) = / 119
19 (3) = (ISS) FB / g(x) = 0 J x = λ g x J ext x =0, J ext = J + λg J ext / λ =0 J(x) / 119
20 = (ISS) FB ( T, ): : ẋ = f(x)+g(x)u (Bolza ): J(x(0); u( )) = F (x(t )) + : K(u) 0 R l : x(0) = x 0 x(0) u(t) T 0 L(x, u)dt : / 119
21 (1) = (ISS) F (x) : F (x + dx) =F (x)+ df dx dx + O(dx2 ) dx dx F (x( )) x + dx x(t)+δx(t) FB ±x(t) x(t) x(t) + ±x(t) t / 119
22 (2) = (ISS) FB F (x( )) = φ(x(t )) + F (x + δx) F (x) = φ δx(t )+ x(t ) = φ δx(t )+ x(t ) T 0 [ L ẋ δx ( ) T 0 0 L(x(t), ẋ(t), t)dt L L δx + x ẋ δẋdt+ O(δx2 ) ] T T [ L + x d dt 0 ( )] L δx dt + O(δx 2 ) ẋ L φ (x(0), ẋ(0), 0) = 0, ẋ x(t ) + L (x(t ), ẋ(t ),T)=0, ẋ L x d ( ) L =0 (Euler-Lagrange ) dt ẋ / 119
23 (1) = (ISS) FB ( : Brachistocrone): ( ) A B y(x) (Bernoulli Challenge 1696) y A mg T = a 0 B x 1+y 2 dx min 2gy v = 2gy = 1+y 2 ẋ 1+y dt = 2 2gy dx / 119
24 (2) = (ISS) Euler-Lagrange : 1+y 2 +2yy =0 (Cycloid) : A FB B / 119
25 (1) = (ISS) FB p(t) R n, μ(t) R l, γ(t) R l : J ext = F (x(t )) T 0 T 0 T 0 L(x, u)dt p(t) T (f(x)+g(x)u ẋ)dt μ(t) T (K(u) (γ 2 1,...,γ 2 l ) T )dt / 119
26 (2) = (ISS) FB : ( ) H(x, p, u) =L(x, u)+p T (f(x)+g(x)u) J ext = F (x(t )) + T 0 p T ẋdt + T 0 T 0 H(x, p, u)dt μ(t) T (K(u) diag(γ)γ)dt / 119
27 (1) = (ISS) : x, p, u, μ, γ x(t) =x (t)+δx(t), μ(t) =μ (t)+δμ(t) p(t) =p (t)+δp(t), γ(t) =γ (t)+δγ(t) u(t) =u (t)+δu(t) Jext FB J ext = F (x (T )+δx(t )) + T 0 T 0 H(x + δx, p + δp, u + δu)dt (p + δp) T (ẋ + δẋ)dt + T 0 ((γ 1 δγ 1 ) 2,...,(γ l δγ l ) 2 ) T }dt (μ + δμ) T {K(u + δu) / 119
28 (2) = (ISS) FB 1 δx(0) = 0 J ext = Jext + F x (x (T ))δx(t) T ( ) H H H + δx + δp + x p u δu 0 p T (T )δx(t ) T μ T 0 T 0 T 0 K u T 0 u=u ṗ T δxdt δudt {K(u ) diag(γ )γ } T δμdt μ T diag(γ )δγdt + dt x=x,p=p,u=u T 0 ẋ T δpdt / 119
29 (3) = (ISS) FB [ H(x δp : ẋ,p,u ] T ) = p [ H(x δx : ṗ,p,u ) = x [ F(x δx(t ) : p (T )) (T )= x (T ) : x (0) = x 0 δu : H(x,p,u ) u + μ T K(u ) u =0 δμ : K(u ) diag(γ )γ =0 δγ : μ T diag(γ )=0 ] T ] T / 119
30 (1) = (ISS) FB x p : : : [ H ẋ = p [ H ṗ = x p(t )= ] T x(0) = x 0 ] T [ F ] T x(t ) / 119
31 (2) = (ISS) FB u Remark: H K + μt u u =0, K(u) diag(γ)γ =0, μ T diag(γ) =0 x p H(x, p, u) min, subject to K(u) 0 u : u K(u) 0 H Hamiltonian L abnormal multiplier / 119
32 (1) = (ISS) FB Finite Horizon x(0) = x 0 : 1. u x p u (x, p) 2. : ] T [ H ẋ = p [ H ṗ = x ] T x(0) = x 0, p(t )= [ F ] x(t ) u = u (x, p) 3. u (x, p) / 119
33 (2) = (ISS) FB 2. 2 u u 1 u 0 ( = ) / 119
34 (1) = (ISS) FB (u ) Ḣ = H x ẋ + H p ṗ + H u u ( = H x H p T = μ T dk(u) dt + H p H x K i (u) > 0 μ i =0 K i (u) =0dK i /dt =0 t Ḣ =0 T ) μ T K u u () u H u H? / 119
35 (2) = (ISS) FB x p () u H u H H Infinite Horizon H = / 119
36 = (ISS) ( ) H HJ Riccati OFP FB / 119
37 ( ) = (ISS) ( T, ): ( ) H HJ Riccati OFP : ẋ = f(x)+g(x)u (Bolza ): J(x(0); u( )) = F (x(t )) + : K(u) 0 R l : x(0) = x 0 T 0 L(x, u)dt min FB / 119
38 = (ISS) ( ) H HJ Riccati OFP [0,T] {x ( ),u ( )} [τ,t], x (τ) {ˆx ( ), û ( )} ˆx (t) =x (t), û (t) =u (t), t [τ,t] u*(t) x*( ) b = x*( ) u*(t) b FB 0 T u (t) T t x(0) / 119
39 = (ISS) ( ) H HJ Riccati OFP FB : u = α(x, t) (T = ) (F (x(t )) = 0) u = α(x) f(0) = 0, L(0, 0) = 0, L(x, u) 0 T = u U(x(0)) = {u( ) : x(0) x(t) 0 } u / 119
40 = (ISS) ( ) H HJ Riccati OFP FB L(x, u) =L 0 (x)+ 1 2 ut Ru L 0 (x) ( ), R ( ) : 0 H(x, p, u) =p T (f(x)+g(x)u)+l 0 (x)+ 1 2 ut Ru p = V x T, V(0) = 0 H(x, p, u)dt + V (x(0)) = J(x(0); u), u U(x(0)) / 119
41 H = (ISS) ( ) H HJ Riccati OFP FB u H(x, p, u) =p T f(x)+l 0 (x) 1 2 pt g(x)r 1 g(x) T p (u + R 1 g(x) T p) T R(u + R 1 g(x) T p) ( ) H(x, p, u) u = R 1 g(x) T p H (x, p) =p T f(x)+l 0 (x) 1 2 pt g(x)r 1 g(x) T p H(x, p, u) H (x, p) / 119
42 Hamilton-Jacobi (1) = (ISS) ( ) H HJ Riccati OFP FB Hamilton-Jacobi (HJ ): ( H x, V x T ) = V x f(x)+l 0(x) 1 2 V x g(x)r 1 g(x) T V x V (x) T =0 p T = V/ x / 119
43 Hamilton-Jacobi (2) = (ISS) ( ) H HJ Riccati OFP FB V (x(0)) 1 (u + R 1 g(x) T V ) T R(u + R 1 g(x) T V 2 0 x x + V (x(0)) = J(x(0),u), u U(x(0)) J(x(0),u) u = u (x) = R 1 g(x) T V x J V (x(0)) V (x) T T T )dt / 119
44 (1) = (ISS) ( ) H HJ Riccati OFP V (x) Lyapunov V (x) : u = R 1 g(x) T ( V/ x) T V (x(t)) : V = V x (f(x)+g(x)u )= L(x, u ) = L 0 (x) 1 2 u (x) T Ru (x) 0 FB / 119
45 (2) = (ISS) ( ) H HJ Riccati OFP FB L(x, u (x)) = 0 u (x) =0and L 0 (x) =0 : L(x, u) L 0 (x) () L 0 (x) V (x) HJ u = u (x) = R 1 g(x) T V/ x T J(x(0); u )=V (x(0)) V (x) / 119
46 = (ISS) ( ) H HJ Riccati OFP FB 1. u =0 L 0 (x) 2. Hamilton-Jacobi : H (x, V x )=0 V (x) 3. u = u (x) = R 1 g(x) T V x T / 119
47 (1) = (ISS) ( ) H HJ Riccati OFP : ẋ = Ax + Bu : J = 1 2 : V (x) = 1 2 xt Px, P > 0 Hamilton-Jacobi : 0 x T Qx + u T Ru dt min, 1 2 xt ( PA+ A T P PBR 1 B T P + Q ) x =0 Q > 0,R>0 FB / 119
48 (2) = (ISS) ( ) H HJ Riccati OFP Riccati : PA+ A T P PBR 1 B T P + Q =0 P (> 0) u = R 1 B T Px (A, B) Riccati FB / 119
49 (1) = (ISS) ( ) H HJ Riccati OFP FB : J = T 0 L 0 (x)+ 1 2 ut Rudt, R > 0 u: u (x, p) = R 1 g(x) T p ẋ = Ax + Bu + O((x, u) 2 ) J = 1 2 T (A, B), Q>0 0 x T Qx + u T Ru dt + O(x 3 ) : ( ) ( ) [ d x x A BR A dt p H = 1 B T p Q A T ]( ) x p A H / 119
