Nonlinear hyperbolic balance laws coupled with ordinary differential - - PowerPoint PPT Presentation

Nonlinear hyperbolic balance laws coupled with ordinary differential equations

Mauro Garavello

Department of Mathematics and Applications University of Milano Bicocca mauro.garavello@unimib.it Joint works with R. Borsche and R. M. Colombo

June 28, 2012 HYP2012

PDE-ODE Model

                   ∂tu + ∂xf(u) = g(u) t > 0, x > γ(t) b (u(t, γ(t))) = B (t, w(t)) t > 0 ˙ w(t) = F (t, u(t, γ(t)), w(t)) t > 0 ˙ γ(t) = Π(w(t)) t > 0 u(0, x) = uo(x) x > xo w(0) = wo γ(0) = xo

• t

x uo

∂tu + ∂xf(u) = g(u) b(u) = B(t, w(t))

γ

June 28, 2012 HYP2012

PDE-ODE Model

                   ∂tu + ∂xf(u) = g(u) t > 0, x > γ(t) b (u(t, γ(t))) = B (t, w(t)) t > 0 ˙ w(t) = F (t, u(t, γ(t)), w(t)) t > 0 ˙ γ(t) = Π(w(t)) t > 0 u(0, x) = uo(x) x > xo w(0) = wo γ(0) = xo

u ∈ Ω ⊆ Rn, w ∈ Rm

• t

x uo

∂tu + ∂xf(u) = g(u) b(u) = B(t, w(t))

γ

June 28, 2012 HYP2012

PDE-ODE Model

                   ∂tu + ∂xf(u) = g(u) t > 0, x > γ(t) b (u(t, γ(t))) = B (t, w(t)) t > 0 ˙ w(t) = F (t, u(t, γ(t)), w(t)) t > 0 ˙ γ(t) = Π(w(t)) t > 0 u(0, x) = uo(x) x > xo w(0) = wo γ(0) = xo

u ∈ Ω ⊆ Rn, w ∈ Rm f smooth and Df(u) strictly hyperbolic ∀ u ∈ Ω

• t

x uo

∂tu + ∂xf(u) = g(u) b(u) = B(t, w(t))

γ

June 28, 2012 HYP2012

PDE-ODE Model

                   ∂tu + ∂xf(u) = g(u) t > 0, x > γ(t) b (u(t, γ(t))) = B (t, w(t)) t > 0 ˙ w(t) = F (t, u(t, γ(t)), w(t)) t > 0 ˙ γ(t) = Π(w(t)) t > 0 u(0, x) = uo(x) x > xo w(0) = wo γ(0) = xo

u ∈ Ω ⊆ Rn, w ∈ Rm f smooth and Df(u) strictly hyperbolic ∀ u ∈ Ω λ1(u) < · · · < λℓ(u) < inf Π − c < sup Π + c < λℓ+1(u) < · · · < λn(u) for all u ∈ Ω

• t

x uo

∂tu + ∂xf(u) = g(u) b(u) = B(t, w(t))

γ

June 28, 2012 HYP2012

PDE-ODE Model

                   ∂tu + ∂xf(u) = g(u) t > 0, x > γ(t) b (u(t, γ(t))) = B (t, w(t)) t > 0 ˙ w(t) = F (t, u(t, γ(t)), w(t)) t > 0 ˙ γ(t) = Π(w(t)) t > 0 u(0, x) = uo(x) x > xo w(0) = wo γ(0) = xo

u ∈ Ω ⊆ Rn, w ∈ Rm f smooth and Df(u) strictly hyperbolic ∀ u ∈ Ω λ1(u) < · · · < λℓ(u) < inf Π − c < sup Π + c < λℓ+1(u) < · · · < λn(u) for all u ∈ Ω b : Ω → Rn−ℓ smooth such that det [Dub(u) rℓ+1(u)· · ·Dub(u) rn(u)] = 0

