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Introduction Another approach Necessary conditions Approximation of optimal control problems for conservation laws via vanishing viscosity and relaxation Andrea Marson University of Padova Work in progress with F . Ancona (Padova) and M.

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SLIDE 1 Introduction Another approach Necessary conditions

Approximation of optimal control problems for conservation laws via vanishing viscosity and relaxation

Andrea Marson University of Padova Work in progress with F . Ancona (Padova) and M. Herty (Aachen)

International Workshop on Hyperbolic and Kinetic Problems: Theory and Applications Andrea Marson Approximation of optimal control problems for conservation laws
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SLIDE 2 Introduction Another approach Necessary conditions

Outline

1 Introduction Optimal control problem for conservation laws What is known 2 Another approach Γ convergence Approximation via vanishing viscosity Approximation via relaxation 3 Necessary conditions Linearized and adjoint equations An alternative attempt Andrea Marson Approximation of optimal control problems for conservation laws
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SLIDE 3 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Outline

1 Introduction Optimal control problem for conservation laws What is known 2 Another approach Γ convergence Approximation via vanishing viscosity Approximation via relaxation 3 Necessary conditions Linearized and adjoint equations An alternative attempt Andrea Marson Approximation of optimal control problems for conservation laws
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SLIDE 4 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

A story started 27 years ago...

Andrea Marson Approximation of optimal control problems for conservation laws
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SLIDE 5 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

A story started 27 years ago...

  • ∂tu + ∂xf(u) = 0
u(0, x) = u0(x) x ∈ R , t ∈ [0, T] (1) Andrea Marson Approximation of optimal control problems for conservation laws
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SLIDE 6 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

A story started 27 years ago...

  • ∂tu + ∂xf(u) = 0
u(0, x) = u0(x) x ∈ R , t ∈ [0, T] (1) u = u(t, x) ∈ Rn Andrea Marson Approximation of optimal control problems for conservation laws
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SLIDE 7 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

A story started 27 years ago...

  • ∂tu + ∂xf(u) = 0
u(0, x) = u0(x) x ∈ R , t ∈ [0, T] (1) u = u(t, x) ∈ Rn f : Rn → Rn, smooth flux Andrea Marson Approximation of optimal control problems for conservation laws
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SLIDE 8 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

A story started 27 years ago...

  • ∂tu + ∂xf(u) = 0
u(0, x) = u0(x) x ∈ R , t ∈ [0, T] (1) u = u(t, x) ∈ Rn f : Rn → Rn, smooth flux u0 control variable u0 ∈ Uad =
  • u0 ∈ L1(R) : TV u0 ≤ δ
  • , set of admissible controls
Andrea Marson Approximation of optimal control problems for conservation laws
slide-9
SLIDE 9 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

A story started 27 years ago...

  • ∂tu + ∂xf(u) = 0
u(0, x) = u0(x) x ∈ R , t ∈ [0, T] (1) u = u(t, x) ∈ Rn f : Rn → Rn, smooth flux u0 control variable u0 ∈ Uad =
  • u0 ∈ L1(R) : TV u0 ≤ δ
  • , set of admissible controls
We would like to solve min R J(x, u(T, x)) dx : u = u(t, x) fulfills (1), u0 ∈ Uad
  • where J : R × Rn → R is a smooth lagrangian
Andrea Marson Approximation of optimal control problems for conservation laws
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SLIDE 10 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Other optimal control problems

Boundary control
  • ∂tu + ∂xf(u) = 0
u(0, x) = u0(x) , “u(t, 0) = ub(t)” x ≥ 0 , t ∈ [0, T] (2) ub ∈ BV(0, T) control variable Andrea Marson Approximation of optimal control problems for conservation laws
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SLIDE 11 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Other optimal control problems

Boundary control
  • ∂tu + ∂xf(u) = 0
u(0, x) = u0(x) , “u(t, 0) = ub(t)” x ≥ 0 , t ∈ [0, T] (2) ub ∈ BV(0, T) control variable Distributed control
  • ∂tu + ∂xf(u) = h
  • t, x, u, z(t, x)
  • u(0, x) = u0(x)
x ∈ R , t ∈ [0, T] (3) z ∈ L∞([0, T] × R) control variable Andrea Marson Approximation of optimal control problems for conservation laws
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SLIDE 12 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Other optimal control problems

Boundary control
  • ∂tu + ∂xf(u) = 0
u(0, x) = u0(x) , “u(t, 0) = ub(t)” x ≥ 0 , t ∈ [0, T] (2) ub ∈ BV(0, T) control variable Distributed control
  • ∂tu + ∂xf(u) = h
  • t, x, u, z(t, x)
  • u(0, x) = u0(x)
x ∈ R , t ∈ [0, T] (3) z ∈ L∞([0, T] × R) control variable min R J(x, u(T, x)) dx : u = u(t, x) fulfills (2) or (3)
  • Andrea Marson
Approximation of optimal control problems for conservation laws
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SLIDE 13 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Other optimal control problems

Boundary control
  • ∂tu + ∂xf(u) = 0
u(0, x) = u0(x) , “u(t, 0) = ub(t)” x ≥ 0 , t ∈ [0, T] (2) ub ∈ BV(0, T) control variable Distributed control
  • ∂tu + ∂xf(u) = h
  • t, x, u, z(t, x)
  • u(0, x) = u0(x)
x ∈ R , t ∈ [0, T] (3) z ∈ L∞([0, T] × R) control variable min R J(x, u(T, x)) dx : u = u(t, x) fulfills (2) or (3)
  • We can also consider a mixed of (2) and (3)
Andrea Marson Approximation of optimal control problems for conservation laws
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SLIDE 14 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Problems

