A Control Method of the Plate Equation by Power Series of Gevrey Class

A Control Method of the Plate Equation by Power Series of Gevrey Class

Shouta MORI^* , Hisashi MORIOKA^**, Hideki TAKUWA^***

(Received October 20, 2017)

Motion planning which is construction of an input implementing a desired output on a system is a fundamental problem on both of the theory of control and its practical applications. In many cases, a system is represented as an ordinary differential equation or a partial differential equation. Here let us deal final states of a system with outputs.

Then we can consider some control problems. A typical example of some control problems is the control by boundary values. Laroche-Martin-Rouchon¹⁾considered an approximate motion planning as a boundary control problem on the heat equation using Gevrey class functions. In this paper, we study an approximate motion planning on the one dimensional plate equation by boundary control using Gevrey class functions. More precisely, we consider the initial-boundary value problem (1)-(3). The output is a given final state of the plate at timeT >0, and the input is the pair of the Dirichlet boundary value and the Neumann boundary value at an endpoint of the plate. We construct this input using finitely truncated Gevrey functions so that the associated solution of the plate equation approximates the desired final state. Our main result is Theorem 6.

Key wordsɿplate equation, Gevrey class, control problem, power series Ωʔϫʔυ ɿྊৼಈͷํఔࣜ, Gevrey class,੍ޚ໰୊,΂͖ڃ਺

ྊৼಈͷํఔࣜͷ Gevrey class ʹΑΔ΂͖ڃ਺Λ༻͍ͨղʹΑΔ੍ޚํ๏

৿ɹকଠ,৿Ԭɹ༔,ଟٱ࿨ӳथ

1. ॹݴ

ಈ࡞ܭը(Motion planning),͢ͳΘͪ,͋Δܥʹ͓͍

ͯࢦఆ͞Εͨग़ྗಈ࡞Λ࣮ݱ͢ΔೖྗΛઃܭ͢Δ໰୊

͸,੍ޚཧ࿦ʹ͓͍ͯ,࣮༻্͓Αͼཧ࿦తͳ؍఺͔Β جຊతͳ໰୊Ͱ͋Δ. ޻ֶతͳ໰୊Ͱ͸,ଟ͘ͷ৔߹,ѻ

͏ܥ͸ৗඍ෼ํఔࣜ͋Δ͍͸ภඍ෼ํఔࣜʹΑͬͯϞσ ϧԽ͞ΕΔ. ैͬͯ, ಈ࡞ܭը໰୊ͱͯ͠͸, ग़ྗ͸͋

Δ࣌ࠁʹ͓͚Δղͷऴঢ়ଶ, ೖྗ͸, ॳظঢ়ଶʹΑΔ੍

* Department of Mechanical and Systems Engineering, Doshisha University, Kyoto Telephone : +81-06-6845-8669, E-mail : [email protected]

** Department of Energy and Mechanical Engineering, Doshisha University, Kyoto Telephone : +81-0774-65-6492, E-mail : [email protected]

*** Department of Mechanical and Systems Engineering, Doshisha University, Kyoto Telephone : +81-0774-65-6431, E-mail : [email protected]

ޚ΍ڥք੍ޚͳͲ͕ߟ͑ΒΕΔ. ภඍ෼ํఔࣜͰهड़͞

ΕΔಈ࡞ܭըͱͯ͠͸, Laroche-Martin-Rouchon¹⁾ʹ ΑΔ೤ํఔࣜͷݚڀ͕͋Δ. ຊݚڀͰ͸,͜ͷख๏Λۭ

ؒҰ࣍ݩͷ௕͞Lͷྊͷৼಈݱ৅ͷ৔߹ʹద༻͠,ྊৼ ಈͷ੍ޚख๏ΛఏҊ͢Δ.

ྊͷยํͷ୺Λݻఆͨ͠৔߹ͷϞσϧ͸,࣍ͷॳظ஋ɾ ڥք஋໰୊ͱͯ͠ఆࣜԽͰ͖Δ. (t, x)∈[0, T]×[0, L]

ʹର͠,u(t, x)ͰྊͷมҐΛද͢ͱ͢Δ.

͜ͷͱ͖,u͸(0, T)×(0, L)ʹ͓͍ͯํఔࣜ

∂_t²u+∂⁴_xu= 0, (1) ʹै͏. ͞Βʹ,u͸ॳظ৚݅

u(0, x) =u0(x), ∂tu(0, x) =u¹₀(x), x∈(0, L), (2)

͓Αͼڥք৚݅

u(t,0) =∂xu(t,0) = 0,

u(t, L) =h(t), ∂xu(t, L) =I(t), t∈(0, T), (3) ΛΈͨ͢ͱ͢Δ. ͜͜Ͱ, h, I͸े෼ͳΊΒ͔ͳؔ਺Ͱ

͋Γ,u0∈H²(0, L), u¹₀∈L²(0, L)ͱ͢Δ.

ؔ਺uT(x), u¹_T(x)͕༩͑ΒΕͨͱ͢Δ. ྊৼಈͷॳ ظ஋ɾڥք஋໰୊(1)-(3)ʹ͓͍ͯ,࣌ࠁT >0Ͱྊͷ ঢ়ଶ͕u(T,·) =uT,∂tu(T,·) =u¹_TͱͳΔΑ͏,ڥք஋