• t

x uo

∂tu + ∂xf(u) = g(u) b(u) = B(t, w(t))

γ

June 28, 2012 HYP2012

Boundary hypotheses

b : Ω → Rn−ℓ smooth such that det [Dub(u) rℓ+1(u) · · · Dub(u) rn(u)] = 0

June 28, 2012 HYP2012

Boundary hypotheses

b : Ω → Rn−ℓ smooth such that det [Dub(u) rℓ+1(u) · · · Dub(u) rn(u)] = 0 Dub(u) rℓ+1(u) = 0Rn−ℓ: the boundary datum “acts” on waves of ℓ + 1 family

June 28, 2012 HYP2012

Boundary hypotheses

b : Ω → Rn−ℓ smooth such that det [Dub(u) rℓ+1(u) · · · Dub(u) rn(u)] = 0 Dub(u) rℓ+1(u) = 0Rn−ℓ: the boundary datum “acts” on waves of ℓ + 1 family . . .

June 28, 2012 HYP2012

Boundary hypotheses

b : Ω → Rn−ℓ smooth such that det [Dub(u) rℓ+1(u) · · · Dub(u) rn(u)] = 0 Dub(u) rℓ+1(u) = 0Rn−ℓ: the boundary datum “acts” on waves of ℓ + 1 family . . . Dub(u) rn(u) = 0Rn−ℓ: the boundary datum “acts” on waves

• f n family

June 28, 2012 HYP2012

Boundary hypotheses

b : Ω → Rn−ℓ smooth such that det [Dub(u) rℓ+1(u) · · · Dub(u) rn(u)] = 0 Dub(u) rℓ+1(u) = 0Rn−ℓ: the boundary datum “acts” on waves of ℓ + 1 family . . . Dub(u) rn(u) = 0Rn−ℓ: the boundary datum “acts” on waves

• f n family

b “prescribes” the boundary data for waves with speed greater than sup Π + c.

June 28, 2012 HYP2012

Similar problems

1

Lattanzio-Maurizi-Piccoli and Delle Monache-Goatin: “Moving bottlenecks” for traffic.

June 28, 2012 HYP2012

Similar problems

1

Lattanzio-Maurizi-Piccoli and Delle Monache-Goatin: “Moving bottlenecks” for traffic.

       ρt + f(x, y(t), ρ)x = 0 ˙ y(t) = w(ρ(t, y(t))) ρ(0, x) = ρo(x) y(0) = yo

June 28, 2012 HYP2012

Similar problems

1

Lattanzio-Maurizi-Piccoli and Delle Monache-Goatin: “Moving bottlenecks” for traffic.

           ρt + f(ρ)x = 0 ˙ y(t) = w(ρ(t, y(t)+)) ρ(t, y(t)) ≤ αR ρ(0, x) = ρo(x) y(0) = yo

June 28, 2012 HYP2012

Similar problems

1

Lattanzio-Maurizi-Piccoli and Delle Monache-Goatin: “Moving bottlenecks” for traffic.

           ρt + f(ρ)x = 0 ˙ y(t) = w(ρ(t, y(t)+)) ρ(t, y(t)) ≤ αR ρ(0, x) = ρo(x) y(0) = yo

2

Lagouti` ere-Seguin-Takahashi and Andreianov-Lagouti` ere-Seguin-Takahashi: interactions between fluids and solids.

         ut +

• u2

2

• x = (h′(t) − u(t, h(t)))δh(t)(x)

h′′(t) = −h′(t) + u(t, h(t)) u(0, x) = uo(x) (h(0), h′(0)) = (0, vo)

June 28, 2012 HYP2012

• Applications. 1 - Gas-particle interaction

X(t)

ρ → density q → linear momentum density p → pressure X and V → particle position and speed m → particle mass g → gravity

         ∂tρ − ∂xq = 0 ∂tq + ∂x

• q2

ρ + p(ρ)