Existence of a solution: via direct method of calculus of variation, provided the compactness of the set of admissible controls Andrea Marson Approximation of optimal control problems for conservation laws
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SLIDE 15 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Problems

Existence of a solution: via direct method of calculus of variation, provided the compactness of the set of admissible controls Compute the optimal solution: difficult task Andrea Marson Approximation of optimal control problems for conservation laws
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SLIDE 16 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Problems

Existence of a solution: via direct method of calculus of variation, provided the compactness of the set of admissible controls Compute the optimal solution: difficult task Need of necessary conditions for optimality Andrea Marson Approximation of optimal control problems for conservation laws
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SLIDE 17 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

An hitchikers’ guide to necessary conditions

Assume to have an optimal solution u = u(t, x)
  • R
J(x, u(T, x)) dx = min R J(x, u(T, x)) dx : u fulfills ∂tu + ∂xf(u) = 0 ,
  • mltitutt
, .)+wrH , .)+o(w ) multi ) w

{ f.

nothyo

pm

,

Fiat

  • n
  • o|
ulti . ) optimal trajectory no t Andrea Marson Approximation of optimal control problems for conservation laws
slide-18
SLIDE 18 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

An hitchikers’ guide to necessary conditions

Assume to have an optimal solution u = u(t, x)
  • R
J(x, u(T, x)) dx = min R J(x, u(T, x)) dx : u fulfills ∂tu + ∂xf(u) = 0 ,
  • Perturbe the optimal initial datum, u(0, ·) = u0 + hv0
mltitutt , .)+wrH , .)+o(w ) multi ) w

{ f.

nothyo

pm

,

Fiat

  • n
  • o|
ulti . ) optimal trajectory no t Andrea Marson Approximation of optimal control problems for conservation laws
slide-19
SLIDE 19 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

An hitchikers’ guide to necessary conditions

Assume to have an optimal solution u = u(t, x)
  • R
J(x, u(T, x)) dx = min R J(x, u(T, x)) dx : u fulfills ∂tu + ∂xf(u) = 0 ,
  • Perturbe the optimal initial datum, u(0, ·) = u0 + hv0
Transport the tangent vector v = v(t, x) along the optimal solution by means of linearized equations mltitutt , .)+wrH , .)+o(w ) multi ) w

{ f.

nothyo

pm

,

Fiat

  • n
  • o|
ulti . ) optimal trajectory no t Andrea Marson Approximation of optimal control problems for conservation laws
slide-20
SLIDE 20 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

An hitchikers’ guide to necessary conditions

At the final time T
  • R
∇J(x, u(T, x)) · v(T, x) dx ≥ 0 Andrea Marson Approximation of optimal control problems for conservation laws
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SLIDE 21 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

An hitchikers’ guide to necessary conditions

At the final time T
  • R
∇J(x, u(T, x)) · v(T, x) dx ≥ 0 "Reverse" the time Andrea Marson Approximation of optimal control problems for conservation laws
slide-22
SLIDE 22 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

An hitchikers’ guide to necessary conditions

At the final time T
  • R
∇J(x, u(T, x)) · v(T, x) dx ≥ 0 "Reverse" the time Choose an adjoint vector p = p(t, x) to the tangent vector v(t, x) such that p(T, x) = ∇J(x, u(T, x)) Andrea Marson Approximation of optimal control problems for conservation laws
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SLIDE 23 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

An hitchikers’ guide to necessary conditions

At the final time T
  • R
∇J(x, u(T, x)) · v(T, x) dx ≥ 0 "Reverse" the time Choose an adjoint vector p = p(t, x) to the tangent vector v(t, x) such that p(T, x) = ∇J(x, u(T, x)) d dt
  • R
p(t, x) · v(t, x) dx = 0 We get
  • R
p(0, x) · v0(x) dx =
  • R
∇J(x, u(T, x)) · v(T, x) dx ≥ 0 ∀v0 and hence p(0, x) ≡ 0. Andrea Marson Approximation of optimal control problems for conservation laws
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SLIDE 24 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

An hitchikers’ guide to necessary conditions transport

p

backwardly

f-

  • pet
. )

/

. UCT . ) / pH , . )

÷

Pai ) µH , . ) optimal trajectory no t Andrea Marson Approximation of optimal control problems for conservation laws
slide-25
SLIDE 25 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

An hitchikers’ guide to necessary conditions transport

p

backwardly

f-

  • pet
. )

/

. UCT . ) / pH , . )

÷

Pai ) µH , . ) optimal trajectory no t Adjoint equations to be fulfilled by p are needed Andrea Marson Approximation of optimal control problems for conservation laws
slide-26
SLIDE 26 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Obstructions

Scalar cases: shift differentiability for L1-solutions A.Bressan, G. Guerra, DCDS, 1997 Andrea Marson Approximation of optimal control problems for conservation laws
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SLIDE 27 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Obstructions

Scalar cases: shift differentiability for L1-solutions A.Bressan, G. Guerra, DCDS, 1997 Unfortunately, in general, it can not be extended to systems
  • S. Bianchini, DCDS, 2000
Andrea Marson Approximation of optimal control problems for conservation laws
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SLIDE 28 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Obstructions