h,IʹΑ੍ͬͯޚ͢Δ໰୊Λߟ͑Δ. ޻ֶతͳ੍ޚ໰୊

ʹ͓͍ͯ͸, ೖྗʹ͋ͨΔh, I͸۩ମతʹ, ͔ͭͰ͖Δ

͚ͩ؆୯ͳؔ਺ʹΑͬͯߏ੒Ͱ͖Δ͜ͱ͕๬·͍͠. ͦ

͜Ͱ,ຊݚڀͰ͸, ·ͣ΂͖ڃ਺Λ༻͍ͯॳظ஋ɾڥք

஋໰୊(1)-(3)ͷۙࣅղΛ۩ମతʹߏ੒͢Δ. ͞Βʹ,͜

ͷ΂͖ڃ਺Λ༗ݶͷ߲Ͱଧͪ੾Δ͜ͱʹΑΓ,े෼খ͞

ͳϵ >0ʹରͯ͠,

||uT −u(T,˜ ·)||H²(0,L)< ϵ, (4)

||u¹_T −∂tu(T,˜ ·)||L²(0,L)< ϵ, (5) ͷҙຯͰۙࣅ੍ޚͱͳΔΑ͏ͳೖྗ˜h,I˜Λߏ੒͢Δ. ͜

͜Ͱu˜͸,ॳظ஋ɾڥք஋໰୊(1)-(3)ʹ͓͍ͯ,h= ˜h, I = ˜Iͱͨ͠ͱ͖ͷղͰ͋Δ. Ҏ্ʹड़΂ͨղu˜ͱೖྗ

h,˜ I˜͸, Gevreyؔ਺Λ༻͍ͯߦ͏. ৄࡉ͸§2Ͱड़΂Δ.

ຊ࿦จͷߏ੒͸ҎԼͷ௨ΓͰ͋Δ. §2Ͱ͸,্ʹड़΂

ͨΑ͏ʹGevreyؔ਺Λ༻͍ͨղͷߏ੒ʹ͍ͭͯड़΂Δ.

§3Ͱ͸, (4)-(5)ͷධՁʹඞཁͳ,ྊৼಈํఔࣜͷॳظ஋

໰୊ʹؔ͢ΔΤωϧΪʔෆ౳ࣜΛಋग़͢Δ. §4Ͱ͸,ओ

݁ՌͷओுΛड़΂,ͦͷূ໌Λߦ͏. ຊ࿦จͷओ݁Ռ͸

ఆཧ6Ͱ͋Δ. §5͸ิҨͱͯ͠,ιϘϨϑۭؒH²ͷҙ ຯͰͷۙࣅଟ߲ࣜͷߏ੒ํ๏ʹ͍ͭͯิ଍Λड़΂Δ.

2. ΂͖ڃ਺ղͱGevreyؔ਺

ఆٛ 1 y :t ∈[0, T]→y(t)∈R͸C^∞ڃͷؔ਺ͱ͢

Δ. ͜ͷͱ͖೚ҙͷඇෛͷ੔਺mʹରͯؔ͠਺y͕

sup

0≤t≤T

��

�z^(m)(t)���≤M(m!)^s1 R^m ,

Λຬͨ͢ਖ਼ͷ਺M, R͕ଘࡏͨ͠ͱ͢Δ. ͜ͷͱ͖ؔ਺

y͸tʹ͍ͭͯs1 ∈ [1,+∞)ΫϥεͷGevreyؔ਺Ͱ

͋Δ.

ఆٛ 2 γ∈(0,+∞), T >0ͱͨ͠ͱ͖ؔ਺ψγ Λ

ψγ(t) =

{ 0 t= 0, T, exp(

−1 ((T−t)t)^γ

)

t= (0, T),

Ͱఆٛ͢Δ. ͜ͷͱ͖ؔ਺ψγ͸GevreyΫϥε1 + (¹_γ) Ͱ͋Δ. ͞Βʹؔ਺ΨγΛ

Ψγ(t) =

∫t

0ψγ(τ)dτ

∫T

0 ψγ(τ)dτ t∈[0, T],

Ͱఆٛ͢Δ. ͜ͷͱ͖ؔ਺Ψγ͸GevreyΫϥε1 + (¹_γ) ͷؔ਺Ͱ͋Δ.

ఆٛ 3 z: (t, x)∈[0, T]×[0, L]→z(t, x)∈R͸C^∞ ڃͷؔ਺ͱ͢Δ.͜ͷͱ͖೚ҙͷඇෛͷ੔਺m, nʹର͠

ͯؔ਺z͕

sup

0≤t≤T,0≤x≤L

��

��∂^m+nz

∂^m_t ∂_xⁿ (t, x)

��

��≤M(m!)^s1(n!)^s2 R^m₁Rⁿ₂ , Λຬͨ͢ਖ਼ͷ਺M, R1, R2͕ଘࡏͨ͠ͱ͢Δ.͜ͷͱ͖

ؔ਺z͸tʹ͍ͭͯs1∈[1,+∞)ΫϥεͰ͋Γ,xʹͭ

͍ͯs2∈[1,+∞)ΫϥεͷGevreyؔ਺Ͱ͋Δ.

࣍ʹ΂͖ڃ਺Λ༻͍ͯ,ۙࣅ੍ޚΛߦ͏͜ͱ͕Ͱ͖Δ

ೖྗ˜h, ˜IͷٻΊΔͨΊ,ํఔࣜ(1)-(3)ͷۙࣅղΛٻΊ Δ. P0, P₀¹, PT, P_T¹Λ,ͦΕͧΕu0, u¹₀, uT, u¹_T ͷۙࣅଟ

߲ࣜ,͢ͳΘͪ,े෼খ͞ͳϵ >0ʹରͯ͠

||P0−u0||H²(0,L)< ϵ, ||P0¹−u¹₀||L²(0,L)< ϵ, (6)

||PT −uT||H²(0,L)< ϵ, ||P_T¹−u¹_T||L²(0,L)< ϵ, (7) Λຬͨ͢΋ͷͱ͢Δ. ͜ͷ৚݅ (6)-(7) Λຬͨ͢ଟ

߲ࣜͷߏ੒ʹ͍ͭͯ͸, §5 Ͱएׯͷ஫ऍΛड़΂Δ.