50 (2) = (ISS) ( ) H HJ Riccati OFP FB : A H A H (λi A H ) A H λ λ ( ) f g =0 λ (f T,g T ) T ( g T,f T )( λi A H )=0 A H λ ( g T,f T ) (H ) n n / 119
51 (1) = (ISS) ( ) H HJ Riccati OFP n n p(t ) = 0 p x(0) = x 0 x Stable manifold T = p( ) =0 FB Unstable manifold / 119
52 (2) = (ISS) ( ) H HJ Riccati OFP FB x p T = ω(x) : u = R 1 g(x) T ω(x) : p T = ω(x) ω ω = V x V (x) p =[ V/ x] T / 119
53 Riccati (1) = (ISS) ( ) H HJ Riccati OFP FB Riccati PA+ A T P PBR 1 B T P + Q =0 ( = ) ( ) : ( ) ( ) [ ]( d x x A BR = A dt p H = 1 B T x p Q A p) T p =[ V/ x] T = Px P / 119
54 Riccati (2) = (ISS) ( ) H HJ Riccati OFP FB 1. A H : [ ] [ ] S1 S1 A H = Λ S 2 S 2 Λ: n 2. (x T,p T ) T ( ) ( ) [ ] x x S1 = = k p Px x = S 1 k, Px = S 2 k k Px = S 2 S 1 1 x Riccati P = S 2 S 1 1 S / 119
55 OFP (1) = (ISS) ( ) H HJ Riccati OFP FB HJ V = L f V + L g V u L g V u L gv (x)r 1 L g V (x) T û = R 1/2 u, ŷ = R 1/2 L g V (x) T = R 1/2 u (x) 0 V û T ŷ + 1 2ŷT ŷ û ŷ OFP( 1/2) + { u^ u(= u*) x y^ R {1/2 R {1/2 L g V(x) T Plant :, :60, : / 119
56 OFP (2) = (ISS) ( ) H HJ Riccati OFP FB : 2 (1) 1 ẋ = f(x)+g(x)u u = k(x) L 0 (x) 0 (L 0 (0) = 0) J = (2) 2 0 L 0 (x)+ 1 2 ut Ru dt ẋ = f(x)+ĝ(x)û = f(x)+g(x)r 1/2 û y = R 1/2 k(x) OFP( 1/2) / 119
57 OFP (3) = (ISS) ( ) H HJ Riccati OFP FB (1) 2 OFP(-1/2) (1) 2 ẋ = f(x) g(x)k(x) y = R 1/2 k(x) ẋ = f(x), y = R 1/2 k(x) (1) (2) (2) (1) (2) Hill=Moylan q(x) S(x) L f S = 1 2 q(x)t q(x)+ 1 2 k(x)t Rk(x) L g S = k(x) T R 1/2 S(x) =V (x), 2L 0 (x) =q(x) T q(x) HJ k(x) =0 L 0 (x) =0 u = k(x) / 119
58 (1) = (ISS) ( ) H HJ Riccati OFP L 0 (x) OFP( 1/2) y u = y L 0 (x) : L 0 (x) FB / 119
59 (2) = (ISS) ( ) H HJ Riccati OFP FB ( ): clf V (x) u = k 1 (x) = (1/2)R(x) 1 (L g V ) T (R(x) ) V (x) 2 u = k 2 (x) = R(x) 1 (L g V ) T : u = k 1 (x) V = L f V + L g V k 1 = W (x) < 0 (x 0) V = L f V + LĝV û = W (x)+lĝv(û ŷ/2) < ŷ (û T 1 ) (x 0) 2ŷ OFP( 1/2) / 119
60 (3) = (ISS) ( ) H HJ Riccati OFP FB u = R(x) 1 L G V T (R(x) > 0) Jurdjevic-Quinn Sontag Jurdjevic-Quinn L g V (x) =0 x R(x) 1 R(x) / 119
61 (4) = (ISS) ( ) H HJ Riccati OFP FB Sontag clf V = 1 { } L f V L f V 2 2 +(L g V L g V T ) 2 < 0 (x 0) L g V (x) =0 Sontag : u = α S(x) =α S (x) c L g V T (c>0) (û ŷ ) u α S (x) c / 119
62 = (ISS) L 2 = H HJI FB H FB / 119
63 (1) = (ISS) : L 2 = H HJI FB H : (=H ) H L 2 L 2 FB / 119
64 (2) = (ISS) L 2 = H HJI FB FB (s) (s) G 0 (s) G 0 (s) { : Δ(s) Δ(jω) <h(ω) h(ω) Δ(s) Δ(s) =0 1+G 0 (jω) >h(ω) / 119
65 (3) = (ISS) + { (s) L 2 = H HJI FB FB G 0 (s) Δ(jω) 1 1+G 0 (jω) < 1 Δ(jω) < 1+G 0(jω) 1+G 0 (jω) >h(ω) / 119
66 (4) = (ISS) L 2 = H HJI FB FB h(ω) L L / 119
67 L 2 = (ISS) L 2 = H HJI FB FB? : ẋ = f(x)+g(x)w z = h(x) x R n :, w R m :, z R p : f, g, h f(0) = 0, h(0) = 0 L 2 : t t 0 z 2 dt γ 2 γ>0 x(t 0 )=0 t t 0 w 2 dt, w L 2 L c, t>t 0 c γ L 2 L 2 γ / 119
68 (1) = (ISS) L 2 = H HJI FB FB V (x) s(w, z) x(t 0 )=0 t t 0 s(w, z)dt 0, t t 0 γ L 2 V (x) s(w, z) =γ 2 w 2 z / 119
69 (2) = (ISS) L 2 = H HJI FB FB ( ) : ( V s(...)) L f V + L g V w γ 2 w 2 h(x) 2 = γ 2 w T w h(x) T h(x) w L f V +h T h+ 1 4γ 2 L gv L g V T γ 2 (w 1 2γ 2 L gv T ) T (w 1 2γ 2 L gv T ) 0 w =(1/2γ 2 )L g V T () L f V + h T h +(1/4γ 2 )L g V L g V T w L f V + h T h + 1 4γ 2 L gv L g V T 0 = / 119
70 = = (ISS) L 2 = H HJI FB FB = L f V + h T h + 1 4γ 2 L gv L g V T 0 V (x) γ L 2 1. HJ / 119
71 (1) = (ISS) L 2 = H HJI FB FB V (x) = V (x) w =0 V = L f V h T h 1 4γ 2 L gv L g V T h(x) T h(x) z = h(x) w =0 w =0 HJ L f V + h T h + 1 4γ 2 L gv L g V T < 0 (x 0) w =0 V < 0 (x 0) / 119
72 (2) = (ISS) L 2 = H HJI FB FB z = h(x) w =0 HJ (HJ ) V (x) γ L 2 (V (x) ) V (x) Hamiltonian HJ / 119
73 = (ISS) L 2 = H HJI FB FB ẋ = Ax + Bw, z = Cx γ L 2 : PA+ A T P + CC T + 1 γ 2 PBBT P 0 P γ L 2 : PA+ A T P + CC T + 1 γ 2 PBBT P<0 (x 0) P V (x) =x T Px / 119
74 H (1) = (ISS) L 2 = H HJI FB FB : ẋ = f(x)+g 1 (x)w + g 2 (x)u z = h(x)+j 1 (x)w + j 2 (x)u x R n :, w R m :, u R l :, z R p : x =0 f(0) = 0, h 1 (0) = 0 : w( ) z( ) L 2 γ (> 0) u / 119
75 H (2) = (ISS) L 2 = H HJI FB J(x 0,w,u)= 0 z(τ) 2 γ 2 w(τ) 2 dτ, x(0) = x 0 x 0 =0 J w( ) L 2 u = k 2 (x) FB / 119
76 = (ISS) L 2 = H HJI FB H : u w (= ) w = k 1(x), u = k 2(x) FB J(x 0, w,k 2) J(x 0,k 1,k 2) J(x 0,k 1, u), w; u U(x 0,k 1) k 1, k 2 U(x 0,k 1) w = k 1(x) x 0(t ) u( ) / 119
77 (1) = (ISS) L 2 = H HJI FB FB : H(x, p, w, u) = h(x)+j 1 (x)w + j 2 (x)u 2 γ 2 w 2 + p(f(x)+g 1 (x)w + g 2 (x)u) (w, u) (w (x, p),u (x, p)) (x, p) H(x, p, w, u (x, p)) H(x, p, w (x, p),u (x, p)) H(x, p, w (x, p),u) : x R 1 (x) =γ 2 I j 1 (x) T j 1 (x) > 0 R 2 (x) =j 2 (x) T j 2 (x) > / 119
78 (2) = (ISS) L 2 = H HJI FB FB w (x, p) = 1 2 Λ 1 {(g T 1 Ξg T 2 )p T +2(j T 1 Ξj T 2 )h} u (x, p) = 1 2 Γ 1 {(Ωg T 1 + g T 2 )p T +2(Ωj T 1 + j T 2 )h} Ξ=j T 1 j 2 R 1 2, Ω=jT 2 j 1 R 1 1, Λ=R 1 +Ξj T 2 j 1 > 0, Γ=R 2 +Ωj T 1 j 2 > 0 H(x, p, w, u) =H (x, p) (w w ) T Λ(w w ) + {u u +Ξ T (w w )} T R 2 {u u +Ξ T (w w )} = H (x, p)+(u u ) T Γ(u u ) {w w +Ω T (u u )} T R 1 {w w +Ω T (u u )} / 119
79 Hamilton-Jacobi-Isaacs = (ISS) L 2 = H HJI FB FB Hamilton-Jacobi-Isaacs : H H H ( x, V x ( x, V x,w,u ( x, V x,w ) ( x, V x ) ( x, V x =0 )) 0, ),u 0, w u / 119