Scalar cases: shift differentiability for L1-solutions A.Bressan, G. Guerra, DCDS, 1997 Unfortunately, in general, it can not be extended to systems
  • S. Bianchini, DCDS, 2000
Systems of conservation/balance laws: We know how to perform a variational calculus for systems only for piecewise Lipschitz solutions and perturbations that preserve the regularity
  • A. Bressan, A.M., Comm. PDEs, 1995
Andrea Marson Approximation of optimal control problems for conservation laws
slide-29
SLIDE 29 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Obstructions

Scalar cases: shift differentiability for L1-solutions A.Bressan, G. Guerra, DCDS, 1997 Unfortunately, in general, it can not be extended to systems
  • S. Bianchini, DCDS, 2000
Systems of conservation/balance laws: We know how to perform a variational calculus for systems only for piecewise Lipschitz solutions and perturbations that preserve the regularity
  • A. Bressan, A.M., Comm. PDEs, 1995
Some hope from lagrangian flow
  • S. Bianchini, S. Modena, Comm. Math. Physics, 2015
  • S. Bianchini, E. Marconi, ARMA, 2017
  • S. Bianchini, P
. Bonicatto, E. Marconi, preprint, 2017 Andrea Marson Approximation of optimal control problems for conservation laws
slide-30
SLIDE 30 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Outline

1 Introduction Optimal control problem for conservation laws What is known 2 Another approach Γ convergence Approximation via vanishing viscosity Approximation via relaxation 3 Necessary conditions Linearized and adjoint equations An alternative attempt Andrea Marson Approximation of optimal control problems for conservation laws
slide-31
SLIDE 31 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Known results on optimal controllability

Scalar conservation/balance laws: necessary conditions F . James, M. Sepulveda, SIAM J. Control Optimization, 1999
  • S. Ulbrich, SIAM J. Control Optimization, 2002
  • S. Ulbrich, Systems & Control Letters, 2003
R.M. Colombo, A. Groli, J. Math. Anal. Appl., 2004
  • A. Bressan, K. Han, SIAM J. Math. Anal., 2011
  • S. Pfaff, S. Ulbrich, SIAM J. Control Optim., 2015
  • S. Pfaff, S. Ulbrich, Optimization Methods & Software, 2017
Andrea Marson Approximation of optimal control problems for conservation laws
slide-32
SLIDE 32 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Known results on optimal controllability

Scalar conservation/balance laws: necessary conditions F . James, M. Sepulveda, SIAM J. Control Optimization, 1999
  • S. Ulbrich, SIAM J. Control Optimization, 2002
  • S. Ulbrich, Systems & Control Letters, 2003
R.M. Colombo, A. Groli, J. Math. Anal. Appl., 2004
  • A. Bressan, K. Han, SIAM J. Math. Anal., 2011
  • S. Pfaff, S. Ulbrich, SIAM J. Control Optim., 2015
  • S. Pfaff, S. Ulbrich, Optimization Methods & Software, 2017
Scalar conservation/balance laws: existence of optimal solutions
  • C. Castro, F
. Palacios, E. Zuazua, Mathematical Models & Methods in Applied Sciences, 2008 R.M. Colombo, P . Goatin, M.D. Rosini, ESAIM Mat. Model. Numer. Anal., 2011
  • A. Bressan, K. Han, ESAIM Control Optim. Calc. Var., 2012
  • A. Bressan, K. Han, Netw. Heterog. Media, 2013
  • A. Bressan, K.T. Nguyen, Netw. Heterog. Media, 2015
R.M. Colombo, E. Rossi, Math. Methods Appl. Sci., 2018 Andrea Marson Approximation of optimal control problems for conservation laws
slide-33
SLIDE 33 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Known results on optimal controllability

Scalar conservation/balance laws: necessary conditions F . James, M. Sepulveda, SIAM J. Control Optimization, 1999
  • S. Ulbrich, SIAM J. Control Optimization, 2002
  • S. Ulbrich, Systems & Control Letters, 2003
R.M. Colombo, A. Groli, J. Math. Anal. Appl., 2004
  • A. Bressan, K. Han, SIAM J. Math. Anal., 2011
  • S. Pfaff, S. Ulbrich, SIAM J. Control Optim., 2015
  • S. Pfaff, S. Ulbrich, Optimization Methods & Software, 2017
Scalar conservation/balance laws: existence of optimal solutions
  • C. Castro, F
. Palacios, E. Zuazua, Mathematical Models & Methods in Applied Sciences, 2008 R.M. Colombo, P . Goatin, M.D. Rosini, ESAIM Mat. Model. Numer. Anal., 2011
  • A. Bressan, K. Han, ESAIM Control Optim. Calc. Var., 2012
  • A. Bressan, K. Han, Netw. Heterog. Media, 2013
  • A. Bressan, K.T. Nguyen, Netw. Heterog. Media, 2015
R.M. Colombo, E. Rossi, Math. Methods Appl. Sci., 2018 Scalar flows on networks: existence of optimal solutions R.M. Colombo, M. Garavello, M. Herty, V. Schleper SIAM J. Math. Anal., 2009 R.M. Colombo, M. Garavello, Math. Biosci. Eng., 2015 F . Ancona, A. Cesaroni, G.M. Coclite, M. Garavello, SIAM J. Control Optimization, 2018 Andrea Marson Approximation of optimal control problems for conservation laws
slide-34
SLIDE 34 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Known results on optimal controllability

Nonlocal fluxes for supply chains models ∂tu + ∂x
  • λ(W(t)) u
  • = 0 ,
W(t) = 1 u(t, x) dx Andrea Marson Approximation of optimal control problems for conservation laws
slide-35
SLIDE 35 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Known results on optimal controllability