P0, P₀¹, PT, P_T¹Λ,े෼େ͖ͳࣗવ਺Nʹରͯ͠,

P0(x) =

∑N i=0

P0,i x⁴ⁱ⁺² (4i+ 2)!, P₀¹(x) =

∑N i=0

P_0,i¹ x⁴ⁱ⁺³ (4i+ 3)!, PT(x) =

∑N i=0

PT,i x⁴ⁱ⁺² (4i+ 2)!, P_T¹(x) =

∑N i=0

P_T,i¹ x⁴ⁱ⁺³ (4i+ 3)!,

ͱද͢.





































∂_t²u¯+∂⁴_xu¯= 0, (t, x)∈(0, T)×(0, L),

¯

u(0, x) =P0(x),

∂tu(0, x) =¯ P₀¹(x), x∈(0, L),

¯

u(t,0) = 0,

¯

u(t, L) = ¯h(t), t∈(0, T),

∂xu(t,¯ 0) = 0,

∂xu(t, L) = ¯¯ I(t),

(8)

Λߟ͑Δ.

͜ͷͱ͖ͷ¯h, I¯ΛٻΊΔ. ͦͷͨΊʹํఔࣜ(8)ͷ ղu¯Λu(t, x) =¯ ∑_∞

i=0ai(t)^x_i!ⁱͷ΂͖ڃ਺ͷܗͰٻΊΔ.

͜͜Ͱ,೚ҙͷiʹରͯ͠ai∈C^∞([0, T])ͱ͢Δ.

ํఔࣜ∂_t²u¯+∂⁴_xu¯= 0ͱڥք৚݅

¯

u(t,0) = 0, ∂xu(t,¯ 0) = 0, ΑΓ,೚ҙͷiʹରͯ͠

a4i(t) = 0, a4i+1(t) = 0,

͕ಘΒΕΔ. ͜͜Ͱ,

∂_x²u(t,¯ 0) =Y(t), ∂³_xu(t,¯ 0) =Z(t),

ͱ͓͘. ͜ͷͱ͖͜ͷY, Zͱํఔࣜ∂_t²u¯+∂_x⁴u¯= 0Λ

༻͍ͯܗࣜతͳղͷ࢒Γͷ܎਺ΛܾΊΔͱ,೚ҙͷiʹ ରͯ͠

a4i+2(t) = (−1)ⁱY⁽²ⁱ⁾(t), a4i+3(t) = (−1)ⁱZ⁽²ⁱ⁾(t), Ͱܾ·Δ. ͜ΕΒΛ༻͍Δ͜ͱͰܗࣜతͳղ͸

¯ u(t, x) =

∑∞ i=0

(−1)ⁱY⁽²ⁱ⁾(t) x⁴ⁱ⁺² (4i+ 2)!

+

∑∞ i=0

(−1)ⁱZ⁽²ⁱ⁾(t) x⁴ⁱ⁺³ (4i+ 3)!,

ͱͳΔ. ͜ͷແݶ࿨Ͱද͞Εͨؔ਺͕ऩଋ͢Δͱ͸ݶΒ ͳ͍ͷͰ,ऩଋ͢ΔͨΊͷؔ਺Y, Zͷ৚݅ΛٻΊΔ.

ิ୊ 4 ܗࣜతͳղΛදͨ͢Ίʹ༻͍ͨؔ਺ Y, Z ͕ Gevrey Ϋϥε α∈[1,2)Ͱ͋ΔͱԾఆ͢Δ.

͜ͷͱ͖΂͖ڃ਺Ͱද͞Εͨܗࣜతͳղu¯͸ऩଋ͠, tʹ͍ͭͯαͰ͋Γxʹ͍ͭͯ1ͷGevreyؔ਺Ͱ͋Δ.

ূ໌. ¯u͕ऩଋ͢Δ͜ͱΛࣔ͢. ܗࣜతͳղu¯Λ

¯ u(t, x) =

∑∞ i=0

(−1)ⁱY⁽²ⁱ⁾(t) x⁴ⁱ⁺² (4i+ 2)!

+

∑∞ i=0

(−1)ⁱZ⁽²ⁱ⁾(t) x⁴ⁱ⁺³ (4i+ 3)!

= ¯u¹(t, x) + ¯u²(t, x),

ͱද͢. ͜ͷͱ͖,

��

��∂^m+nu¯

∂t^m∂xⁿ

(t, x)

��

��

≤

��

��∂^m+nu¯¹

∂^m_t ∂_xⁿ (t, x)

��

��+

��

��∂^m+nu¯²

∂_t^m∂_xⁿ (t, x)

��

��,

Ͱ͋Δ͔Β, ¯u¹,u¯²͕tʹ͍ͭͯαͰ͋Γxʹ͍ͭͯ1 ͷGevreyؔ਺Ͱ͋Δ͜ͱΛࣔͤ͹Α͍. ҎԼu¯¹Λධ Ձ͢Δ.

��

��∂^m+nu¯¹

∂_t^m∂ⁿ_x (t, x) Lⁿ (m!)^α(n!)

��

��

=

��

��

��

∑∞ 4i+2≥n

(−1)ⁱY^(2i+m)(t) x⁴ⁱ⁺²⁻ⁿ (4i+ 2−n)!

Lⁿ (m!)^α(n!)

��

��

��

<

∑∞ 4i+2≥n

��

��Y^(2i+m)(t) x⁴ⁱ⁺²⁻ⁿ (4i+ 2−n)!

Lⁿ (m!)^α(n!)

��

��,

࿨ͷҰൠ߲Λܭࢉ͢Δͱ,

��

��Y^(2i+m)(t) x⁴ⁱ⁺²⁻ⁿ (4i+ 2−n)!

Lⁿ (m!)^α(n!)

