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 ) = x T J x x T J {u(t)} λ(t) ẋ = f(x, u) L = = V (x, u)dt + λ[f(x, u) ẋ]dt [V (x, u) + λf(x, u) λẋ]dt ( ) H(x, u) = V (x, u) + λf(x, u) () L = [H(x, u) λẋ]dt (3)
(3) λẋdt = L L = λxdt [λ(t )x(t ) λ()x()] (4) [H(x, u) + λx]dt [λ(t )x(t ) λ()x()] (5) {u(t)} {u(t) + u(t)} {x(t)} {x(t) + x(t)}. L L = = [ dx + x u [ u du + ] du + λdx dt λ(t )dx T ( x + λ ) ] dx dt λ(t )dx T du dx ± L =. u = t T λ = x t T (iii) λ(t ) = ( x T x(t ) = x T dx T = ) λ (iii) x(t ) = x T dx T =. ẋ = f(x, u) = λ λ H(x, u) = V (x, u) + λf(x, u) {u(t)}, {λ(t)}, {x(t)} u = t T λ = x t T (iii) ẋ = = f(x, u) λ t T (6) (iv) x() = x (v) λ(t ) = ( x(t ) = x T ) 3
x (t),, x n (t) λ (t),, λ n (t) u (t),, u m (t) (iii) (iv) u i = i =,, m t T λi = x i i =,, n t T ẋ i = λ i = f i (x, u) t T (7) x i () = x i (v) λ i (T ) = ( x i (T ) = x T i ), x() = x x(t ) = x T T : u = u λ = λ = x = (iii) ẋ = λ = u (iv) x() = (v) x() = max {u} ẋ = u u dt x() =, x() = H(x, u) = V (x, u) + λf(x, u) = u + λ( u) u = λ u = λ ẋ = λ, λ = x(t) = c λt, λ(t) = c (c, c ) 4
x() = c =., x() =, : x() = λ = λ = c = x (t) = t = t u (t) = λ = max {u} 4 (x + u )dt ẋ = x + u x() =, x() = H(x, u) = V (x, u) + λf(x, u) = 4 (x + u ) + λ(x + u) (iii) u = u + λ = λ = x = x λ ẋ = λ = x + u = u = λ = ẋ = x + λ (iii) u, x λ ẋ = x + λ λ = x λ, A = [ ] r =, r r = r =. 5
i) r = ii) r = [ ] [ = [ v v [ + ] [ + = [ v v v v ] v v ] ] [ ( ] [ ] )v = + v v ( + )v = = [ ] ] [ ( + ] [ ] )v = + v v + ( + )v = = [ x(t) = c e t + c e t λ(t) = c ( )e + ] t c ( + )e x() = x() = c c x() = c + c = x() = c e + c e = t. c = e e e.56, c = e e e.56 : () () u = u λ (3) : x λ (4) (5) x() x(t ) (6) λ u u 6
3 : max {u} ẋ = x + u x() =, : 5xdx u(t) [, 3] x() H(x, u) = V (x, u) + λf(x, u) = 5x + λ(x + u) = (5 + λ)x + λu u = λ λ = = (5 + λ) x (iii) ẋ = λ = x + u (iv) x() = (v) λ() = u. λ > H u = 3 ( ), u (t) = 3. λ = λ 5 λ + λ = 5 e t λe t + λe t = 5e t (λe t ) λe t = 5e t + k (k ) λ (t) = ke t 5 λ () = ke 5 = k = 5e λ (t) = 5e t 5 u (t) = 3 (λ (t) >, t < ) ẋ = x + 3 ẋ x = 3 7
e t ẋe t xe t = 3e t (xe t ) x (t) = ke t 3 x() = ke 3 = k = 5 x (t) = 5e t 3 x() =. ( ) ( ) ( ) T Z T Z x(t ) = Z λ(t ) = x(t ) = x ( ) g(x) λ g(x ) = g(x ) > x λ g(x ) = (tranversality condition) max J = V (x, u)dt {u(t)} s.t.ẋ = f(x, u) x() = x (8) x(t ) 8
L = = V (x, u)dt + [V (x, u) + λf(x, u)]dt + λ[f(x, u) ẋ]dt + µ x(t ) λxdt λ(t )x(t ) + λ()x() + µ x(t ) L x(t ) L = λ(t ) + µ = µ = λ(t ) x(t ) µ x(t ) = λ(t ) x(t ) = : λ (T ) >, x (T ) = λ (T ) =, x (T ) 3 V (x, u)., V (x, u) δ max J = {u(t)} e δt V (x, u)dt (9) max J = {u(t)} e δt V (x, u)dt ẋ = f(x, u) () x() = x x(t ) = x T L = {e δt V (x, u) + λ[f(x, u) ẋ]}dt H(x, u) = e δt V (x, u) + λf(x, u) 9
( ) H c H c (x, u) = V (x, u) + µf(x, u) H c = He δt H = H c e δt µ = λe δt λ = µe δt µ = λe δt 5 e δt c =. u λ = x = c x e δt λ = µe δt λ = µe δt δµe δt c x e δt = µe δt δµe δt µ = c x + δµ (iii) (iv) ẋ = λ = c λ e δt = c µ (v) λ(t ) = µ(t )e δt = = f(x, u) H c (x, u) = H(x, u)e δt = V (x, u) + µf(x, u) λ = µe δt. : (iii) c u = µ = c x + δµ ẋ = c µ (iv) x() = x t T t T = f(x, u) t T () (v) µ(t )e δt = ( x(t ) = x T ) u : x, µ.
4 max J = {u(t)} ẋ = u x() = x() = u e.t dt x (t). H c = u + µu (iii) c u = u + µ = µ = c x ẋ = c µ = u + δµ = +.µ u =.5µ, (iii) ẋ =.5µ. ẋ =.5µ µ =.µ µ µ =. ln µ =.t + k (k ) µ = e.t e k = k e.t ( e k = k ) ẋ =.5k e.t x = 5k e.t + k k x(t) = c + c e.t µ(t) = c e.t x() = x() = c c c + c = c + c e =
c = e 58.9767 c = 58.9767. e x (t) = e + e e.t = e (e.t ) 58.9767 + 58.9767e.t = 58.9767(e.t ). max {u} ẋ = u udt x() =, x() =. max ( x + u ) dt {u} 4 ẋ = x + u x() =, x() = 3. max {u} ẋ = x + u ( x 4 u 9 ) dt x() = 5, x() = 4. max {u} ẋ = x + u (3x u )dt x() =, x() = 5 5. max {u} ẋ = u u e.t dt x() =, x() =