Nonlocal fluxes for supply chains models ∂tu + ∂x
  • λ(W(t)) u
  • = 0 ,
W(t) = 1 u(t, x) dx No loss of regularity in the solution, no need of entropy conditions Andrea Marson Approximation of optimal control problems for conservation laws
slide-36
SLIDE 36 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Known results on optimal controllability

Nonlocal fluxes for supply chains models ∂tu + ∂x
  • λ(W(t)) u
  • = 0 ,
W(t) = 1 u(t, x) dx No loss of regularity in the solution, no need of entropy conditions Existence of optimal controls J.M. Coron, M. Kawski, Z. Wang, DCDS Series B, 2010 P . Shang, Z. Wang, J. Differential Equations, 2011 Andrea Marson Approximation of optimal control problems for conservation laws
slide-37
SLIDE 37 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Known results on optimal controllability

Nonlocal fluxes for supply chains models ∂tu + ∂x
  • λ(W(t)) u
  • = 0 ,
W(t) = 1 u(t, x) dx No loss of regularity in the solution, no need of entropy conditions Existence of optimal controls J.M. Coron, M. Kawski, Z. Wang, DCDS Series B, 2010 P . Shang, Z. Wang, J. Differential Equations, 2011 Necessary conditions for optimality R.M. Colombo, M. Herty, M. Mercier, ESAIM Control Optim. Calc. Var., 2011
  • M. Groschel, A. Keimer, G. Leugering, Z. Wang, SIAM J. Control Optimization 2014
Andrea Marson Approximation of optimal control problems for conservation laws
slide-38
SLIDE 38 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Known results on optimal controllability

Nonlocal fluxes for supply chains models ∂tu + ∂x
  • λ(W(t)) u
  • = 0 ,
W(t) = 1 u(t, x) dx No loss of regularity in the solution, no need of entropy conditions Existence of optimal controls J.M. Coron, M. Kawski, Z. Wang, DCDS Series B, 2010 P . Shang, Z. Wang, J. Differential Equations, 2011 Necessary conditions for optimality R.M. Colombo, M. Herty, M. Mercier, ESAIM Control Optim. Calc. Var., 2011
  • M. Groschel, A. Keimer, G. Leugering, Z. Wang, SIAM J. Control Optimization 2014
Systems of conservation/balance laws
  • A. Bressan, A.M., Rendiconti Seminario Matematico di Padova, 1995
R.M. Colombo, A. Groli, Nonlinear Analysis, 2004
  • A. Bressan, W. Shen, Contemporary Mathematics, 2007
F .M. Hante, G. Leugering, A. Martin & others, Industrial and Applied Mathematics, 2017 Andrea Marson Approximation of optimal control problems for conservation laws
slide-39
SLIDE 39 Introduction Another approach Necessary conditions Optimal control problem for conservation laws What is known

Known results on optimal controllability

Nonlocal fluxes for supply chains models ∂tu + ∂x
  • λ(W(t)) u
  • = 0 ,
W(t) = 1 u(t, x) dx No loss of regularity in the solution, no need of entropy conditions Existence of optimal controls J.M. Coron, M. Kawski, Z. Wang, DCDS Series B, 2010 P . Shang, Z. Wang, J. Differential Equations, 2011 Necessary conditions for optimality R.M. Colombo, M. Herty, M. Mercier, ESAIM Control Optim. Calc. Var., 2011
  • M. Groschel, A. Keimer, G. Leugering, Z. Wang, SIAM J. Control Optimization 2014
Systems of conservation/balance laws
  • A. Bressan, A.M., Rendiconti Seminario Matematico di Padova, 1995
R.M. Colombo, A. Groli, Nonlinear Analysis, 2004
  • A. Bressan, W. Shen, Contemporary Mathematics, 2007
F .M. Hante, G. Leugering, A. Martin & others, Industrial and Applied Mathematics, 2017 I apologize if someone is missing Andrea Marson Approximation of optimal control problems for conservation laws
slide-40
SLIDE 40 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Outline

1 Introduction Optimal control problem for conservation laws What is known 2 Another approach Γ convergence Approximation via vanishing viscosity Approximation via relaxation 3 Necessary conditions Linearized and adjoint equations An alternative attempt Andrea Marson Approximation of optimal control problems for conservation laws
slide-41
SLIDE 41 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Γ convergence

Assume to have a functional j : X → R and want to solve min x∈X j(x) Andrea Marson Approximation of optimal control problems for conservation laws
slide-42
SLIDE 42 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Γ convergence

Assume to have a functional j : X → R and want to solve min x∈X j(x) Find a sequence jn : X → R such that you can find xn = argmin
  • jn(x) : x ∈ X
  • Andrea Marson
Approximation of optimal control problems for conservation laws
slide-43
SLIDE 43 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Γ convergence

Assume to have a functional j : X → R and want to solve min x∈X j(x) Find a sequence jn : X → R such that you can find xn = argmin
  • jn(x) : x ∈ X
  • up to a subsequence
xn → x0 = argmin
  • j(x) : x ∈ X
  • G. Dal Maso, An introduction to Γ-convergence, Birkhäuser, Basel, 2003
Andrea Marson Approximation of optimal control problems for conservation laws
slide-44
SLIDE 44 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Outline

1 Introduction Optimal control problem for conservation laws What is known 2 Another approach Γ convergence Approximation via vanishing viscosity Approximation via relaxation 3 Necessary conditions Linearized and adjoint equations An alternative attempt Andrea Marson Approximation of optimal control problems for conservation laws
slide-45
SLIDE 45 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Vanishing viscosity approximations