��

��

≤

��

��Y^(2i+m)(t) L⁴ⁱ⁺² (4i+ 2−n)!

1 (m!)^α(n!)

��

��,

ͱͳΔ. ؔ਺Y ͸Gevrey Ϋϥε αͱ͍ͯ͠ΔͷͰ,

GevreyΫϥεͷධՁࣜΑΓ, ͋Δਖ਼ͷఆ਺M1, A1͕ ଘࡏͯ͠,

��

��Y^(2i+m)(t) L⁴ⁱ⁺² (4i+ 2−n)!

1 (m!)^α(n!)

��

��,

≤M1(2i+m)!^α A^2i+m₁

L⁴ⁱ⁺² (4i+ 2−n)!

1 (m!)^α(n!)

≤M1L⁴ⁱ⁺² A^2i+m₁

(2i)!^α (4i+ 2−n)!(n!)

(2i+m)!

(m!)(2i)!

α

≤M1L⁴ⁱ⁺² A^2i+m₁

(2i)!^α−4(2i)!⁴ (4i+ 2−n)!(n!)

(2i+m)!

(m!)(2i)!

α

,

ͱͳΔ. ελʔϦϯάͷෆ౳ࣜΑΓ M1L⁴ⁱ⁺²

A^2i+m₁

(2i)!^α−4(2i)!⁴ (4i+ 2−n)!(n!)

(2i+m)!

(m!)(2i)!

α

∼M1L⁴ⁱ⁺² A^2i+m₁

(2i)!^α−4 (4i+ 2−n)!(n!)

× (2i+m)!

(m!)(2i)!

α(8i)!

2

(4πi)³² 4⁸ⁱ

≤M1L⁴ⁱ⁺²⁻ⁿ

A^2i+m₁ (2i)!^α−4(2^2i+m)^α4²(4i)!(4πi)³² 2

=M1L⁴ⁱ⁺² 1 R₁²ⁱ

1

R^m₁ (2i)!^α−4(4i)!8(4πi)³²

=M1

vi

R^m₁ ,

͕ಘΒΕΔ. ͜͜Ͱ, R1= A1

2^α, vi=L⁴ⁱ⁺² 1

R²ⁱ₁ (2i)!^α−4(4i)!8(4πi)³², ͱ͓͍ͨ. ͜ͷࣜʹରͯ͠,μϥϯϕʔϧͷ൑ఆ๏ΑΓ

ilim→∞

vi+1

vi

= lim

i→∞

L⁴ R²₁

(4i+ 4)(4i+ 3)(4i+ 2)(4i+ 1) (2i+ 1)^4−α(2i+ 2)^4−α (i+ 1

i )³², ͱͳΔ. ԾఆΑΓα∈[1,2)Ͱ͋ΔͷͰ, ¯u¹͸ऩଋ͢Δ.

࣍ʹu¯¹͕tʹ͍ͭͯαͰ͋Γxʹ͍ͭͯ1ͷGevrey

ؔ਺Ͱ͋Δ͜ͱΛࣔ͢.

M¯1=M1

∑∞ 4i+2≥n

vi,

ͱ͓͘. ¯u¹͸α∈[1,2)ͷͱ͖,

��

��∂^m+nu¯¹

∂t^m∂ⁿ_x (t, x)

��

��<M¯1(m!)^α(n!) R^m₁ Lⁿ ,

ͱͳΔ. ¯u²ʹରͯ͠΋ಉ༷ʹͯ͠, ͋Δਖ਼ͷఆ਺M¯2, R2͕ଘࡏͯ͠,

��

��∂^m+nu¯²

∂_t^m∂ⁿ_x (t, x)

��

��<M¯2

(m!)^α(n!) R^m₂ Lⁿ , ͱٻΊΒΕΔ. Αͬͯ,͋Δਖ਼ͷఆ਺M,RʹΑΓ,

��

��∂^m+nu¯

∂t^m∂xⁿ

(t, x)

��

��

≤

��

��∂^m+nu¯¹

∂_t^m∂_xⁿ (t, x)

��

��+

��

��∂^m+nu¯²

∂_t^m∂ⁿ_x (t, x)

��

��

≤M(m!)^α(n!) R^mLⁿ , ͱͳΔ.

Αͬͯ,΂͖ڃ਺Ͱදͨ͠ղ͕ऩଋ͢Δ৚݅͸xʹͭ

͍ͯGevrey Ϋϥε 1ͱͨ͠ͱ͖,tʹ͍ͭͯͷ৚݅͸

Gevrey Ϋϥε α∈[1,2)ͱͳΔ.ɹ(ূ໌ऴ)

͜ͷิ୊ʹΑΓऩଋ͢Δ৚݅͸෼͔ͬͨͷͰ,ͦΕΛ

ຬͨ͢Α͏ʹؔ਺Y, ZΛܾఆ͢Δ. ͦͷͨΊʹ, ؔ਺

P0, P₀¹, PT, P_T¹ ͷ܎਺ͱఆٛ2ͰܾΊͨGevrey Ϋϥ ε 1 + (_γ¹)ͷؔ਺ΨγΛ༻͍ͯ,Y, ZΛ࣍ͷΑ͏ʹఆٛ

͢Δ.