80 (1) = (ISS) L 2 = H HJI FB FB : H V (x(t 1 )) V (x 0 )+ V (x(t 1 )) V (x 0 )+ ( x, V ) x,w,u t1 0 t1 0 = dv dt + z 2 γ 2 w 2 z 2 γ 2 w 2 dt 0, z 2 γ 2 w (x, V x ) 2 dt 0, u; w = w (x, V x ) u U(x 0,w (x, V/ x)) V (x) 0 w; u = u (x, V x ) J(x 0,w,u ) V (x 0 ) J(x 0,w,u), w, u U(x 0,w ) / 119
81 (2) = (ISS) L 2 = H HJI FB u U(x 0,w ) u, w V (x) =J(x, u,w ) k 1(x) =w (x, V x ), J(0,w,u ) 0 k 2(x) =u (x, V x ) FB / 119
82 (3) = (ISS) L 2 = H HJI FB FB Hamilton-Jacobi-Isaacs V x f (x)+h T (x)q(x)h(x)+ 1 4 V x { (g1 (x) g 2 (x)ξ T (x)) Λ(x) 1 (g T 1 (x) Ξ(x)g T 2 (x)) g 2 (x)r 2 (x) 1 g T 2 (x) } V x T =0 V (0) = 0, V (x) 0 V (x) f = f g 2 R2 1 jt 2 h 1 +(g 1 g 2 Ξ T )Λ 1 (j1 T Ξj2 T )h Q = I j 2 R2 1 jt 2 +(j 1 j 2 Ξ T )Λ 1 (j1 T Ξj2 T ) 0 u = k 2(x) =u (x, V/ x) w z L 2 γ>0 Hamilton-Jacobi- Isaacs V (x) / 119
83 H (1) = (ISS) L 2 = H HJI FB FB Hamilton-Jacobi-Isaacs Hamilton-Jacobi-Isaacs H (x, V x ) 0, V(0) = 0, V(x) 0 V x (f(x)+g 1(x)w + g 2 (x)u (x, V x )) + z 2 γ 2 w 2 = H (x, V x ) (w w (x, V x ))T R 1 (x)(w w (x, V x )) 0 V (x) γ 2 w 2 z 2 u = k 2 (x) =u (x, V/ x) γ 2 w 2 z 2 V (x) / 119
84 H (2) = (ISS) L 2 = H HJI FB FB z Hamilton-Jacobi-Isaacs V (x) u = k 2 (x) =u (x, V/ x) γ L 2 w =0 γ L 2 w =0 V (x) z 2 z 0 (t ) z =0 x 0 (t ) V (x) HJI / 119
85 = (ISS) FB HJ HJ H / 119
86 (1) = (ISS) FB HJ HJ : L 2? L 2 L 2 Disturbance w Control input u Generalized Plant Evaluated Output z Measurment y State-FB Controller Estimated State x» Observer / 119
87 (2) = (ISS) FB HJ HJ : ẋ = f(x)+g 1 (x)w + g 2 (x)u z = h 1 (x)+j 11 (x)w + j 12 (x)u y = h 2 (x)+j 21 (x)w + j 22 (x)u x R n :, w R m :, u R l : z R p :, y R q : f(0) = 0, h 1 (0) = 0, h 2 (0) = 0 : : ξ = δ(ξ,y) u = α(ξ) w z L 2 γ / 119
88 (3) = (ISS) FB HJ HJ : u = k 2 (ξ) ξ = f(ξ)+g 1 (ξ)k 1 (ξ)+g 2 (ξ)k 2 (ξ) + G(ξ)(y h 2 (ξ) j 21 (ξ)k 1 (ξ) j 22 (ξ)k 2 (ξ)) k 1 (ξ) =w (ξ, V (ξ)/ ξ) G( ) j 11 =0, j 22 = / 119
89 (1) = (ISS) : FB HJ HJ ξ = f(ξ)+g 2 (ξ)k 2 (ξ)+g(ξ)(y h 2 (ξ)) f(ξ) =f(ξ)+g 1 (ξ)k 1 (ξ) h 2 (ξ) =h 2 (ξ)+j 21 (ξ)k 1 (ξ) / 119
90 (2) = (ISS) FB HJ HJ (= + + ): ẋ E = f E (x E )+g E (x E )r z = h E1 (x E ) v = h E2 (x E ) x E =(x T,ξ T ) T :, v: u, r = w k 1 (x): w ( ) f(x)+g f E (x E )= 2 (x)k 2 (ξ) f(ξ)+g 2 (ξ)k 2 (ξ)+g(ξ)( h 2 (x) h 2 (ξ)) [ ] g g E (x E )= 1 (x) G(ξ)j 21 (x) h E1 (x E )=h 1 (x)+j 12 (x)k 2 (ξ) h E2 (x E )=k 2 (x) k 2 (ξ) / 119
91 (1) = (ISS) FB HJ HJ FB Hamilton-Jacobi-Isaacs : V x (f g 2R 1 2 jt 12h 1 )+h T 1 (I j 12 R 1 2 jt 12)h V x { 1 γ 2 g 1g T 1 g 2 R 1 2 gt 2 } V x T 0 V (x) u = k 1 (ξ) V x (f(x)+g 1(x)w + g 2 (x)u)+ z 2 γ 2 w 2 = H(x, V x,w,u)=h (x, V x )+ j 12(x)v 2 γ 2 r 2 j 12 (x)v 2 γ 2 r 2 ( ) / 119
92 (2) = (ISS) FB HJ HJ r j 12 (x)v L 2 γ : W x E (f E (x E )+g E (x E )r) γ 2 r 2 j 12 (x)v 2 2n Hamilton-Jacobi : W f E (x E )+h T x E2(x E )R 2 (x)h E2 (x E ) E + 1 W 4γ 2 g E (x E )g x E(x T E ) W T E x E W (x E ) 0, W(0) = 0 0 G( ) / 119
93 (1) = (ISS) : FB HJ HJ : U(x E )=W (x E )+V (x) U x E (f E (x E )+g E (x E )r) γ 2 w 2 + z 2 U(x E ) γ 2 w 2 z 2 = γ L / 119
94 (2) = (ISS) FB HJ HJ : 1. FB, Hamilton-Jacobi-Isaacs V (x) 2. W (x E ), G(ξ) 2n Hamilton-Jacobi 3. ξ = f(ξ) G(ξ) h 2 (ξ) ξ =0 γ L 2 w =0 x E =(0, 0) T V (x) / 119
95 (3) = (ISS) FB HJ HJ : γ L 2 w =0 w =0 du(x E (t)) dt z 2 = h 1 (x)+j 12 (x)k 2 (ξ) 2 U 0 Ω Ω z 0 Ω x 0 (t ) j 12 (x) Ω h 1 (x)+ j 12 (x)k 2 (ξ) =0(x 0) k 2 (ξ(t)) 0 (t ) Ω ξ = f(ξ) G(ξ) h 2 (ξ) 3 Ω ξ 0 (t ) / 119
96 Hamilton-Jacobi (1) = (ISS) Hamilton-Jacobi 2n Hamilton-Jacobi G( ) FB HJ HJ W (x E )=Q(x ξ) Q(0) = 0, Q(e) 0 e = x ξ Q e ( f(x) f(ξ)+(g 2 (x) g 2 (ξ))k 2 (ξ) G(ξ)( h 2 (x) h 2 (ξ))) +(k 2 (x) k 2 (ξ)) T R 2 (x)(k 2 (x) k 2 (ξ)) + 1 Q 4γ 2 e (g 1(x) j21(x)g T T (ξ)) (g 1 (x) j21(x)g T T (ξ)) T Q T 0 e / 119
97 Hamilton-Jacobi (2) = (ISS) FB HJ HJ x = ξ + e K(e, ξ) Q/ e e=0 =0 K K(0,ξ)=0, e =0 e=0 K(e, ξ) (0, 0) Hessian ( 2 Q f ) e 2 e=0 x (0) G(0) h 2 x (0) + k T 2 (0)R 2 (0) k 2 x x (0) Q 4γ 2 e 2 (g 1 (0) j21(0)g T T (0)) e=0 (g 1 (0) j21(0)g T T (0)) T 2 T Q e 2 e= / 119
98 Hamilton-Jacobi (3) = (ISS) FB HJ HJ 1. FB Hamilton-Jacobi-Isaacs C 3 V (x) 2. ɛ>0 S(e) = Q e ( f(e) G(e) h 2 (e)) + k2 T (e)r 2 (e)k 2 (e) + 1 Q 4γ 2 e (g 1(e) G(e)j 21 (e)) (g 1 (e) G(e)j 21 (e)) T Q e ɛe T e<0, (e 0) T C 3 Q(e) ( 0) W (x E )=Q(x ξ) 2n 2, / 119
99 Hamilton-Jacobi (4) = (ISS) FB HJ HJ 2n Hamilton-Jacobi K(0,ξ)=0, ( K/ e) e=0 =0 R(e, ξ) K(e, ξ) =e T R(e, ξ)e R(e, ξ) < 0 2n Hamilton-Jacobi R(0, 0) (0, 0) K(e, ξ) e Hessian n Hamilton-Jacobi e =0 Hessian 2 S/ e 2 (0, 0) K(e, ξ) e Hessian R(e, ξ) K(e, ξ) e =0 2n Hamilton-Jacobi Hamilton-Jacobi Q e ( f(ξ) G(ξ) h 2 (ξ)) < 0, (ξ 0) e=ξ Q Ω Lyapunov / 119
100 (1) = (ISS) FB HJ HJ HJ S G( ) : R 3 (x) =j 21 (x)j T 21(x) ( ) G(x) : Q(x) x f(x)+k T 2 (x)r 2 (x)k 2 (x)+ 1 Q(x) 4γ 2 x g 1(x)g1 T (x) Q(x) T x + 1 4γ 2 (λ(x) BT (x)r3 1 (x))r 3(x)(λ(x) B T (x)r3 1 (x))t 1 (x)b(x) < 0, (x 0) 4γ 2 BT (x)r3 1 e x λ(x) = Q(x) x G(x), B(x) =2γ2 h2 (x)+j 21 (x)g1 T (x) Q(x) T x / 119
101 (2) = (ISS) FB HJ HJ G(x) HJ S(x) λ(x) B T (x)r 1 3 (x) = Q(x) x G(x) (2γ 2 ht 2 (x)+ Q(x) x G(x) ( ) G(x) Q x L(x) = h T 2 (x) g 1(x)j21(x))R T 3 1 (x) =0 n q L(x) / 119