• = −gρ

V = v(t, 0+) ˙ V = −g − p(ρ+)−p(ρ−)

m

June 28, 2012 HYP2012

• Applications. 2 - The piston problem
• τ

P V

τ → specific volume v → Lagrangian speed p → pressure V → speed of the piston P → pressure to the left of the piston        ∂tτ − ∂xv = 0 ∂tv + ∂xp(τ) = 0 V = v(t, 0+) ˙ V = (P(t) − p (τ(t, 0+)))

June 28, 2012 HYP2012

Applications. 3 - Moving bottleneck

X(t) ρ → traffic density v → average speed p → “pressure” X → truck position      ∂tρ + ∂x (ρv) = 0 ∂t (ρ (v + p(ρ))) + ∂x (ρv (v + p(ρ))) = 0 ¨ X = − 1

T ∗

• ˙

X − V∗

• 1 − ρ+(t,X(t)+)

R

• June 28, 2012

HYP2012

Applications. 4 - Supply chains

q1 q2 A(t)

ρi → material density vi → production speed qi → queue load A(t) → distribution matrix µk → maximal production capacity        ∂tρi(t, x) + ∂x (viρi(t, x)) = 0 i = 1, . . . , n vk ρk(t, 0+) = min qk (t)

ǫ

, µk

• k = ℓ + 1, . . . , n

˙ qk(t) = ℓ

j=1 ajk(t) vk ρk(t, 0) − min

qk (t)

ǫ

, µk

• k = ℓ + 1, . . . , n .

June 28, 2012 HYP2012

Definition of solution

The triple (u, w, γ) such that

       u ∈ C0 R+; L1(R+; Ω)

• u(t) ∈ BV(R+; Ω) for a.e. t > 0

w ∈ W1,1(R+; Rm) γ ∈ W1,∞(R+; R)

is a solution of

                 ∂tu + ∂xf(u) = g(u) t > 0, x > 0 b (u(t, γ(t))) = B (t, w(t)) t > 0 ˙ w = F (t, u(t, 0), w) t > 0 ˙ γ = Π(w) t > 0 u(0, x) = uo(x) x > 0 w(0) = wo γ(0) = xo

if

June 28, 2012 HYP2012

Definition of solution

The triple (u, w, γ) such that

       u ∈ C0 R+; L1(R+; Ω)

• u(t) ∈ BV(R+; Ω) for a.e. t > 0

w ∈ W1,1(R+; Rm) γ ∈ W1,∞(R+; R)

is a solution of

                 ∂tu + ∂xf(u) = g(u) t > 0, x > 0 b (u(t, γ(t))) = B (t, w(t)) t > 0 ˙ w = F (t, u(t, 0), w) t > 0 ˙ γ = Π(w) t > 0 u(0, x) = uo(x) x > 0 w(0) = wo γ(0) = xo

if u solves ∂tu + ∂xf(u) = g(u) b (u(t, γ(t))) = B∗(t) with B∗(t) = B (t, w(t))

June 28, 2012 HYP2012

Definition of solution

The triple (u, w, γ) such that

       u ∈ C0 R+; L1(R+; Ω)

• u(t) ∈ BV(R+; Ω) for a.e. t > 0

w ∈ W1,1(R+; Rm) γ ∈ W1,∞(R+; R)

is a solution of

                 ∂tu + ∂xf(u) = g(u) t > 0, x > 0 b (u(t, γ(t))) = B (t, w(t)) t > 0 ˙ w = F (t, u(t, 0), w) t > 0 ˙ γ = Π(w) t > 0 u(0, x) = uo(x) x > 0 w(0) = wo γ(0) = xo

if u solves ∂tu + ∂xf(u) = g(u) b (u(t, γ(t))) = B∗(t) with B∗(t) = B (t, w(t)) w(t) = wo + t