Y(t) =

∑N i=0

ʢ−1ʣⁱP0,i

t²ⁱ

(2i)!(1−Ψγ(t)) +

∑N i=0

ʢ−1ʣⁱPT,i(t−T)²ⁱ (2i)! Ψγ(t), Z(t) =

∑N i=0

ʢ−1ʣⁱP_0,i¹ t²ⁱ⁺¹

(2i+ 1)!(1−Ψγ(t)) +

∑N i=0

ʢ−1ʣⁱP_T,i¹ (t−T)²ⁱ⁺¹ (2i+ 1)! Ψγ(t),

͜͜Ͱ,γ∈(1,∞)ͱ͢Δ. ͜ͷؔ਺Λ༻͍Δ͜ͱͰ,ܗ

ࣜతͳղu¯Λऩଋ͢Δ΋ͷͱͯ͠ද͢͜ͱ͕Ͱ͖ͨ. ͜ ͷແݶ࿨Ͱද͞Εͨղu¯Λ͋Δ༗ݶͳN߲໨·Ͱͷ࿨

ͱͯ͠

sup

0≤t≤T,0≤x≤L|u¯−uˆ|< ϵ, (9) Λຬͨؔ͢਺

ˆ u(t, x) =

∑N i=0

(−1)ⁱY⁽²ⁱ⁾(t) x⁴ⁱ⁺² (4i+ 2)!

+

∑N i=0

(−1)ⁱZ⁽²ⁱ⁾(t) x⁴ⁱ⁺³ (4i+ 3)!, ͱͯ͠ఆٛ͢Δ.

ํఔࣜ(8)ΑΓ¯h(t) = ¯u(t, L), ¯I(t) =∂xu(t, L)¯ ͸ٻ

·Δ. ༗ݶ࿨Ͱද͞Εͨؔ਺uˆ͔Βٻ·Δ˜h,I˜͸, sup

0≤t≤T

��

�h¯−˜h���< ϵ, (10) sup

0≤t≤T

��

�I¯−I˜���< ϵ, (11) Λຬͨ͢΋ͷͱ͢Δ. ৚݅(9), (6)-(7)ͱ(10)-(11)Λຬ

ͨ͢Α͏ʹNΛऔΓ௚͢. Αͬͯ,

˜h(t) =

∑N i=0

(−1)ⁱY⁽²ⁱ⁾(t) L⁴ⁱ⁺² (4i+ 2)!

+

∑N i=0

(−1)ⁱZ⁽²ⁱ⁾(t) L⁴ⁱ⁺³ (4i+ 3)!, I˜(t) =

∑N i=0

(−1)ⁱY⁽²ⁱ⁾(t) L⁴ⁱ⁺¹ (4i+ 1)!

+

∑N i=0

(−1)ⁱZ⁽²ⁱ⁾(t) L⁴ⁱ⁺² (4i+ 2)!, ͱٻΊΒΕΔ. ͜ͷؔ਺Λೖྗͱͯ͠༻͍Δ.

3. ྊৼಈͷํఔࣜͷऑܗࣜͱධՁࣜ

͜͜Ͱ͸, Evans²⁾ ͷ§7.2ͷٞ࿦Λྊৼಈํఔࣜͷ

৔߹ʹద༻͠, ΤωϧΪʔෆ౳ࣜΛಋ͘. ྊৼಈͷํఔ

͕ࣜ





































∂_t²w+∂_x⁴w=f, (t, x)∈(0, T)×(0, L), w(0, x) =w0(x),

∂tw(0, x) =w₀¹(x), x∈(0, L), w(t,0) = 0,

w(t, L) = 0, t∈(0, T),

∂xw(t,0) = 0,

∂xw(t, L) = 0,

(12)

Ͱ༩͑ΒΕͨͱ͢Δ.

͜͜Ͱf ∈L²(

0, T;L²(0, L))

, w0 ∈H₀²(0, L), w₀¹ ∈ L²(0, L)ͷؔ਺ͱ͢Δ. ͜ͷͱ͖ؔ਺wΛ

w∈L²(

0, T;H₀²(0, L)) , ͱ͢Δ. ͨͩ͠, ∂tw ∈ L²(

0, T;L²(0, L))

, ∂²_tw ∈ L²(

0, T;H⁻¹(0, L))

ͱ͢Δ. ͜ͷͱ͖೚ҙͷ v ∈ H₀²(0, L)ʹରͯ͠t∈[0, T]Ͱ

(∂_t²w, v)

L²(0,L)+B[w, v;t] = (f, v)_L2(0,L), (13) Λߟ͑ͨͱ͖,͜Εͷ͕ࣜྊৼಈͷํఔࣜ(12)ͷऑܗࣜ

Ͱ͋Δ. ͜͜ͰB[u, v;t]͸೚ҙͷu, v∈H₀²(0, L)ʹର

ͯ͠

B[u, v;t] :=

∫ L 0

∂_x²u ∂_x²v dx, ͱ͢Δ.

ิ୊ 5 ྊৼಈͷํఔࣜͷऑܗࣜ(13)͕༩͑ΒΕͨͱ͖,

ͦͷऑղ͸t∈[0, T]Ͱఆ਺C ʹରͯ͠,

||∂tw||²L²(0,L)+||w||²H²(0,L)

≤C(

||w¹0||²L²(0,L)+||w0||H²²(0,L)+||f||²L²(0,T;L²(0,L))

) , (14) Λຬͨ͢.

ূ໌. GalerkinۙࣅΛ༻͍ͯূ໌͢Δ. ͦͷͨΊʹ׈

Β͔ͳؔ਺ͱͯ͠,θk=θk(x)(k= 1,2, . . .)ΛH0²(0, L) Ͱ௚ަجఈͱ͠,L²(0, L)Ͱਖ਼ن௚ަجఈͰ͋Δ΋ͷͱ

͢Δ. ͜ͷͱ͖,͋Δࣗવ਺mΛ༻͍ͯ৽ͨʹؔ਺wm

Λ

wm(t, x) :=

∑m k=1

d^km(t)θk(x),

ͱ͢Δ. ͜͜Ͱ܎਺d^k_m(k = 1,2,· · ·)͸೚ҙͷk = 1,2, . . .ʹରͯ͠t∈[0, T]Ͱ

d^m_k(0) = (w0, θk), d

dtd^m_k(0) =( w¹₀, θk

),

(∂_t²wm, θk) +B[wm, θk;t] = (f, θk),

Λຬͨؔ͢਺ͱ͢Δ. ͜ͷ3ͭ໨ͷࣜͷ྆ลʹ_dt^dd^m_k(t) Λֻ͚k= 1͔Βk=m·Ͱͷ࿨ΛͱΔ͜ͱͰ,

(∂_t²wm, ∂twm)