102 (3) = (ISS) FB HJ HJ Q(x)/ x Q(x) x = x T M(x) h T 2 (x) =x T N(x) G(x): M(x) x =0 L(x) =M 1 (x)n(x) G(x) =(2γ 2 L(x)+g 1 (x)j T 21(x))R 1 3 (x) G(x) / 119
103 Hamilton-Jacobi (1) = (ISS) FB HJ HJ G(x) Hamilton-Jacobi : Q(x) x f(x)+k2 T (x)r 2 (x)k 2 (x)+ 1 Q(x) 4γ 2 x 1 4γ 2 BT (x)r3 1 (x)b(x) < 0, (x 0) Q(x) x ( f(x) g 1 (x)j T 21(x)R 1 3 (x) h 2 (x)) + k T 2 (x)r 2 (x)k 2 (x) γ 2 ht 2 (x)r 1 3 (x) h 2 (x) + 1 Q(x) 4γ 2 x < 0 (x 0) g 1(x)g1 T (x) Q(x) T x g 1(x)(I m j T 21(x)R 1 3 (x)j 21(x))g T 1 (x) Q(x) x T / 119
104 Hamilton-Jacobi (2) = (ISS) FB HJ HJ Hamilton-Jacobi Q( ) M(x) G(x) (Q( ),G(x)) n Hamilton- Jacobi / 119
105 = (ISS) FB St.fdbk.sys. Σ 1 Σ 2 Σ / 119
106 () = (ISS) FB St.fdbk.sys. Σ 1 Σ 2 Σ 3 Strict feedback system Hessian / 119
107 Strict feedback system = (ISS) FB St.fdbk.sys. Σ 1 Σ 2 Σ 3 Strict feedback system: ẋ 1 = f 1 (x 1 )+g 1 (x 1 )x 2 ẋ 2 = f 2 (x 1,x 2 )+g 2 (x 1,x 2 )x 3 ẋ 3 = f 3 (x 1,x 2,x 3 )+g 3 (x 1,x 2,x 3 )x 4. ẋ s 1 = f s 1 (x 1,...,x s 1 )+g s 1 (x 1,...,x s 1 )x n ẋ s = f s (x)+g s (x)u Σ k x k+1 Σ n (=) u dim x 1 dim x 2 =dimx 3 = =dimx s. g 2 (x 1,x 2 ),...,g s (x) / 119
108 Strict feedback system = (ISS) FB St.fdbk.sys. Σ 1 Σ 2 Σ 3 Strict feedback system: ẋ 1 = f 1 (x 1 )+g 1 (x 1 )x 2 ẋ 2 = f 2 (x 1,x 2 )+g 2 (x 1,x 2 )x 3 ẋ 3 = f 3 (x 1,x 2,x 3 )+g 3 (x 1,x 2,x 3 )x 4. ẋ s 1 = f s 1 (x 1,...,x s 1 )+g s 1 (x 1,...,x s 1 )x n ẋ s = f s (x)+g s (x)u Σ k x k+1 Σ n (=) u dim x 1 dim x 2 =dimx 3 = =dimx s. g 2 (x 1,x 2 ),...,g s (x) / 119
109 Strict feedback system = (ISS) FB St.fdbk.sys. Σ 1 Σ 2 Σ 3 Strict feedback system: ẋ 1 = f 1 (x 1 )+g 1 (x 1 )x 2 ẋ 2 = f 2 (x 1,x 2 )+g 2 (x 1,x 2 )x 3 ẋ 3 = f 3 (x 1,x 2,x 3 )+g 3 (x 1,x 2,x 3 )x 4. ẋ s 1 = f s 1 (x 1,...,x s 1 )+g s 1 (x 1,...,x s 1 )x n ẋ s = f s (x)+g s (x)u Σ k x k+1 Σ n (=) u dim x 1 dim x 2 =dimx 3 = =dimx s. g 2 (x 1,x 2 ),...,g s (x) / 119
110 Strict feedback system = (ISS) FB St.fdbk.sys. Σ 1 Σ 2 Σ 3 Strict feedback system: ẋ 1 = f 1 (x 1 )+g 1 (x 1 )x 2 ẋ 2 = f 2 (x 1,x 2 )+g 2 (x 1,x 2 )x 3 ẋ 3 = f 3 (x 1,x 2,x 3 )+g 3 (x 1,x 2,x 3 )x 4. ẋ s 1 = f s 1 (x 1,...,x s 1 )+g s 1 (x 1,...,x s 1 )x n ẋ s = f s (x)+g s (x)u Σ k x k+1 Σ n (=) u dim x 1 dim x 2 =dimx 3 = =dimx s. g 2 (x 1,x 2 ),...,g s (x) / 119
111 Strict feedback system = (ISS) FB St.fdbk.sys. Σ 1 Σ 2 Σ 3 Strict feedback system: ẋ 1 = f 1 (x 1 )+g 1 (x 1 )x 2 ẋ 2 = f 2 (x 1,x 2 )+g 2 (x 1,x 2 )x 3 ẋ 3 = f 3 (x 1,x 2,x 3 )+g 3 (x 1,x 2,x 3 )x 4. ẋ s 1 = f s 1 (x 1,...,x s 1 )+g s 1 (x 1,...,x s 1 )x n ẋ s = f s (x)+g s (x)u Σ k x k+1 Σ n (=) u dim x 1 dim x 2 =dimx 3 = =dimx s. g 2 (x 1,x 2 ),...,g s (x) / 119
112 Σ 1 = (ISS) Σ 1 FB St.fdbk.sys. Σ 1 Σ 2 Σ 3 ẋ 1 = f 1 (x 1 )+g 1 (x 1 )v 1 (x 2 v 1 ) : Σ 1 v 1 = α 1 (x 1 ) V 1 (x 1 ) V 1 x 1 (f 1 + g 2 α 1 ) < 0 (x 1 0) / 119
113 Σ 2 (1) = (ISS) FB St.fdbk.sys. Σ 1 Σ 2 Σ 3 Σ 2 : ẋ 1 = f 1 (x 1 )+g 1 (x 1 )x 2 ẋ 2 = f 2 (x 1,x 2 )+g 2 (x 1,x 2 )v 2 Σ 2 Σ 1 v 1 x 2 z 2 = x 2 α 1 (x 1 ) V 2 (x 1,z 2 )=V 1 (x 1 )+ γ 2 2 zt 2 z / 119
114 Σ 2 (2) : V 2 = V 1 {f 1 + g 1 α 1 } + V 1 g 1 z 2 + γ 2 z2 T x 1 x 1 { = V 1 x 1 {f 1 + g 1 α 1 } + z T 2 g T 1 [ V1 x 1 { (f 2 + g 2 v 2 ) α 1 } (f 1 + g 1 x 2 ) x 1 ] T + γ 2 (f 2 + g 2 v 2 ) γ 2 α 1 x 1 (f 1 + g 1 x 2 ) < 0 (x 1 0) k 2 z 2 } g T 1 [ V1 x 1 ] T + γ 2 (f 2 + g 2 v 2 ) γ 2 α 1 x 1 (f 1 + g 1 x 2 )= k 2 z 2 = k 2 (x 2 + α(x 1 )) v 2 Σ 2 v 2 = α 2 (x 1,x 2 ) / 119
115 Σ 3 (1) = (ISS) FB St.fdbk.sys. Σ 1 Σ 2 Σ 3 Σ 3 : : : ẋ 1 = f 1 (x 1 )+g 1 (x 1 )x 2 ẋ 2 = f 2 (x 1,x 2 )+g 2 (x 1,x 2 )x 3 ẋ 3 = f 2 (x 1,x 2,x 3 )+g 2 (x 1,x 2,x 3 )v 3 z 3 = x 3 α 2 (x 1,x 2 ) V 3 (x 1,z 2,z 3 )=V 2 (x 1,z 2 )+ γ 3 2 zt 3 z 3 ( ) ( ) ( x1 f1 + g x 2 =, f2 (x 1,x 2 )= 1 x 1 0, ḡ 2 (x 1,x2) = g2) x 2 f / 119
116 Σ 3 (2) : V 2 = V 2 { x f 2 +ḡ 2 α 2 } + V 2 ḡ 2 z 3 + γ 3 z3 T 2 x 2 { = V 2 x 2 { f 2 +ḡ 2 α 2 } + z T 3 ḡ T 2 [ V2 x 2 { (f 3 + g 3 v 3 ) α 2 } ( x f 2 +ḡ 2 x 3 ) 2 ] T + γ 3 (f 3 + g 3 v 3 ) γ 3 α 2 x 2 ( f 2 +ḡ 2 x 3 ) < 0 ( x 2 0) k 3 z 3 } ḡ T 2 [ V2 x 2 ] T + γ 3 (f 3 + g 3 v 3 ) γ 3 α 2 x 2 ( f 2 +ḡ 2 x 3 )= k 3 z 3 = k 3 (x 3 + α 2 ) v 3 Σ 3 v 3 = α 3 (x 1,x 2,x 3 ) / 119
117 = (ISS) FB St.fdbk.sys. Σ 1 Σ 2 Σ 3 V 1 Hessian V s Hessian Σ 1 α 1 (x 1 ) Σ 1 u = α s (x) / 119
118 = (ISS) FB ( ) 1 ISS / 119
119 ( ) = (ISS) Vidyasagar : FB ( ) 1 ISS ẋ = f(x) ż = g(z)+γ(x, z)x ẋ = f(x) ż = g(z) γ(x, z) ẋ = f(x), ż = g(z) / 119
120 = (ISS) System 1 System 2 FB ( ) 1 ISS ẋ = f(x) ż = g(z)+γ(x, z)x 1 2 ISS / 119
121 ( ) = (ISS) FB ( ) 1 ISS : Lyapunov V 1 (x), V 2 (z) V 1 = L f V 1 c( x ) V 2 = L g V 2 + L γ V 2 x a( z )+b( x ) a, b, c K K k 1, k 2, k 3, k 4 k 1 ( x ) V 1 (x) k 2 ( x ), k 3 ( z ) V 2 (z) k 4 ( z ) K χ a 1 b( x ) <χ( x ), z 0 Lyapunov W = F (V 1 )+G(V 2 )+V 1 F (v) = v 0 b k 1 2 (s)ds, G(v) = v 0 c χ 1 k 1 4 (s)ds / 119
122 1 = (ISS) FB ( ) 1 ISS : ẋ = f(x) ż = g(z)+γ(x, z)x 1. ẋ = f(x) α( ), M 0 ( ) K, k 0 > 0 x(t) α(m 0 (x(0)) exp( k 0 t)) 2. 2 iiss iiss V 2 (z) V 2 (x, z) a( z )+b( x ), a( ) :, b( ) class-k 3. b(s) : 1 0 b(α(s)) ds < s 1 (α =Id) 2 x / 119