0 F∗(τ)dτ with F∗(t) = F (t, u(t, γ(t)+), w)

June 28, 2012 HYP2012

Definition of solution

The triple (u, w, γ) such that

       u ∈ C0 R+; L1(R+; Ω)

• u(t) ∈ BV(R+; Ω) for a.e. t > 0

w ∈ W1,1(R+; Rm) γ ∈ W1,∞(R+; R)

is a solution of

                 ∂tu + ∂xf(u) = g(u) t > 0, x > 0 b (u(t, γ(t))) = B (t, w(t)) t > 0 ˙ w = F (t, u(t, 0), w) t > 0 ˙ γ = Π(w) t > 0 u(0, x) = uo(x) x > 0 w(0) = wo γ(0) = xo

if u solves ∂tu + ∂xf(u) = g(u) b (u(t, γ(t))) = B∗(t) with B∗(t) = B (t, w(t)) w(t) = wo + t

0 F∗(τ)dτ with F∗(t) = F (t, u(t, γ(t)+), w)

γ(t) = xo + t

0 Π∗(τ)dτ with Π∗(t) = Π (w(t))

June 28, 2012 HYP2012

Theorem: Well posedness of the system

                 ∂t u + ∂x f(u) = g(u) t > 0, x > γ(t) b (u(t, γ(t))) = B (t, w(t)) t > 0 ˙ w = F (t, u(t, γ(t)), w) t > 0 ˙ γ(t) = Π(w(t)) u(0, x) = uo(x) x > 0 w(0) = wo γ(0) = xo

Suppose that

June 28, 2012 HYP2012

Theorem: Well posedness of the system

                 ∂t u + ∂x f(u) = g(u) t > 0, x > γ(t) b (u(t, γ(t))) = B (t, w(t)) t > 0 ˙ w = F (t, u(t, γ(t)), w) t > 0 ˙ γ(t) = Π(w(t)) u(0, x) = uo(x) x > 0 w(0) = wo γ(0) = xo

Suppose that (B) B ∈ C1(R+ × Rm; Rn−ℓ) locally Lipschitz continuous

June 28, 2012 HYP2012

Theorem: Well posedness of the system

                 ∂t u + ∂x f(u) = g(u) t > 0, x > γ(t) b (u(t, γ(t))) = B (t, w(t)) t > 0 ˙ w = F (t, u(t, γ(t)), w) t > 0 ˙ γ(t) = Π(w(t)) u(0, x) = uo(x) x > 0 w(0) = wo γ(0) = xo

Suppose that (B) B ∈ C1(R+ × Rm; Rn−ℓ) locally Lipschitz continuous (F) F : R+ × Ω × Rm − → Rm satisfies

June 28, 2012 HYP2012

Theorem: Well posedness of the system

                 ∂t u + ∂x f(u) = g(u) t > 0, x > γ(t) b (u(t, γ(t))) = B (t, w(t)) t > 0 ˙ w = F (t, u(t, γ(t)), w) t > 0 ˙ γ(t) = Π(w(t)) u(0, x) = uo(x) x > 0 w(0) = wo γ(0) = xo

Suppose that (B) B ∈ C1(R+ × Rm; Rn−ℓ) locally Lipschitz continuous (F) F : R+ × Ω × Rm − → Rm satisfies

1

w → F(t, u, w) continuous

June 28, 2012 HYP2012

Theorem: Well posedness of the system

                 ∂t u + ∂x f(u) = g(u) t > 0, x > γ(t) b (u(t, γ(t))) = B (t, w(t)) t > 0 ˙ w = F (t, u(t, γ(t)), w) t > 0 ˙ γ(t) = Π(w(t)) u(0, x) = uo(x) x > 0 w(0) = wo γ(0) = xo

Suppose that (B) B ∈ C1(R+ × Rm; Rn−ℓ) locally Lipschitz continuous (F) F : R+ × Ω × Rm − → Rm satisfies