+B[wm, ∂twm;t] = (f, ∂twm), ΛಘΔ͜ͱ͕Ͱ͖Δ. ͜ͷࣜͷ3ͭͷ߲ʹ͍ͭͯͦΕͧ

Εߟ͑Δ.·ͣ,ࠨลͷୈ1߲໨͔Β (∂t²wm, ∂twm

)=

∫ L 0

∂t²wm∂twmdx

=

∫ L 0

∂

∂t 1

2(∂twm)²dx

= 1 2

∂

∂t

∫ L 0

(∂twm)²dx

= 1 2

∂

∂t||∂twm||²L²(0,L),

͕ٻ·Δ. ࣍ʹୈ2߲໨Λܭࢉ͢Δͱ, B[wm, ∂twm;t] =

∫ L 0

∂_x²wm∂²_x∂twmdx

=

∫ L 0

∂

∂t 1 2

(∂_x²wm

)2

dx

= 1 2

∂

∂t

∫ L 0

(∂_x²wm)²dx

= 1 2

∂

∂t||∂_x²wm||²L²(0,L),

͕ٻ·Δ. ӈลΛܭࢉ͢Δͱ, (f, ∂twm) =

∫ L 0

f ∂twmdx

≤

∫ L

0 |f| |∂twm|dx

≤ 1 2

∫ L

0 |f|²dx+ 1 2

∫ L

0 |∂twm|²dx

= 1

2||f||²L²(0,L)+1

2||∂twm||²L²(0,L),

͕ٻ·Δ. Αͬͯ͜ΕΒͷ͔ࣜΒऑܗࣜΑΓ, 1

2

∂

∂t (

||∂twm||²L²(0,L)+||∂_x²wm||²L²(0.L)

)

≤1 2

(||∂twm||²L²(0,L)+||f||²L²(0,L)

)

≤1 2

(||∂twm||²L²(0,L)+||∂x²wm||²L²(0.L)+||f||²L²(0,L)

) ,

ͱͳΔ. ৽ͨʹ

η(t) :=||∂twm||²L²(0,L)+||∂x²wm||²L²(0.L), ξ(t) :=||f||²L²(0,L),

ͱ͢Δ. ͜ͷͱ͖,΋ͱͷෆ౳ࣜ͸

∂tη(t)≤η(t) +ξ(t),

ͱͳΔ. ͜ͷࣜʹάϩϯ΢Υʔϧͷෆ౳ࣜΛ༻͍Δͱ, t∈[0, T]Ͱ

η(t)≤e^t(η(0) +

∫ t 0

ξ(s)ds),

͕ಘΒΕΔ. η(0)͸

η(0) =||∂twm(0,·)||²L²(0,L)+||∂x²wm(0,·)||²L²(0,L)

=||w¹₀||²L²(0,L)+||∂²_xwm(0,·)||²L²(0,L)

≤ ||w¹₀||²L²(0,L)+||wm(0,·)||²L²(0,L)

+||∂xwm(0,·)||²L²(0,L)+||∂x²wm(0,·)||²L²(0,L)

=||w¹₀||²L²(0,L)+||wm(0,·)||²H²(0,L)

=||w¹₀||²L²(0,L)+||w0||²H²(0,L), ͱͳΔ. ͜ͷࣜΛ༻͍Δ͜ͱͰ

||∂twm||²L²(0,L)+||∂x²wm||²L²(0.L)

≤ e^T (

||w¹0||²L²(0,L)+||w0||²H²(0,L)+

∫ t

0 ||f||²L²(0,L)ds )

≤ e^T(

||w¹₀||²L²(0,L)+||w0||²H²(0,L)+||f||²L²(0,T;L²(0,L))

),

͕ಘΒΕΔ. θk͸H₀²(0, L)ͷ௚ަجఈͰ͔͋ͬͨΒ,ಛ ʹwm, ∂xwm∈H₀¹(0, L)Ͱ͋Δ. Αͬͯ,wm,∂xwmʹ ϙΞϯΧϨͷෆ౳ࣜΛద༻Ͱ͖ͯ,े෼େ͖͍ਖ਼ͷఆ਺

CʹΑΓ,

||∂twm||²L²(0,L)+||wm||²H²(0.L)

≤C(

||w₀¹||²L²(0,L)+||w0||²H²(0,L)+||f||²L²(0,T;L²(0,L))

),

(15) ͱͳΔ.

ࣜ(15)ΑΓ, (wm)^∞m=1͸H²(0, L)Ͱऑऩଋ͢Δ෦෼

ྻ(wml)^∞_l=1Λ࣋ͪ, ऑऩଋۃݶw͕ํఔࣜ(12)ͷऑ ղʹͳ͍ͬͯΔ͜ͱ͕෼͔Δ. ͞Βʹ, ∥w∥H²(0,L) ≤ lim inf_l→∞∥wml∥H²(0,L) ͕੒Γཱͭ(ྫͱͯ͠, ఆཧ 8.29³⁾). Αͬͯ,

||∂tw||L²(0,L)+||w||H²(0.L)

≤C(

||w¹0||L²(0,L)+||w0||H²(0,L)+||f||L²(0,T;L²(0,L))

),

ΛಘΔ͜ͱ͕Ͱ͖Δ. (ূ໌ऴ)

4. ओ݁Ռ

§2-§3Λ༻͍Δ͜ͱͰํఔࣜ(1)-(3)ͷۙࣅ੍ޚ໰୊Λ ղ͘͜ͱ͕Ͱ͖ͨ.