123 ISS = (ISS) w 1 System1 x 1 = f 1 (x 1, x 2, w 1 ) FB ( ) 1 ISS 1,2 ISS K χ 1, χ 1w, χ 2, χ 2w KL β 1, β 2 x 2 x 1 System2 x 2 = f 2 (x 2, x 1, w 2 ) x 1 (t) β 1 (x 1 (0),t)+χ 1 (sup x 2 )+χ 1w (sup w 1 ) x 2 (t) β 2 (x 2 (0),t)+χ 2 (sup x 1 )+χ 2w (sup w 2 ) K η 1 ( ), η 2 ( ) (Id + η 1 ) χ 1 (Id + η 2 ) χ 2 (s) s (Id + η 2 ) χ 2 (Id + η 1 ) χ 1 (s) s s>0 w 2 w 1, w 2 ISS ISS 1 ISS [1] Z.P. Jiang, A.R. Teel, and L. Praly: Small-gain theorem for ISS systems and applications, Mathe. Contr. Signals and Syst., 7, 95/120 (1994) / 119
??
() 2014,,,,, - (ISS) Sontag Hamilton-Jacobi H L 2-, Hamilton-Jacobi-Issacs H,, ISS, ISS Strict Feedback Form,, 2014 1/154 2014 2/154 http://stlab.ssi.ist.hokudai.ac.jp/yuhyama/lecture/tokuron/ 3 2014 3/154
2011de.dvi
211 ( 4 2 1. 3 1.1............................... 3 1.2 1- -......................... 13 1.3 2-1 -................... 19 1.4 3- -......................... 29 2. 37 2.1................................ 37
(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y
[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.
9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,
X G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2
( ) ( 40 )+( 60 ) Schrödinger 3. (a) (b) (c) yoshioka/education-09.html pdf 1
2009 1 ( ) ( 40 )+( 60 ) 1 1. 2. Schrödinger 3. (a) (b) (c) http://goofy.phys.nara-wu.ac.jp/ yoshioka/education-09.html pdf 1 1. ( photon) ν λ = c ν (c = 3.0 108 /m : ) ɛ = hν (1) p = hν/c = h/λ (2) h
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d
S I.. http://ayapin.film.s.dendai.ac.jp/~matuda /TeX/lecture.html PDF PS.................................... 3.3.................... 9.4................5.............. 3 5. Laplace................. 5....
I, II 1, A = A 4 : 6 = max{ A, } A A 10 10%
1 2006.4.17. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 1. 1. 2. 3. 4. 5. 2. ɛ-δ 1. ɛ-n
S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt
S I. x yx y y, y,. F x, y, y, y,, y n http://ayapin.film.s.dendai.ac.jp/~matuda n /TeX/lecture.html PDF PS yx.................................... 3.3.................... 9.4................5..............
f(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f
22 A 3,4 No.3 () (2) (3) (4), (5) (6) (7) (8) () n x = (x,, x n ), = (,, n ), x = ( (x i i ) 2 ) /2 f(x) R n f(x) = f() + i α i (x ) i + o( x ) α,, α n g(x) = o( x )) lim x g(x) x = y = f() + i α i(x )
No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1.
1.1 1. 1.3.1..3.4 3.1 3. 3.3 4.1 4. 4.3 5.1 5. 5.3 6.1 6. 6.3 7.1 7. 7.3 1 1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N
9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L) du (L) = f (9.3) dx (9.) P
9 (Finite Element Method; FEM) 9. 9. P(0) P(x) u(x) (a) P(L) f P(0) P(x) (b) 9. P(L) 9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L)
201711grade1ouyou.pdf
2017 11 26 1 2 52 3 12 13 22 23 32 33 42 3 5 3 4 90 5 6 A 1 2 Web Web 3 4 1 2... 5 6 7 7 44 8 9 1 2 3 1 p p >2 2 A 1 2 0.6 0.4 0.52... (a) 0.6 0.4...... B 1 2 0.8-0.2 0.52..... (b) 0.6 0.52.... 1 A B 2
n ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................
23 7 28 i i 1 1 1.1................................... 2 1.2............................... 3 1.2.1.................................... 3 1.2.2............................... 4 1.2.3 SI..............................
4................................. 4................................. 4 6................................. 6................................. 9.................................................... 3..3..........................
() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.
![() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.](/thumbs/89/98384840.jpg "() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.")
() 6 f(x) [, b] 6. Riemnn [, b] f(x) S f(x) [, b] (Riemnn) = x 0 < x < x < < x n = b. I = [, b] = {x,, x n } mx(x i x i ) =. i [x i, x i ] ξ i n (f) = f(ξ i )(x i x i ) i=. (ξ i ) (f) 0( ), ξ i, S, ε >
1 c Koichi Suga, ISBN
c Koichi Suga, 4 4 6 5 ISBN 978-4-64-6445- 4 ( ) x(t) t u(t) t {u(t)} {x(t)} () T, (), (3), (4) max J = {u(t)} V (x, u)dt ẋ = f(x, u) x() = x x(t ) = x T (), x, u, t ẋ x t u u ẋ = f(x, u) x(t ) = x T x(t
I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT
I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345
( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes )
( 3 7 4 ) 2 2 ) 8 2 954 2) 955 3) 5) J = σe 2 6) 955 7) 9) 955 Statistical-Mechanical Theory of Irreversible Processes 957 ) 3 4 2 A B H (t) = Ae iωt B(t) = B(ω)e iωt B(ω) = [ Φ R (ω) Φ R () ] iω Φ R (t)
II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google
I4 - : April, 4 Version :. Kwhir, Tomoki TA (Kondo, Hirotk) Google http://www.mth.ngoy-u.c.jp/~kwhir/courses/4s-biseki.html pdf 4 4 4 4 8 e 5 5 9 etc. 5 6 6 6 9 n etc. 6 6 6 3 6 3 7 7 etc 7 4 7 7 8 5 59
* 1 2014 7 8 *1 iii 1. Newton 1 1.1 Newton........................... 1 1.2............................. 4 1.3................................. 5 2. 9 2.1......................... 9 2.2........................
K E N Z OU
K E N Z OU 11 1 1 1.1..................................... 1.1.1............................ 1.1..................................................................................... 4 1.........................................