  • ∂tuε + ∂xf(uε) = ε ∂xxuε
uε(0, ·) = u0 control variable (4) Andrea Marson Approximation of optimal control problems for conservation laws
slide-46
SLIDE 46 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Vanishing viscosity approximations

  • ∂tuε + ∂xf(uε) = ε ∂xxuε
uε(0, ·) = u0 control variable (4) Convergence as ε → 0: S. Bianchini, A. Bressan, Annals of Mathematics, 2005 Andrea Marson Approximation of optimal control problems for conservation laws
slide-47
SLIDE 47 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Vanishing viscosity approximations

  • ∂tuε + ∂xf(uε) = ε ∂xxuε
uε(0, ·) = u0 control variable (4) Convergence as ε → 0: S. Bianchini, A. Bressan, Annals of Mathematics, 2005 u0 ∈ Uad =
  • u0 ∈ L1(R) : TV u0 ≤ δ
  • set of admissible controls
Same set of control of inviscid problem Andrea Marson Approximation of optimal control problems for conservation laws
slide-48
SLIDE 48 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Vanishing viscosity approximations

  • ∂tuε + ∂xf(uε) = ε ∂xxuε
uε(0, ·) = u0 control variable (4) Convergence as ε → 0: S. Bianchini, A. Bressan, Annals of Mathematics, 2005 u0 ∈ Uad =
  • u0 ∈ L1(R) : TV u0 ≤ δ
  • set of admissible controls
Same set of control of inviscid problem we take advantage of the regularizing effect of the equations Andrea Marson Approximation of optimal control problems for conservation laws
slide-49
SLIDE 49 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Vanishing viscosity approximations

  • ∂tuε + ∂xf(uε) = ε ∂xxuε
uε(0, ·) = u0 control variable (4) Convergence as ε → 0: S. Bianchini, A. Bressan, Annals of Mathematics, 2005 u0 ∈ Uad =
  • u0 ∈ L1(R) : TV u0 ≤ δ
  • set of admissible controls
Same set of control of inviscid problem we take advantage of the regularizing effect of the equations Uad is compact in L1(R) Andrea Marson Approximation of optimal control problems for conservation laws
slide-50
SLIDE 50 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Vanishing viscosity approximations

  • ∂tuε + ∂xf(uε) = ε ∂xxuε
uε(0, ·) = u0 control variable (4) Convergence as ε → 0: S. Bianchini, A. Bressan, Annals of Mathematics, 2005 u0 ∈ Uad =
  • u0 ∈ L1(R) : TV u0 ≤ δ
  • set of admissible controls
Same set of control of inviscid problem we take advantage of the regularizing effect of the equations Uad is compact in L1(R) min R J(x, uε(T, x)) dx : uε = uε(t, x) fulfills (4), u0 ∈ Uad
  • Andrea Marson
Approximation of optimal control problems for conservation laws
slide-51
SLIDE 51 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Vanishing viscosity approximations

  • ∂tuε + ∂xf(uε) = ε ∂xxuε
uε(0, ·) = u0 control variable (4) Convergence as ε → 0: S. Bianchini, A. Bressan, Annals of Mathematics, 2005 u0 ∈ Uad =
  • u0 ∈ L1(R) : TV u0 ≤ δ
  • set of admissible controls
Same set of control of inviscid problem we take advantage of the regularizing effect of the equations Uad is compact in L1(R) min R J(x, uε(T, x)) dx : uε = uε(t, x) fulfills (4), u0 ∈ Uad
  • Existence of a minimizer uε
0: ok via direct method of calculus of variationss Andrea Marson Approximation of optimal control problems for conservation laws
slide-52
SLIDE 52 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Γ-convergence of vanishing viscosity approximations

uε 0 = argmin R J(x, uε(T, x)) dx : uε solves the viscid Cauchy problem
  • Up to a subsequence
uε 0 → u0 uε(t, ·) → u(t, ·) in L1(R) ,
  • ∂tu + ∂xf(u) = 0
u(0, ·) = u0 ,
  • ptimal for inviscid problem
Andrea Marson Approximation of optimal control problems for conservation laws
slide-53
SLIDE 53 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Γ-convergence of vanishing viscosity approximations

uε 0 = argmin R J(x, uε(T, x)) dx : uε solves the viscid Cauchy problem
  • Up to a subsequence
uε 0 → u0 uε(t, ·) → u(t, ·) in L1(R) ,
  • ∂tu + ∂xf(u) = 0
u(0, ·) = u0 ,
  • ptimal for inviscid problem
Choose w0 ∈ Uad, and let wε = wε(t, x), w = w(t, x) be such that
  • ∂tw + ∂xf(w) = 0
w(0, ·) = w0
  • ∂twε + ∂xf(wε) = ε ∂xxwε
wε(0, ·) = w0 Andrea Marson Approximation of optimal control problems for conservation laws
slide-54
SLIDE 54 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Γ-convergence of vanishing viscosity approximations

uε 0 = argmin R J(x, uε(T, x)) dx : uε solves the viscid Cauchy problem
  • Up to a subsequence
uε 0 → u0 uε(t, ·) → u(t, ·) in L1(R) ,
  • ∂tu + ∂xf(u) = 0
u(0, ·) = u0 ,
  • ptimal for inviscid problem
Choose w0 ∈ Uad, and let wε = wε(t, x), w = w(t, x) be such that
  • ∂tw + ∂xf(w) = 0
w(0, ·) = w0
  • ∂twε + ∂xf(wε) = ε ∂xxwε
wε(0, ·) = w0 As ε → 0: wε(T, ·) → w(T, ·) in L1(R), and Andrea Marson Approximation of optimal control problems for conservation laws
slide-55
SLIDE 55 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Γ-convergence of vanishing viscosity approximations