1

w → F(t, u, w) continuous

2

F locally Lipschitz continuous w.r.t. u and w

June 28, 2012 HYP2012

Theorem: Well posedness of the system

                 ∂t u + ∂x f(u) = g(u) t > 0, x > γ(t) b (u(t, γ(t))) = B (t, w(t)) t > 0 ˙ w = F (t, u(t, γ(t)), w) t > 0 ˙ γ(t) = Π(w(t)) u(0, x) = uo(x) x > 0 w(0) = wo γ(0) = xo

Suppose that (B) B ∈ C1(R+ × Rm; Rn−ℓ) locally Lipschitz continuous (F) F : R+ × Ω × Rm − → Rm satisfies

1

w → F(t, u, w) continuous

2

F locally Lipschitz continuous w.r.t. u and w

3

F(t, u, w) ≤ C(t) (1 + w)

June 28, 2012 HYP2012

Theorem: Well posedness of the system

                 ∂t u + ∂x f(u) = g(u) t > 0, x > γ(t) b (u(t, γ(t))) = B (t, w(t)) t > 0 ˙ w = F (t, u(t, γ(t)), w) t > 0 ˙ γ(t) = Π(w(t)) u(0, x) = uo(x) x > 0 w(0) = wo γ(0) = xo

Suppose that (B) B ∈ C1(R+ × Rm; Rn−ℓ) locally Lipschitz continuous (F) F : R+ × Ω × Rm − → Rm satisfies

1

w → F(t, u, w) continuous

2

F locally Lipschitz continuous w.r.t. u and w

3

F(t, u, w) ≤ C(t) (1 + w) (Π) Π Lipschitz continuous

June 28, 2012 HYP2012

Theorem: Well posedness of the system

                 ∂t u + ∂x f(u) = g(u) t > 0, x > γ(t) b (u(t, γ(t))) = B (t, w(t)) t > 0 ˙ w = F (t, u(t, γ(t)), w) t > 0 ˙ γ(t) = Π(w(t)) u(0, x) = uo(x) x > 0 w(0) = wo γ(0) = xo

Suppose that (B) B ∈ C1(R+ × Rm; Rn−ℓ) locally Lipschitz continuous (F) F : R+ × Ω × Rm − → Rm satisfies

1

w → F(t, u, w) continuous

2

F locally Lipschitz continuous w.r.t. u and w

3

F(t, u, w) ≤ C(t) (1 + w) (Π) Π Lipschitz continuous (f) f smooth, Df strictly hyperbolic

June 28, 2012 HYP2012

Theorem: Well posedness of the system

                 ∂t u + ∂x f(u) = g(u) t > 0, x > γ(t) b (u(t, γ(t))) = B (t, w(t)) t > 0 ˙ w = F (t, u(t, γ(t)), w) t > 0 ˙ γ(t) = Π(w(t)) u(0, x) = uo(x) x > 0 w(0) = wo γ(0) = xo

Suppose that (B) B ∈ C1(R+ × Rm; Rn−ℓ) locally Lipschitz continuous (F) F : R+ × Ω × Rm − → Rm satisfies

1

w → F(t, u, w) continuous

2

F locally Lipschitz continuous w.r.t. u and w

3

F(t, u, w) ≤ C(t) (1 + w) (Π) Π Lipschitz continuous (f) f smooth, Df strictly hyperbolic (g) g(u) − g(u′)L1 ≤ L1u − u′L1 and TV(g(u)) ≤ L2.