A 2 3. m S m = {x R m+1 x = 1} U + k = {x S m x k > 0}, U k = {x S m x k < 0}, ϕ ± k (x) = (x 0,..., ˆx k,... x m ) 1. {(U ± k, ϕ± k ) 0 k m} S m 1.2.
A A 1 A 5 A 6 1 2 3 4 5 6 7 1 1.1 1.1 (). Hausdorff M R m M M {U α } U α R m E α ϕ α : U α E α U α U β = ϕ α (ϕ β ϕβ (U α U β )) 1 : ϕ β (U α U β ) ϕ α (U α U β ) C M a m dim M a U α ϕ α {x i, 1 i m} {U,
9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x
2009 9 6 16 7 1 7.1 1 1 1 9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x(cos y y sin y) y dy 1 sin
1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =
1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A
Morse ( ) 2014
Morse ( ) 2014 1 1 Morse 1 1.1 Morse................................ 1 1.2 Morse.............................. 7 2 12 2.1....................... 12 2.2.................. 13 2.3 Smale..............................
2011 8 26 3 I 5 1 7 1.1 Markov................................ 7 2 Gau 13 2.1.................................. 13 2.2............................... 18 2.3............................ 23 3 Gau (Le vy
DVIOUT
A. A. A-- [ ] f(x) x = f 00 (x) f 0 () =0 f 00 () > 0= f(x) x = f 00 () < 0= f(x) x = A--2 [ ] f(x) D f 00 (x) > 0= y = f(x) f 00 (x) < 0= y = f(x) P (, f()) f 00 () =0 A--3 [ ] y = f(x) [, b] x = f (y)
Untitled
II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j
08-Note2-web
r(t) t r(t) O v(t) = dr(t) dt a(t) = dv(t) dt = d2 r(t) dt 2 r(t), v(t), a(t) t dr(t) dt r(t) =(x(t),y(t),z(t)) = d 2 r(t) dt 2 = ( dx(t) dt ( d 2 x(t) dt 2, dy(t), dz(t) dt dt ), d2 y(t) dt 2, d2 z(t)
IA hara@math.kyushu-u.ac.jp Last updated: January,......................................................................................................................................................................................
25 7 18 1 1 1.1 v.s............................. 1 1.1.1.................................. 1 1.1.2................................. 1 1.1.3.................................. 3 1.2................... 3
2019 1 5 0 3 1 4 1.1.................... 4 1.1.1......................... 4 1.1.2........................ 5 1.1.3................... 5 1.1.4........................ 6 1.1.5......................... 6 1.2..........................
Grushin 2MA16039T
Grushin 2MA1639T 3 2 2 R d Borel α i k (x, bi (x, 1 i d, 1 k N d N α R d b α = α(x := (αk(x i 1 i d, 1 k N b = b(x := (b i (x 1 i d X = (X t t x R d dx t = α(x t db t + b(x t dt ( 3 u t = Au + V u, u(,
http://www.ns.kogakuin.ac.jp/~ft13389/lecture/physics1a2b/ pdf I 1 1 1.1 ( ) 1. 30 m µm 2. 20 cm km 3. 10 m 2 cm 2 4. 5 cm 3 km 3 5. 1 6. 1 7. 1 1.2 ( ) 1. 1 m + 10 cm 2. 1 hr + 6400 sec 3. 3.0 10 5 kg
2.2 ( y = y(x ( (x 0, y 0 y (x 0 (y 0 = y(x 0 y = y(x ( y (x 0 = F (x 0, y(x 0 = F (x 0, y 0 (x 0, y 0 ( (x 0, y 0 F (x 0, y 0 xy (x, y (, F (x, y ( (
(. x y y x f y = f(x y x y = y(x y x y dx = d dx y(x = y (x = f (x y = y(x x ( (differential equation ( + y 2 dx + xy = 0 dx = xy + y 2 2 2 x y 2 F (x, y = xy + y 2 y = y(x x x xy(x = F (x, y(x + y(x 2
dynamics-solution2.dvi
1 1. (1) a + b = i +3i + k () a b =5i 5j +3k (3) a b =1 (4) a b = 7i j +1k. a = 14 l =/ 14, m=1/ 14, n=3/ 14 3. 4. 5. df (t) d [a(t)e(t)] =ti +9t j +4k, = d a(t) d[a(t)e(t)] e(t)+ da(t) d f (t) =i +18tj
I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )
I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17
webkaitou.dvi
( c Akir KANEKO) ).. m. l s = lθ m d s dt = mg sin θ d θ dt = g l sinθ θ l θ mg. d s dt xy t ( d x dt, d y dt ) t ( mg sin θ cos θ, sin θ sin θ). (.) m t ( d x dt, d y dt ) = t ( mg sin θ cos θ, mg sin
gr09.dvi
.1, θ, ϕ d = A, t dt + B, t dtd + C, t d + D, t dθ +in θdϕ.1.1 t { = f1,t t = f,t { D, t = B, t =.1. t A, tdt e φ,t dt, C, td e λ,t d.1.3,t, t d = e φ,t dt + e λ,t d + dθ +in θdϕ.1.4 { = f1,t t = f,t {
構造と連続体の力学基礎
II 37 Wabash Avenue Bridge, Illinois 州 Winnipeg にある歩道橋 Esplanade Riel 橋6 6 斜張橋である必要は多分無いと思われる すぐ横に道路用桁橋有り しかも塔基部のレストランは 8 年には営業していなかった 9 9. 9.. () 97 [3] [5] k 9. m w(t) f (t) = f (t) + mg k w(t) Newton
v er.1/ c /(21)
12 -- 1 1 2009 1 17 1-1 1-2 1-3 1-4 2 2 2 1-5 1 1-6 1 1-7 1-1 1-2 1-3 1-4 1-5 1-6 1-7 c 2011 1/(21) 12 -- 1 -- 1 1--1 1--1--1 1 2009 1 n n α { n } α α { n } lim n = α, n α n n ε n > N n α < ε N {1, 1,
2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) =
1 1 1.1 I R 1.1.1 c : I R 2 (i) c C (ii) t I c (t) (0, 0) c (t) c(i) c c(t) 1.1.2 (1) (2) (3) (1) r > 0 c : R R 2 : t (r cos t, r sin t) (2) C f : I R c : I R 2 : t (t, f(t)) (3) y = x c : R R 2 : t (t,
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
基礎数学I
I & II ii ii........... 22................. 25 12............... 28.................. 28.................... 31............. 32.................. 34 3 1 9.................... 1....................... 1............
43433 8 3 . Stochastic exponentials...................................... 3. Girsanov s theorem......................................... 4 On the martingale property of stochastic exponentials 5. Gronwall
I, II 1, 2 ɛ-δ 100 A = A 4 : 6 = max{ A, } A A 10
1 2007.4.13. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 0. 1. 1. 2. 3. 2. ɛ-δ 1. ɛ-n
1. x { e 1,..., e n } x = x1 e1 + + x n en = (x 1,..., x n ) X, Y [X, Y ] Intrinsic ( ) Intrinsic M m P M C P P M P M v 3 v : C P R 1
![1. x { e 1,..., e n } x = x1 e1 + + x n en = (x 1,..., x n ) X, Y [X, Y ] Intrinsic ( ) Intrinsic M m P M C P P M P M v 3 v : C P R 1](/thumbs/96/129392387.jpg "1. x { e 1,..., e n } x = x1 e1 + + x n en = (x 1,..., x n ) X, Y [X, Y ] Intrinsic ( ) Intrinsic M m P M C P P M P M v 3 v : C P R 1")
1. x { e 1,..., e n } x = x1 e1 + + x n en = (x 1,..., x n ) X, Y [X, Y ] Intrinsic ( ) Intrinsic M m P M C P P M P M v 3 v : C P R 1 f, g C P, λ R (1) v(f + g) = v(f) + v(g) (2) v(λf) = λv(f) (3) v(fg)
Microsoft Word - 信号処理3.doc
Junji OHTSUBO 2012 FFT FFT SN sin cos x v ψ(x,t) = f (x vt) (1.1) t=0 (1.1) ψ(x,t) = A 0 cos{k(x vt) + φ} = A 0 cos(kx ωt + φ) (1.2) A 0 v=ω/k φ ω k 1.3 (1.2) (1.2) (1.2) (1.1) 1.1 c c = a + ib, a = Re[c],
6kg 1.1m 1.m.1m.1 l λ ϵ λ l + λ l l l dl dl + dλ ϵ dλ dl dl + dλ dl dl 3 1. JIS 1 6kg 1% 66kg 1 13 σ a1 σ m σ a1 σ m σ m σ a1 f f σ a1 σ a1 σ m f 4
35-8585 7 8 1 I I 1 1.1 6kg 1m P σ σ P 1 l l λ λ l 1.m 1 6kg 1.1m 1.m.1m.1 l λ ϵ λ l + λ l l l dl dl + dλ ϵ dλ dl dl + dλ dl dl 3 1. JIS 1 6kg 1% 66kg 1 13 σ a1 σ m σ a1 σ m σ m σ a1 f f σ a1 σ a1 σ m
C : q i (t) C : q i (t) q i (t) q i(t) q i(t) q i (t)+δq i (t) (2) δq i (t) δq i (t) C, C δq i (t 0 )0, δq i (t 1 ) 0 (3) δs S[C ] S[C] t1 t 0 t1 t 0
![C : q i (t) C : q i (t) q i (t) q i(t) q i(t) q i (t)+δq i (t) (2) δq i (t) δq i (t) C, C δq i (t 0 )0, δq i (t 1 ) 0 (3) δs S[C ] S[C] t1 t 0 t1 t 0](/thumbs/92/108844778.jpg "C : q i (t) C : q i (t) q i (t) q i(t) q i(t) q i (t)+δq i (t) (2) δq i (t) δq i (t) C, C δq i (t 0 )0, δq i (t 1 ) 0 (3) δs S[C ] S[C] t1 t 0 t1 t 0")
1 2003 4 24 ( ) 1 1.1 q i (i 1,,N) N [ ] t t 0 q i (t 0 )q 0 i t 1 q i (t 1 )q 1 i t 0 t t 1 t t 0 q 0 i t 1 q 1 i S[q(t)] t1 t 0 L(q(t), q(t),t)dt (1) S[q(t)] L(q(t), q(t),t) q 1.,q N q 1,, q N t C :
B ver B
B ver. 2017.02.24 B Contents 1 11 1.1....................... 11 1.1.1............. 11 1.1.2.......................... 12 1.2............................. 14 1.2.1................ 14 1.2.2.......................