uε 0 = argmin R J(x, uε(T, x)) dx : uε solves the viscid Cauchy problem
  • Up to a subsequence
uε 0 → u0 uε(t, ·) → u(t, ·) in L1(R) ,
  • ∂tu + ∂xf(u) = 0
u(0, ·) = u0 ,
  • ptimal for inviscid problem
Choose w0 ∈ Uad, and let wε = wε(t, x), w = w(t, x) be such that
  • ∂tw + ∂xf(w) = 0
w(0, ·) = w0
  • ∂twε + ∂xf(wε) = ε ∂xxwε
wε(0, ·) = w0 As ε → 0: wε(T, ·) → w(T, ·) in L1(R), and
  • R
J(w(T, x)) dx = lim ε→0
  • R
J(x, wε(T, x)) dx ≥ ≥ lim ε→0
  • R
J(x, uε(T, x)) dx =
  • R
J(x, u(T, x)) dx Andrea Marson Approximation of optimal control problems for conservation laws
slide-56
SLIDE 56 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Γ-convergence of vanishing viscosity approximations

uε 0 = argmin R J(x, uε(T, x)) dx : uε solves the viscid Cauchy problem
  • Up to a subsequence
uε 0 → u0 uε(t, ·) → u(t, ·) in L1(R) ,
  • ∂tu + ∂xf(u) = 0
u(0, ·) = u0 ,
  • ptimal for inviscid problem
Choose w0 ∈ Uad, and let wε = wε(t, x), w = w(t, x) be such that
  • ∂tw + ∂xf(w) = 0
w(0, ·) = w0
  • ∂twε + ∂xf(wε) = ε ∂xxwε
wε(0, ·) = w0 As ε → 0: wε(T, ·) → w(T, ·) in L1(R), and
  • R
J(w(T, x)) dx = lim ε→0
  • R
J(x, wε(T, x)) dx ≥ ≥ lim ε→0
  • R
J(x, uε(T, x)) dx =
  • R
J(x, u(T, x)) dx Burgers’ equation, L2 setting, J(u) = |u − uF |2:
  • C. Castro, F
. Palacios, E. Zuazua, in Integral methods in science and engineering, 2010 Andrea Marson Approximation of optimal control problems for conservation laws
slide-57
SLIDE 57 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Outline

1 Introduction Optimal control problem for conservation laws What is known 2 Another approach Γ convergence Approximation via vanishing viscosity Approximation via relaxation 3 Necessary conditions Linearized and adjoint equations An alternative attempt Andrea Marson Approximation of optimal control problems for conservation laws
slide-58
SLIDE 58 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Jin-Xin relaxation approximation

  • ∂tuε + ∂xf(uε) = ε
  • ∂xxuε − ∂ttuε
uε(0, ·) = u0 control variable , ∂tuε(0, ·) = 0 (5) Andrea Marson Approximation of optimal control problems for conservation laws
slide-59
SLIDE 59 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Jin-Xin relaxation approximation

  • ∂tuε + ∂xf(uε) = ε
  • ∂xxuε − ∂ttuε
uε(0, ·) = u0 control variable , ∂tuε(0, ·) = 0 (5)
  • S. Jin, Z. Xin, Comm. Pure Appl. Math., 1995
Andrea Marson Approximation of optimal control problems for conservation laws
slide-60
SLIDE 60 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Jin-Xin relaxation approximation

  • ∂tuε + ∂xf(uε) = ε
  • ∂xxuε − ∂ttuε
uε(0, ·) = u0 control variable , ∂tuε(0, ·) = 0 (5)
  • S. Jin, Z. Xin, Comm. Pure Appl. Math., 1995
Convergence as ε → 0: S. Bianchini, Comm. Pure Appl. Math., 2006 Andrea Marson Approximation of optimal control problems for conservation laws
slide-61
SLIDE 61 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Jin-Xin relaxation approximation

  • ∂tuε + ∂xf(uε) = ε
  • ∂xxuε − ∂ttuε
uε(0, ·) = u0 control variable , ∂tuε(0, ·) = 0 (5)
  • S. Jin, Z. Xin, Comm. Pure Appl. Math., 1995
Convergence as ε → 0: S. Bianchini, Comm. Pure Appl. Math., 2006 Set of admissible controls u0 ∈ Uε ad =
  • u0 ∈ W 1,1(R) : u0W 1,1 ≤ δ ,
∂xxu0L1 ≤ C/ε , ∂xxxu0L1 ≤ C/ε2 , TV ∂xxxu0 ≤ C/ε3 Andrea Marson Approximation of optimal control problems for conservation laws
slide-62
SLIDE 62 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Jin-Xin relaxation approximation

  • ∂tuε + ∂xf(uε) = ε
  • ∂xxuε − ∂ttuε
uε(0, ·) = u0 control variable , ∂tuε(0, ·) = 0 (5)
  • S. Jin, Z. Xin, Comm. Pure Appl. Math., 1995
Convergence as ε → 0: S. Bianchini, Comm. Pure Appl. Math., 2006 Set of admissible controls u0 ∈ Uε ad =
  • u0 ∈ W 1,1(R) : u0W 1,1 ≤ δ ,
∂xxu0L1 ≤ C/ε , ∂xxxu0L1 ≤ C/ε2 , TV ∂xxxu0 ≤ C/ε3 Andrea Marson Approximation of optimal control problems for conservation laws
slide-63
SLIDE 63 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Jin-Xin relaxation approximation