June 28, 2012 HYP2012

Theorem: Well posedness of the system

there exist T > 0, L > 0, domains Dt and maps P(t, t0) : Dt0 → Dt0+t such that

1

for (uo, wo, xo) ∈ D0, t → P(t, 0) (uo, wo, xo) is a solution

2

for t0 ∈ [0, T[, t1 ∈ [0, T − t0[, t2 ∈ [0, T − t0 − t1[, P(t2, t0 + t1) ◦ P(t1, t0) = P(t1 + t2, t0); P(0, t0) = Id

3

for t0 ∈ [0, T[, t ∈ [0, T − t0], (u, w, x), (¯ u, ¯ w, ¯ x) ∈ Dt0

P(t, t0)(u, w, x) − P(t, t0)(¯ u, ¯ w, ¯ x) ≤ L (u − ¯ u + w − ¯ w + |x − ¯ x|)

June 28, 2012 HYP2012

Proof — 1

• (Colombo-Guerra) Theorem about systems of balance laws with

boundary

June 28, 2012 HYP2012

Proof — 1

• (Colombo-Guerra) Theorem about systems of balance laws with

boundary

   ut + f(u)x = g(u) b(u(t, γ∗(t))) = B∗(t) u(0, x) = uo(x) Under suitable assumptions, there exists a process PB∗(t, to): Dto → Dto+t such that

1

PB∗ (0, to) u = u and PB∗ (t + s, to) u = PB∗ (t, to + s) ◦ PB∗(s, to)u

June 28, 2012 HYP2012

Proof — 1

• (Colombo-Guerra) Theorem about systems of balance laws with

boundary

   ut + f(u)x = g(u) b(u(t, γ∗(t))) = B∗(t) u(0, x) = uo(x) Under suitable assumptions, there exists a process PB∗(t, to): Dto → Dto+t such that

1

PB∗ (0, to) u = u and PB∗ (t + s, to) u = PB∗ (t, to + s) ◦ PB∗(s, to)u

2

Lipschitz estimates

• PB∗ (t, to)u−P¯

B∗ (t, to)v

• ≤

L

• u − v+
• B∗ − ¯

B∗

• +
• γ1

∗ − γ2 ∗

• t0+t

t0

Tr (P1(τ, t0)u1) − Tr (P2(τ, t0)u2)dτ ≤ L

• u − v+
• B∗ − ¯

B∗

• +
• γ1

∗ − γ2 ∗

• June 28, 2012

HYP2012

Proof — 1

• (Colombo-Guerra) Theorem about systems of balance laws with

boundary

   ut + f(u)x = g(u) b(u(t, γ∗(t))) = B∗(t) u(0, x) = uo(x) Under suitable assumptions, there exists a process PB∗(t, to): Dto → Dto+t such that

1

PB∗ (0, to) u = u and PB∗ (t + s, to) u = PB∗ (t, to + s) ◦ PB∗(s, to)u

2

Lipschitz estimates

• PB∗ (t, to)u−P¯

B∗ (t, to)v

• ≤

L

• u − v+
• B∗ − ¯

B∗

• +
• γ1

∗ − γ2 ∗

• t0+t

t0

Tr (P1(τ, t0)u1) − Tr (P2(τ, t0)u2)dτ ≤ L

• u − v+
• B∗ − ¯

B∗

• +
• γ1

∗ − γ2 ∗

• 3

u(t, x) =

• PB∗(t, 0)uo
• (x) is the solution

June 28, 2012 HYP2012

Proof — 1

• (Colombo-Guerra) Theorem about systems of balance laws with

boundary

• Well posedness results for ODEs
• ˙

w(t) = F(t, w(t)) w(0) = wo IP1: F Carath´ eodory IP2: F Lipschitz continuous w.r.t. w IP2: Grow conditions for F TH: Existence + continuous dependence + stability

June 28, 2012 HYP2012

Proof — 1

• (Colombo-Guerra) Theorem about systems of balance laws with

boundary

• Well posedness results for ODEs
• Completeness

June 28, 2012 HYP2012

Proof — 1

• (Colombo-Guerra) Theorem about systems of balance laws with

boundary

• Well posedness results for ODEs
• Completeness
• Stability inequalities

June 28, 2012 HYP2012

Proof — 2

Fix the constant functions: u0(t, x) = uo(x), w0(t) = wo and γ0(t) = xo.