x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s
... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z
y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =
[ ] 9 IC. dx = 3x 4y dt dy dt = x y u xt = expλt u yt λ u u t = u u u + u = xt yt 6 3. u = x, y, z = x + y + z u u 9 s9 grad u ux, y, z = c c : grad u = u x i + u y j + u k i, j, k z x, y, z grad u v =
i
009 I 1 8 5 i 0 1 0.1..................................... 1 0.................................................. 1 0.3................................. 0.4........................................... 3
W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)
3 215 4 27 1 1 u u(x, t) u tt a 2 u xx, a > (1) D : {(x, t) : x, t } u (, t), u (, t), t (2) u(x, ) f(x), u(x, ) t 2, x (3) u(x, t) X(x)T (t) u (1) 1 T (t) a 2 T (t) X (x) X(x) α (2) T (t) αa 2 T (t) (4)
M3 x y f(x, y) (= x) (= y) x + y f(x, y) = x + y + *. f(x, y) π y f(x, y) x f(x + x, y) f(x, y) lim x x () f(x,y) x 3 -
M3............................................................................................ 3.3................................................... 3 6........................................... 6..........................................
20 6 4 1 4 1.1 1.................................... 4 1.1.1.................................... 4 1.1.2 1................................ 5 1.2................................... 7 1.2.1....................................
<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63>
MATLAB/Simulink による現代制御入門 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/9241 このサンプルページの内容は, 初版 1 刷発行当時のものです. i MATLAB/Simulink MATLAB/Simulink 1. 1 2. 3. MATLAB/Simulink
t = h x z z = h z = t (x, z) (v x (x, z, t), v z (x, z, t)) ρ v x x + v z z = 0 (1) 2-2. (v x, v z ) φ(x, z, t) v x = φ x, v z
I 1 m 2 l k 2 x = 0 x 1 x 1 2 x 2 g x x 2 x 1 m k m 1-1. L x 1, x 2, ẋ 1, ẋ 2 ẋ 1 x = 0 1-2. 2 Q = x 1 + x 2 2 q = x 2 x 1 l L Q, q, Q, q M = 2m µ = m 2 1-3. Q q 1-4. 2 x 2 = h 1 x 1 t = 0 2 1 t x 1 (t)
: 2005 ( ρ t +dv j =0 r m m r = e E( r +e r B( r T 208 T = d E j 207 ρ t = = = e t δ( r r (t e r r δ( r r (t e r ( r δ( r r (t dv j =
72 Maxwell. Maxwell e r ( =,,N Maxwell rot E + B t = 0 rot H D t = j dv D = ρ dv B = 0 D = ɛ 0 E H = μ 0 B ρ( r = j( r = N e δ( r r = N e r δ( r r = : 2005 ( 2006.8.22 73 207 ρ t +dv j =0 r m m r = e E(
t, x (4) 3 u(t, x) + 6u(t, x) u(t, x) + u(t, x) = 0 t x x3 ( u x = u x (4) u t + 6uu x + u xxx = 0 ) ( ): ( ) (2) Riccati ( ) ( ) ( ) 2 (1) : f
: ( ) 2008 5 31 1 f(t) t (1) d 2 f(t) + f(t) = 0 dt2 f(t) = sin t f(t) = cos t (1) 1 (2) d dt f(t) + f(t)2 = 0 (1) (2) t (c ) (3) 2 2 u(t, x) c2 u(t, x) = 0 t2 x2 1 (1) (1) 1 t, x (4) 3 u(t, x) + 6u(t,
ばらつき抑制のための確率最適制御
( ) http://wwwhayanuemnagoya-uacjp/ fujimoto/ 2011 3 9 11 ( ) 2011/03/09-11 1 / 46 Outline 1 2 3 4 5 ( ) 2011/03/09-11 2 / 46 Outline 1 2 3 4 5 ( ) 2011/03/09-11 3 / 46 (1/2) r + Controller - u Plant y
第10章 アイソパラメトリック要素
June 5, 2019 1 / 26 10.1 ( ) 2 / 26 10.2 8 2 3 4 3 4 6 10.1 4 2 3 4 3 (a) 4 (b) 2 3 (c) 2 4 10.1: 3 / 26 8.3 3 5.1 4 10.4 Gauss 10.1 Ω i 2 3 4 Ξ 3 4 6 Ξ ( ) Ξ 5.1 Gauss ˆx : Ξ Ω i ˆx h u 4 / 26 10.2.1
i 6 3 ii 3 7 8 9 3 6 iii 5 8 5 3 7 8 v...................................................... 5.3....................... 7 3........................ 3.................3.......................... 8 3 35
2009 IA I 22, 23, 24, 25, 26, a h f(x) x x a h
009 IA I, 3, 4, 5, 6, 7 7 7 4 5 h fx) x x h 4 5 4 5 1 3 1.1........................... 3 1........................... 4 1.3..................................... 6 1.4.............................. 8 1.4.1..............................
2016 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 16 2 1 () X O 3 (O1) X O, O (O2) O O (O3) O O O X (X, O) O X X (O1), (O2), (O3) (O2) (O3) n (O2) U 1,..., U n O U k O k=1 (O3) U λ O( λ Λ) λ Λ U λ O 0 X 0 (O2) n =
2 1 x 2 x 2 = RT 3πηaN A t (1.2) R/N A N A N A = N A m n(z) = n exp ( ) m gz k B T (1.3) z n z = m = m ρgv k B = erg K 1 R =
1 1 1.1 1827 *1 195 *2 x 2 t x 2 = 2Dt D RT D = RT N A 1 6πaη (1.1) D N A a η 198 *3 ( a =.212µ) *1 Robert Brown (1773-1858. *2 Albert Einstein (1879-1955 *3 Jean Baptiste Perrin (187-1942 2 1 x 2 x 2
ft. ft τfτdτ = e t.5.. fx = x [ π, π] n sinnx n n=. π a π a, x [ π, π] x = a n cosnx cosna + 4 n=. 3, x [ π, π] x 3 π x = n sinnx. n=.6 f, t gt n 3 n
![ft. ft τfτdτ = e t.5.. fx = x [ π, π] n sinnx n n=. π a π a, x [ π, π] x = a n cosnx cosna + 4 n=. 3, x [ π, π] x 3 π x = n sinnx. n=.6 f, t gt n 3 n](/thumbs/92/108735706.jpg "ft. ft τfτdτ = e t.5.. fx = x [ π, π] n sinnx n n=. π a π a, x [ π, π] x = a n cosnx cosna + 4 n=. 3, x [ π, π] x 3 π x = n sinnx. n=.6 f, t gt n 3 n")
[ ]. A = IC X n 3 expx = E + expta t : n! n=. fx π x π. { π x < fx = x π fx F k F k = π 9 s9 fxe ikx dx, i =. F k. { x x fx = x >.3 ft = cosωt F s = s4 e st ftdt., e, s. s = c + iφ., i, c, φ., Gφ = lim
,,,17,,, ( ),, E Q [S T F t ] < S t, t [, T ],,,,,,,,
![,,,17,,, ( ),, E Q [S T F t ] < S t, t [, T ],,,,,,,,](/thumbs/91/105754403.jpg ",,,17,,, ( ),, E Q [S T F t ] < S t, t [, T ],,,,,,,,")
14 5 1 ,,,17,,,194 1 4 ( ),, E Q [S T F t ] < S t, t [, T ],,,,,,,, 1 4 1.1........................................ 4 5.1........................................ 5.........................................
2013 25 9 i 1 1 1.1................................... 1 1.2........................... 2 1.3..................................... 3 1.4..................................... 4 2 6 2.1.................................