ఆཧ 6 ͋Δ༩͑ΒΕͨt= 0Ͱͷঢ়ଶu0, u¹₀͔Β, ༩

͑ΒΕͨt=TͰͷঢ়ଶuT =u(T, x), u¹_T =∂tu(T, x)

΁ͷঢ়ଶͷมԽΛߟ͑Δ. ͦͷͱ͖, ํఔࣜ(1)-(3)Λ h= ˜h, I= ˜Iͱͨ͠ͱ͖ͷํఔࣜͷղu˜͸,͋ΔK >0 ʹରͯ͠,

||uT −u(T,˜ ·)||H²(0,L)< Kϵ, (16)

||u¹_T−∂tu(T,˜ ·)||L²(0,L)< Kϵ, (17) ͱͳΔ.

ূ໌. ओఆཧΛূ໌͢ΔͨΊʹ,§3ͷධՁࣜΛ༻͍ͯ

ূ໌͢Δ. ͦͷͨΊʹ,ํఔࣜΛ§3ͷܗͷํఔࣜʹม׵

͢Δ͜ͱΛߟ͑Δ. ؔ਺χΛ

χ(x) =

{ 0 0≤x < δ, 1 L−δ < x≤L,

Ͱ͋ΔC^∞([0, L])ͷؔ਺ͱఆٛ͢Δ. ͜ͷؔ਺ʹΑͬͯ, w(t, x) :=u(t, x)−χ(x)h(t)−χ(x)(x−L)I(t),

ͱ͢Δ. w0, w¹₀, fΛ

w0(x) =u0(x)−χ(x)h(0)−χ(x)(x−L)I(0), w¹₀(x) =u¹₀(x)−χ(x)∂th(0)−χ(x)(x−L)∂tI(0), f(t, x) =−χ(x)∂t²h(t)−χ(x)(x−L)∂_t²I(t)

−∂⁴_xχ(x)h(t)−∂_x⁴χ(x)·(x−L)I(t)

−4∂_x³χ(x)I(t),

ͱͯ͠,w͸§3ͷํఔࣜΛຬͨ͢ܗʹͳ͍ͬͯΔ. ͜ͷ

ؔ܎Λ༻͍ͯෆ౳ࣜ(16)-(17)Λࣔ͢. ̍ͭ໨ͷෆ౳ࣜ

(16)ʹ͍ͭͯߟ͑Δ. ධՁࣜͷࠨลΑΓ,

||uT −u(T,˜ ·)||H²(0,L)

≤||uT −PT||H²(0,L)+||PT−u(T,˜ ·)||H²(0,L), ͱͳΔ. ͜ͷ2߲໨ͷ||PT −u(T,˜ ·)||H²(0,L)ͷPT ͱ

˜

u(T, x)͸ͦΕͧΕํఔࣜ(8)ͷ৚݅ͱํఔࣜ(1)-(3)Λ h= ˜h, I = ˜Iͱͨ͠ͱ͖ͷํఔࣜͷղu˜͔ΒͳΔࣜͰ

͋Δ. ͦΕͧΕͷํఔࣜΛχ(x)Λ༻͍ͯલͱಉ༷ʹ§3 ͷํఔࣜͷܗʹ௚͢͜ͱΛߟ͑Δ.

¯

w(T, x) =PT(x)−χ(x)¯h(T)−χ(x)(x−L) ¯I,

˜

w(T, x) = ˜u(T, x)−χ(x)˜h(T)−χ(x)(x−L) ˜I(T),

ʹΑΓ, 2߲໨͸

||PT −u(T,˜ ·)||H²(0,L)

≤||(PT−χ¯h(T)−χ·(x−L) ¯I(T))

−(˜u(T,·)−χ˜h(T)−χ·(x−L) ˜I(T))||H²(0,L)

+||χ(˜h(T)−¯h(T))||H²(0,L)

+||χ·(x−L)( ˜I(T)−I(T¯ ))||H²(0,L)

≤||w(T,¯ ·)−w(T,˜ ·)||H²(0,L)

+||χ(˜h(T)−¯h(T))||H²(0,L)

+||χ·(x−L)( ˜I(T)−I¯(T))||H²(0,L), ͱͳΔ. Αͬͯ,ෆ౳ࣜ(16)͸

||uT −˜u(T,·)||H²(0,L)

≤||uT −PT||H²(0,L)+||w(T,¯ ·)−w(T,˜ ·)||H²(0,L)

+||χ(˜h(T)−¯h(T))||H²(0,L)

+||χ·(x−L)( ˜I(T)−I¯(T))||H²(0,L)

ͱٻ·Δ. ͜ͷෆ౳ࣜͷӈล͸2߲໨Λআ͖,ͦΕͧΕ Ծఆͱೖྗͷ৚݅(7), (10)ͱ(11)͔Β,͋Δఆ਺A >0 ʹରͯ͠,

||uT−u(T,˜ ·)||H²(0,L)

<||w(T,¯ ·)−w(T,˜ ·)||H²(0,L)+Aϵ

ͱٻ·Δ. ͜ͷ࢒Γͷ߲ʹ͍ͭͯධՁ͢Δ. ͜ͷ߲ʹର

ͯ͠,§3ͷධՁࣜ(14)Λ༻͍Δ͜ͱͰ

||w(T,¯ ·)−w(T,˜ ·)||H²(0,L)

<C(||w(0,¯ ·)−w(0,˜ ·)||H²(0,L)

+||∂tw(0,¯ ·)−∂tw(0,˜ ·)||L²(0,L)