June 28, 2012 HYP2012

Proof — 2

Fix the constant functions: u0(t, x) = uo(x), w0(t) = wo and γ0(t) = xo. Define the sequences γk, uk and wk such that γk as γk = xo + t

0 Π(wk−1(τ))dτ

uk solution of    ∂tu + ∂xf(u) = g(u) b (u(t, γk(t))) = B (wk−1(t)) u(0, x) = uo(x) wk solution of ˙ w = F(t, uk−1(t, γk−1(t)), w) w(0) = wo .

June 28, 2012 HYP2012

Proof — 3

• wk is a Cauchy sequence

wk(t) − wk−1(t) ≤ C1 t wk(τ) − wk−1(τ)dτ +C2 t wk−2(τ) − wk−3(τ)dτ

June 28, 2012 HYP2012

Proof — 3

• wk is a Cauchy sequence

wk(t) − wk−1(t) ≤ C1 t wk(τ) − wk−1(τ)dτ +C2 t wk−2(τ) − wk−3(τ)dτ Gronwall inequality wk(t) − wk−1(t) ≤ C3eC3t t wk−2(τ) − wk−3(τ)dτ

June 28, 2012 HYP2012

Proof — 3

• wk is a Cauchy sequence

wk(t) − wk−1(t) ≤ C1 t wk(τ) − wk−1(τ)dτ +C2 t wk−2(τ) − wk−3(τ)dτ By recursion wk(t) − wk−1(t) ≤ Ck

4

k! [w1 − w0 + w2 − w1]

June 28, 2012 HYP2012

Proof — 3

• wk is a Cauchy sequence: wk → w∗

wk(t) − wk−1(t) ≤ C1 t wk(τ) − wk−1(τ)dτ +C2 t wk−2(τ) − wk−3(τ)dτ By recursion wk(t) − wk−1(t) ≤ Ck

4

k! [w1 − w0 + w2 − w1]

June 28, 2012 HYP2012

Proof — 3

• wk is a Cauchy sequence: wk → w∗
• Lebesgue Theorem: γk → γ∗ = xo +

t

0 Π(w∗(τ))dτ

June 28, 2012 HYP2012

Proof — 3

• wk is a Cauchy sequence: wk → w∗
• Lebesgue Theorem: γk → γ∗ = xo +

t

0 Π(w∗(τ))dτ

• uk satisfies

uk(t) − uh(t)L1 ≤ C sup

τ∈[0,T]

wk−1(τ) − wh−1(τ) +C sup

τ∈[0,T]

γk(τ) − γh(τ)

June 28, 2012 HYP2012

Proof — 3

• wk is a Cauchy sequence: wk → w∗
• Lebesgue Theorem: γk → γ∗ = xo +

t

0 Π(w∗(τ))dτ

• uk → u∗

June 28, 2012 HYP2012

Proof — 3

• wk is a Cauchy sequence: wk → w∗
• Lebesgue Theorem: γk → γ∗ = xo +

t

0 Π(w∗(τ))dτ

• uk → u∗
• (u∗, w∗, γ∗) solves the problem

June 28, 2012 HYP2012

Proof — 4

(uo,1, wo,1, xo,1) and (uo,2, wo,2, xo,2) initial conditions

June 28, 2012 HYP2012

Proof — 4

(uo,1, wo,1, xo,1) and (uo,2, wo,2, xo,2) initial conditions wk,1, uk,1, γk,1 sequences for (uo,1, wo,1, xo,1)

June 28, 2012 HYP2012

Proof — 4

(uo,1, wo,1, xo,1) and (uo,2, wo,2, xo,2) initial conditions wk,1, uk,1, γk,1 sequences for (uo,1, wo,1, xo,1) wk,2, uk,2, γk,2 sequences for (uo,2, wo,2, xo,2)