1W II K =25 A (1) office(a439) (2) A4 etc. 12:00-13:30 Cafe David 1 2 TA appointment Cafe D
1W II K200 : October 6, 2004 Version : 1.2, kawahira@math.nagoa-u.ac.jp, http://www.math.nagoa-u.ac.jp/~kawahira/courses.htm TA M1, m0418c@math.nagoa-u.ac.jp TA Talor Jacobian 4 45 25 30 20 K2-1W04-00
( ) ( )
20 21 2 8 1 2 2 3 21 3 22 3 23 4 24 5 25 5 26 6 27 8 28 ( ) 9 3 10 31 10 32 ( ) 12 4 13 41 0 13 42 14 43 0 15 44 17 5 18 6 18 1 1 2 2 1 2 1 0 2 0 3 0 4 0 2 2 21 t (x(t) y(t)) 2 x(t) y(t) γ(t) (x(t) y(t))
B [ 0.1 ] x > 0 x 6= 1 f(x) µ 1 1 xn 1 + sin sin x 1 x 1 f(x) := lim. n x n (1) lim inf f(x) (2) lim sup f(x) x 1 0 x 1 0 (
![B [ 0.1 ] x > 0 x 6= 1 f(x) µ 1 1 xn 1 + sin sin x 1 x 1 f(x) := lim. n x n (1) lim inf f(x) (2) lim sup f(x) x 1 0 x 1 0 (](/thumbs/97/133036028.jpg "B [ 0.1 ] x > 0 x 6= 1 f(x) µ 1 1 xn 1 + sin sin x 1 x 1 f(x) := lim. n x n (1) lim inf f(x) (2) lim sup f(x) x 1 0 x 1 0 (")
. 28 4 14 [.1 ] x > x 6= 1 f(x) µ 1 1 xn 1 + sin + 2 + sin x 1 x 1 f(x) := lim. 1 + x n (1) lim inf f(x) (2) lim sup f(x) x 1 x 1 (3) lim inf x 1+ f(x) (4) lim sup f(x) x 1+ [.2 ] [, 1] Ω æ x (1) (2) nx(1
1. 1 BASIC PC BASIC BASIC BASIC Fortran WS PC (1.3) 1 + x 1 x = x = (1.1) 1 + x = (1.2) 1 + x 1 = (1.
Section Title Pages Id 1 3 7239 2 4 7239 3 10 7239 4 8 7244 5 13 7276 6 14 7338 7 8 7338 8 7 7445 9 11 7580 10 10 7590 11 8 7580 12 6 7395 13 z 11 7746 14 13 7753 15 7 7859 16 8 7942 17 8 Id URL http://km.int.oyo.co.jp/showdocumentdetailspage.jsp?documentid=
Sturm-Liouville Green KEN ZOU Hermite Legendre Laguerre L L [p(x) d2 dx 2 + q(x) d ] dx + r(x) u(x) = Lu(x) = 0 (1) L = p(x) d2 dx
![Sturm-Liouville Green KEN ZOU Hermite Legendre Laguerre L L [p(x) d2 dx 2 + q(x) d ] dx + r(x) u(x) = Lu(x) = 0 (1) L = p(x) d2 dx](/thumbs/88/116809375.jpg "Sturm-Liouville Green KEN ZOU Hermite Legendre Laguerre L L [p(x) d2 dx 2 + q(x) d ] dx + r(x) u(x) = Lu(x) = 0 (1) L = p(x) d2 dx")
Sturm-Liouville Green KEN ZOU 2006 4 23 1 Hermite Legendre Lguerre 1 1.1 2 L L p(x) d2 2 + q(x) d + r(x) u(x) = Lu(x) = 0 (1) L = p(x) d2 2 + q(x) d + r(x) (2) L = d2 2 p(x) d q(x) + r(x) (3) 2 (Self-Adjoint
1.2 y + P (x)y + Q(x)y = 0 (1) y 1 (x), y 2 (x) y 1 (x), y 2 (x) (1) y(x) c 1, c 2 y(x) = c 1 y 1 (x) + c 2 y 2 (x) 3 y 1 (x) y 1 (x) e R P (x)dx y 2
1 1.1 R(x) = 0 y + P (x)y + Q(x)y = R(x)...(1) y + P (x)y + Q(x)y = 0...(2) 1 2 u(x) v(x) c 1 u(x)+ c 2 v(x) = 0 c 1 = c 2 = 0 c 1 = c 2 = 0 2 0 2 u(x) v(x) u(x) u (x) W (u, v)(x) = v(x) v (x) 0 1 1.2
all.dvi
29 4 Green-Lagrange,,.,,,,,,.,,,,,,,,,, E, σ, ε σ = Eε,,.. 4.1? l, l 1 (l 1 l) ε ε = l 1 l l (4.1) F l l 1 F 30 4 Green-Lagrange Δz Δδ γ = Δδ (4.2) Δz π/2 φ γ = π 2 φ (4.3) γ tan γ γ,sin γ γ ( π ) γ tan
ẍ = kx, (k > ) (.) x x(t) = A cos(ωt + α) (.). d/ = D. d dt x + k ( x = D + k ) ( ) ( ) k k x = D + i D i x =... ( ) k D + i x = or ( ) k D i x =.. k.
K E N Z OU 8 9 8. F = kx x 3 678 ẍ = kx, (k > ) (.) x x(t) = A cos(ωt + α) (.). d/ = D. d dt x + k ( x = D + k ) ( ) ( ) k k x = D + i D i x =... ( ) k D + i x = or ( ) k D i x =.. k. D = ±i dt = ±iωx,
II (10 4 ) 1. p (x, y) (a, b) ε(x, y; a, b) 0 f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a 2 x a 0 A = f x (
II (1 4 ) 1. p.13 1 (x, y) (a, b) ε(x, y; a, b) f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a x a A = f x (a, b) y x 3 3y 3 (x, y) (, ) f (x, y) = x + y (x, y) = (, )
x,, z v = (, b, c) v v 2 + b 2 + c 2 x,, z 1 i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) v 1 = ( 1, b 1, c 1 ), v 2 = ( 2, b 2, c 2 ) v
12 -- 1 4 2009 9 4-1 4-2 4-3 4-4 4-5 4-6 4-7 4-8 4-9 4-10 c 2011 1/(13) 4--1 2009 9 3 x,, z v = (, b, c) v v 2 + b 2 + c 2 x,, z 1 i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) v 1 = ( 1, b 1, c 1 ), v 2
Part () () Γ Part ,
Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
( ) 5. VSS (VIM ) 10. ( ) 11. (ANN ) ( )
1.... ( ) 5. VSS.. 8. 9. (VIM ) 1. ( ) 11. (ANN ) 1. 1. ( ) 1 Lagrange 1..1 Lagrange q, Lagrange D(q)q + C(q; _q)_q + G(q) = (1.1) D(q)q C(q; _q)_q G(q) ( ) D(q) D(q) m ; M < M m (D(q)) (1.) (D(q)) M
2012 IA 8 I p.3, 2 p.19, 3 p.19, 4 p.22, 5 p.27, 6 p.27, 7 p
2012 IA 8 I 1 10 10 29 1. [0, 1] n x = 1 (n = 1, 2, 3,...) 2 f(x) = n 0 [0, 1] 2. 1 x = 1 (n = 1, 2, 3,...) 2 f(x) = n 0 [0, 1] 1 0 f(x)dx 3. < b < c [, c] b [, c] 4. [, b] f(x) 1 f(x) 1 f(x) [, b] 5.
30 I .............................................2........................................3................................................4.......................................... 2.5..........................................
Maxwell
I 2018 12 13 0 4 1 6 1.1............................ 6 1.2 Maxwell......................... 8 1.3.......................... 9 1.4..................... 11 1.5..................... 12 2 13 2.1...................
Note.tex 2008/09/19( )
1 20 9 19 2 1 5 1.1........................ 5 1.2............................. 8 2 9 2.1............................. 9 2.2.............................. 10 3 13 3.1.............................. 13 3.2..................................
IA
IA 31 4 11 1 1 4 1.1 Planck.............................. 4 1. Bohr.................................... 5 1.3..................................... 6 8.1................................... 8....................................
00 3 9 ........................................................................................................................................... 4..3................................. 5.3.......................................
phs.dvi
483F 3 6.........3... 6.4... 7 7.... 7.... 9.5 N (... 3.6 N (... 5.7... 5 3 6 3.... 6 3.... 7 3.3... 9 3.4... 3 4 7 4.... 7 4.... 9 4.3... 3 4.4... 34 4.4.... 34 4.4.... 35 4.5... 38 4.6... 39 5 4 5....
ohp_06nov_tohoku.dvi
2006 11 28 1. (1) ẋ = ax = x(t) =Ce at C C>0 a0 x(t) 0(t )!! 1 0.8 0.6 0.4 0.2 2 4 6 8 10-0.2 (1) a =2 C =1 1. (1) τ>0 (2) ẋ(t) = ax(t τ) 4 2 2 4 6 8 10-2 -4 (2) a =2 τ =1!! 1. (2) A. (2)
I 1
I 1 1 1.1 1. 3 m = 3 1 7 µm. cm = 1 4 km 3. 1 m = 1 1 5 cm 4. 5 cm 3 = 5 1 15 km 3 5. 1 = 36 6. 1 = 8.64 1 4 7. 1 = 3.15 1 7 1 =3 1 7 1 3 π 1. 1. 1 m + 1 cm = 1.1 m. 1 hr + 64 sec = 1 4 sec 3. 3. 1 5 kg
(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
数値計算：有限要素法
( ) 1 / 61 1 2 3 4 ( ) 2 / 61 ( ) 3 / 61 P(0) P(x) u(x) P(L) f P(0) P(x) P(L) ( ) 4 / 61 L P(x) E(x) A(x) x P(x) P(x) u(x) P(x) u(x) (0 x L) ( ) 5 / 61 u(x) 0 L x ( ) 6 / 61 P(0) P(L) f d dx ( EA du dx