  • ∂tuε + ∂xf(uε) = ε
  • ∂xxuε − ∂ttuε
uε(0, ·) = u0 control variable , ∂tuε(0, ·) = 0 (5)
  • S. Jin, Z. Xin, Comm. Pure Appl. Math., 1995
Convergence as ε → 0: S. Bianchini, Comm. Pure Appl. Math., 2006 Set of admissible controls u0 ∈ Uε ad =
  • u0 ∈ W 1,1(R) : u0W 1,1 ≤ δ ,
∂xxu0L1 ≤ C/ε , ∂xxxu0L1 ≤ C/ε2 , TV ∂xxxu0 ≤ C/ε3
  • gain compactness
  • Andrea Marson
Approximation of optimal control problems for conservation laws
slide-64
SLIDE 64 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Jin-Xin relaxation approximation

  • ∂tuε + ∂xf(uε) = ε
  • ∂xxuε − ∂ttuε
uε(0, ·) = u0 control variable , ∂tuε(0, ·) = 0 (5)
  • S. Jin, Z. Xin, Comm. Pure Appl. Math., 1995
Convergence as ε → 0: S. Bianchini, Comm. Pure Appl. Math., 2006 Set of admissible controls u0 ∈ Uε ad =
  • u0 ∈ W 1,1(R) : u0W 1,1 ≤ δ ,
∂xxu0L1 ≤ C/ε , ∂xxxu0L1 ≤ C/ε2 , TV ∂xxxu0 ≤ C/ε3 Different set of admissible controls of the non relaxed problem: the equations do not have a regularizing effect Andrea Marson Approximation of optimal control problems for conservation laws
slide-65
SLIDE 65 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Jin-Xin relaxation approximation

  • ∂tuε + ∂xf(uε) = ε
  • ∂xxuε − ∂ttuε
uε(0, ·) = u0 control variable , ∂tuε(0, ·) = 0 (5)
  • S. Jin, Z. Xin, Comm. Pure Appl. Math., 1995
Convergence as ε → 0: S. Bianchini, Comm. Pure Appl. Math., 2006 Set of admissible controls u0 ∈ Uε ad =
  • u0 ∈ W 1,1(R) : u0W 1,1 ≤ δ ,
∂xxu0L1 ≤ C/ε , ∂xxxu0L1 ≤ C/ε2 , TV ∂xxxu0 ≤ C/ε3 Different set of admissible controls of the non relaxed problem: the equations do not have a regularizing effect min R J(x, uε(T, x)) dx : uε = uε(t, x) fulfills (5), u0 ∈ Uε ad
  • Andrea Marson
Approximation of optimal control problems for conservation laws
slide-66
SLIDE 66 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Jin-Xin relaxation approximation

  • ∂tuε + ∂xf(uε) = ε
  • ∂xxuε − ∂ttuε
uε(0, ·) = u0 control variable , ∂tuε(0, ·) = 0 (5)
  • S. Jin, Z. Xin, Comm. Pure Appl. Math., 1995
Convergence as ε → 0: S. Bianchini, Comm. Pure Appl. Math., 2006 Set of admissible controls u0 ∈ Uε ad =
  • u0 ∈ W 1,1(R) : u0W 1,1 ≤ δ ,
∂xxu0L1 ≤ C/ε , ∂xxxu0L1 ≤ C/ε2 , TV ∂xxxu0 ≤ C/ε3 Different set of admissible controls of the non relaxed problem: the equations do not have a regularizing effect min R J(x, uε(T, x)) dx : uε = uε(t, x) fulfills (5), u0 ∈ Uε ad
  • Existence of a minimizer uε
0: ok, being Uε ad compact in W 3,1(R) Andrea Marson Approximation of optimal control problems for conservation laws
slide-67
SLIDE 67 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Γ-convergence of Jin-Xin approximations

uε 0 = argmin R J(x, uε(T, x)) dx : uε solves the relaxed Cauchy problem
  • Andrea Marson
Approximation of optimal control problems for conservation laws
slide-68
SLIDE 68 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Γ-convergence of Jin-Xin approximations

uε 0 = argmin R J(x, uε(T, x)) dx : uε solves the relaxed Cauchy problem
  • Beware!
Uad =
  • u0 ∈ L1(R) : TV u0 ≤ δ
  • controls non relaxed system

⊂

Uε ad controls relaxed system Andrea Marson Approximation of optimal control problems for conservation laws
slide-69
SLIDE 69 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Γ-convergence of Jin-Xin approximations

uε 0 = argmin R J(x, uε(T, x)) dx : uε solves the relaxed Cauchy problem
  • Beware!
Uad =
  • u0 ∈ L1(R) : TV u0 ≤ δ
  • controls non relaxed system

⊂

Uε ad controls relaxed system But uε 0W 1,1 ≤ δ Andrea Marson Approximation of optimal control problems for conservation laws
slide-70
SLIDE 70 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Γ-convergence of Jin-Xin approximations

uε 0 = argmin R J(x, uε(T, x)) dx : uε solves the relaxed Cauchy problem
  • Beware!
Uad =
  • u0 ∈ L1(R) : TV u0 ≤ δ
  • controls non relaxed system

⊂

Uε ad controls relaxed system But uε 0W 1,1 ≤ δ Up to a subsequence uε 0 → u0 uε(t, ·) → u(t, ·) in L1(R) ,
  • ∂tu + ∂xf(u) = 0
u(0, ·) = u0 Andrea Marson Approximation of optimal control problems for conservation laws
slide-71
SLIDE 71 Introduction Another approach Necessary conditions Γ convergence Approximation via vanishing viscosity Approximation via relaxation