June 28, 2012 HYP2012

Proof — 4

(uo,1, wo,1, xo,1) and (uo,2, wo,2, xo,2) initial conditions wk,1, uk,1, γk,1 sequences for (uo,1, wo,1, xo,1) wk,2, uk,2, γk,2 sequences for (uo,2, wo,2, xo,2) the term uk,1(t) − uk,2(t)1 can be estimated by

L

• u0,1 − u0,2
• 1 + C

t

• wk−1,1(τ) − wk−1,2(τ)
• dτ +
• γk,1(·) − γk,2(·)
• June 28, 2012

HYP2012

Proof — 4

(uo,1, wo,1, xo,1) and (uo,2, wo,2, xo,2) initial conditions wk,1, uk,1, γk,1 sequences for (uo,1, wo,1, xo,1) wk,2, uk,2, γk,2 sequences for (uo,2, wo,2, xo,2) the term uk,1(t) − uk,2(t)1 can be estimated by

L

• u0,1 − u0,2
• 1 + C

t

• wk−1,1(τ) − wk−1,2(τ)
• dτ +
• γk,1(·) − γk,2(·)
• γk,1(t) − γk,2(t)
• ≤
• xo,1 − xo,2
• + L

t

• wk−1,1(τ) − wk−1,2(τ)
• dτ

June 28, 2012 HYP2012

Proof — 4

(uo,1, wo,1, xo,1) and (uo,2, wo,2, xo,2) initial conditions wk,1, uk,1, γk,1 sequences for (uo,1, wo,1, xo,1) wk,2, uk,2, γk,2 sequences for (uo,2, wo,2, xo,2) the term uk,1(t) − uk,2(t)1 can be estimated by

L

• u0,1 − u0,2
• 1 + C

t

• wk−1,1(τ) − wk−1,2(τ)
• dτ +
• γk,1(·) − γk,2(·)
• γk,1(t) − γk,2(t)
• ≤
• xo,1 − xo,2
• + L

t

• wk−1,1(τ) − wk−1,2(τ)
• dτ

the term wk,1(t) − wk,2(t) can be estimated by

M

• w0,1 − w0,2
• +
• u0,1 − u0,2
• 1 +
• xo,1 − xo,2
• +

t

• wk−2,1(s) − wk−2,2(s)
• ds
• June 28, 2012

HYP2012

Proof — 4

(uo,1, wo,1, xo,1) and (uo,2, wo,2, xo,2) initial conditions wk,1, uk,1, γk,1 sequences for (uo,1, wo,1, xo,1) wk,2, uk,2, γk,2 sequences for (uo,2, wo,2, xo,2) the term uk,1(t) − uk,2(t)1 can be estimated by

L

• u0,1 − u0,2
• 1 + C

t

• wk−1,1(τ) − wk−1,2(τ)
• dτ +
• γk,1(·) − γk,2(·)
• γk,1(t) − γk,2(t)
• ≤
• xo,1 − xo,2
• + L

t

• wk−1,1(τ) − wk−1,2(τ)
• dτ

the term wk,1(t) − wk,2(t) can be estimated by

M

• w0,1 − w0,2
• +
• u0,1 − u0,2
• 1 +
• xo,1 − xo,2
• +

t

• wk−2,1(s) − wk−2,2(s)
• ds
• Inductively
• wk,1(t) − wk,2(t)
• ≤ ˜

M

• w0,1 − w0,2
• +
• u0,1 − u0,2
• 1 +
• xo,1 − xo,2
• uk,1 − uk,2
• 1 ≤ ˜

M

• w0,1 − w0,2
• +
• u0,1 − u0,2
• 1 +
• xo,1 − xo,2
• sup

t∈[0,Tδ]

• γk,1(t) − γk,2(t)
• ≤ ˜

M

• w0,1 − w0,2
• +
• u0,1 − u0,2
• 1 +
• xo,1 − xo,2
• June 28, 2012

HYP2012

Piston movie

(Piston)

June 28, 2012 HYP2012

Large vehicle movie

(Piston)

June 28, 2012 HYP2012