+||f¯−f˜||L²(0,T;L²(0,L))), ͱͳΔ. ͜͜Ͱ, ¯f , f˜͸

f(t, x) =¯ −χ(x)∂²_t¯h(t)−χ(x)(x−L)∂_t²I(t)¯

−∂_x⁴χ(x)¯h(t)−∂_x⁴χ(x)·(x−L) ¯I(t)

−4∂³_xχ(x) ¯I(t),

f(t, x) =˜ −χ(x)∂²_t˜h(t)−χ(x)(x−L)∂_t²I(t)˜

−∂_x⁴χ(x)˜h(t)−∂_x⁴χ(x)·(x−L) ˜I(t)

−4∂³_xχ(x) ˜I(t),

Ͱ͋Δ. §3ͷධՁࣜ(14)Λ༻͍ͨࣜͷ߲͢΂ͯ,ೖྗͷ

৚݅(10)-(11)͔Β,ਖ਼ͷఆ਺B, D, Eʹରͯ͠,

||w(0,¯ ·)−w(0,˜ ·)||H²(0,L)< Bϵ,

||∂tw(0,¯ ·)−∂tw(0,˜ ·)||L²(0,L)< Dϵ,

||f¯−f˜||L²(0,T;L²(0,L)))< Eϵ,

ͱͳΔ. Αͬͯ,ਖ਼ͷఆ਺F ʹରͯ͠,

||w(T,¯ ·)−w(T,˜ ·)||H²(0,L)< F ϵ, ͱٻ·Δ. ͜ΕΒΛ·ͱΊΔ͜ͱͰ,

||uT−u(T,˜ ·)||H²(0,L)< Kϵ, ͱٻΊΔ͜ͱ͕Ͱ͖Δ.

࠷ޙʹෆ౳ࣜ(17)͕੒Γཱͭ͜ͱΛߟ͑Δ. ͜ͷෆ

౳ࣜ(17)΋ෆ౳ࣜ(16)ͱಉ༷ʹߦ͏͜ͱͰٻ·Δ. (ূ

໌ऴ)

5. ۙࣅଟ߲ࣜͷߏ੒ʹؔ͢ΔิҨ

͜͜Ͱ͸, ৚݅(6)-(7)Λຬͨ͢ଟ߲ࣜͷߏ੒ʹ͍ͭ

ͯड़΂Δ. ࣍ͷิ୊͕,ߏ੒๏Λ͍ࣔͯ͠Δ.

ิ୊ 7 f ∈H²(0, L)ͱ͢Δ. ೚ҙͷϵ >0ʹରͯ͠,͋ Δଟ߲ࣜp͕ଘࡏ͠,

∥f−p∥H²(0,L)< ϵ, Λຬͨ͢.

ূ໌. C²(0, L)ͷH²(0, L)ʹ͓͚Δ᜚ີੑ͔Β,fΛ

[0, L]Ͱ2֊࿈ଓඍ෼Մೳͳؔ਺ͱͯ͠ҰൠੑΛࣦΘͳ

͍. ϫΠΤϧγϡτϥεͷଟ߲ࣜۙࣅఆཧʹΑΓ, ∂²_xf Λ[0, L]ͰҰ༷ʹۙࣅ͠, ͞Βʹ۠ؒ(0, L)ͷ༗քੑ͔

Β,೚ҙʹখ͍͞ϵ >0ʹରͯ͠

∥∂²xf−p0∥L²(0,L)< ϵ, (18) ͱͳΔଟ߲ࣜp0͕ଘࡏ͢Δ.

p1(x) =

∫ x 0

p0(s)ds+∂xf(0), x∈(0, L), ͱ͓͘ͱ,p1΋ଟ߲ࣜͰ͋Δ. ͞Βʹ,

∥∂xf−p1∥²L²(0,L)

=

∫ L 0

��

��

∫ x 0

(∂_x²f(s)−p0(s)) ds

��

��

2

dx

≤ ∥∂_x²f−p0∥²L²(0,L)

∫ L 0

x dx

≤L²ϵ² 2 ,

(19)

Λຬͨ͢.

p(x) =

∫ x 0

p1(s)ds+f(0), x∈(0, L),

ͱ͓͖, ಉ༷ͷධՁΛ܁Γฦ͢͜ͱͰ, े෼େ͖ͳਖ਼ͷ ఆ਺Cʹର͠

∥f−p∥L²(0,L)< Cϵ (20)

͕੒Γཱͭ͜ͱ΋ࣔͤΔ.

ෆ౳ࣜ(18), (19), (20)ΑΓ,վΊͯখ͞ͳϵ >0Λऔ Γ௚ͤ͹,ิ୊͕ಘΒΕΔ. (ূ໌ऴ)

6. ݁ݴ

ྊৼಈͷํఔࣜʹ͍ͭͯݚڀ͠ҎԼͷ݁ՌΛಘͨ.

1. ྊৼಈͷํఔࣜͷऑܗࣜΛఆٛ͢Δ͜ͱ͕Ͱ͖ͨ.

2. ఆٛͨ͠ऑܗ͔ࣜΒղʹ͍ͭͯͷධՁࣜΛಘΔ͜ͱ

͕Ͱ͖ͨ.

3. ྊৼಈͷํఔࣜʹ͍ͭͯۙࣅ੍ޚ໰୊Λղ͘͜ͱ͕

Ͱ͖ͨ.

ࢀߟจݙ

1) B. Laroche, P. Martin and P. Rouchon, “Motion Plan- ning for the Heat Equation”,Int. J. Robust Nonlinear Control,10, 629-643 (2000).

2) L. C. Evans, Partial Differential Equations , Second Edition, (American Mathematical Society, Rhode Is- land, 2010).

3) ࠇా੒ढ़,ؔ਺ղੳ,ڞཱ਺ֶߨ࠲15, (ڞཱग़൛,౦ژ, 1980), p.191.