Γ-convergence of Jin-Xin approximations

uε 0 = argmin R J(x, uε(T, x)) dx : uε solves the relaxed Cauchy problem
  • Beware!
Uad =
  • u0 ∈ L1(R) : TV u0 ≤ δ
  • controls non relaxed system

⊂

Uε ad controls relaxed system But uε 0W 1,1 ≤ δ Up to a subsequence uε 0 → u0 uε(t, ·) → u(t, ·) in L1(R) ,
  • ∂tu + ∂xf(u) = 0
u(0, ·) = u0 Moreover u0 = argmin R J(x, u(T, x)) dx : u solves the non-relaxed Cauchy problem
  • Andrea Marson
Approximation of optimal control problems for conservation laws
slide-72
SLIDE 72 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

Outline

1 Introduction Optimal control problem for conservation laws What is known 2 Another approach Γ convergence Approximation via vanishing viscosity Approximation via relaxation 3 Necessary conditions Linearized and adjoint equations An alternative attempt Andrea Marson Approximation of optimal control problems for conservation laws
slide-73
SLIDE 73 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

Linearized equations

Let    ∂tuε h + ∂xf(uε h) = ε
  • ∂xxuε
h − ∂ttuε h
  • uε
h(0, ·) = uε 0 + h vε 0 ∈ Uε ad , h → 0 , Andrea Marson Approximation of optimal control problems for conservation laws
slide-74
SLIDE 74 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

Linearized equations

Let    ∂tuε h + ∂xf(uε h) = ε
  • ∂xxuε
h − ∂ttuε h
  • uε
h(0, ·) = uε 0 + h vε 0 ∈ Uε ad , h → 0 , ∂tuε h(0, ·) = 0 Andrea Marson Approximation of optimal control problems for conservation laws
slide-75
SLIDE 75 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

Linearized equations

Let    ∂tuε h + ∂xf(uε h) = ε
  • ∂xxuε
h − ∂ttuε h
  • uε
h(0, ·) = uε 0 + h vε 0 ∈ Uε ad , h → 0 , ∂tuε h(0, ·) = 0 Thanks to the fact that the equations preserve regularity uε h(t, ·) = uε(t, ·) + h vε(t, ·) + o(h) in W 1,1(R) where
  • ∂tvε + ∂x
  • Df(uε) vε
= ε
  • ∂xxvε − ∂ttvε
vε(0, ·) = vε 0 , ∂tvε(0, ·) = 0 Andrea Marson Approximation of optimal control problems for conservation laws
slide-76
SLIDE 76 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

Linearized equations

Let    ∂tuε h + ∂xf(uε h) = ε
  • ∂xxuε
h − ∂ttuε h
  • uε
h(0, ·) = uε 0 + h vε 0 ∈ Uε ad , h → 0 , ∂tuε h(0, ·) = 0 Thanks to the fact that the equations preserve regularity uε h(t, ·) = uε(t, ·) + h vε(t, ·) + o(h) in W 1,1(R) where
  • ∂tvε + ∂x
  • Df(uε) vε
= ε
  • ∂xxvε − ∂ttvε
vε(0, ·) = vε 0 , ∂tvε(0, ·) = ∂tvε In order to determine the adjoint system, take into account that we could choose freely also ∂tvε Andrea Marson Approximation of optimal control problems for conservation laws
slide-77
SLIDE 77 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

Adjoint equations

We look for an adjoint vector (pε, qε) =
  • pε(t, x), qε(t, x)
  • ∈ Rn × Rn such that
d dt
  • R
  • pε(t, x)vε(t, x) + qε(t, x)∂tvε(t, x)
  • dx = 0
Andrea Marson Approximation of optimal control problems for conservation laws
slide-78
SLIDE 78 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

Adjoint equations

We look for an adjoint vector (pε, qε) =
  • pε(t, x), qε(t, x)
  • ∈ Rn × Rn such that
d dt
  • R
  • pε(t, x)vε(t, x) + qε(t, x)∂tvε(t, x)
  • dx = 0
Using the equations, integrating by parts and taking advantage of the fact that the system is in conservation form, we get        ∂tpε + ∂xxqε + 1 ε ∂xqε Df(uε) = 0 ∂tqε + pε − 1 ε qε = 0 Andrea Marson Approximation of optimal control problems for conservation laws
slide-79
SLIDE 79 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

Adjoint equations

We look for an adjoint vector (pε, qε) =
  • pε(t, x), qε(t, x)
  • ∈ Rn × Rn such that
d dt
  • R
  • pε(t, x)vε(t, x) + qε(t, x)∂tvε(t, x)
  • dx = 0
Using the equations, integrating by parts and taking advantage of the fact that the system is in conservation form, we get        ∂tpε + ∂xxqε + 1 ε ∂xqε Df(uε) = 0 ∂tqε + pε − 1 ε qε = 0 Terminal conditions for pε and qε: pε(T, ·) = 1 ε ∇J(x, uε(T, ·)) , qε(T, ·) = 0 .
  • S. Bianchini, Comm. Pure Appl. Math., 2006
Andrea Marson Approximation of optimal control problems for conservation laws
slide-80
SLIDE 80 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

Adjoint equations

We look for an adjoint vector (pε, qε) =
  • pε(t, x), qε(t, x)
  • ∈ Rn × Rn such that
d dt
  • R
  • pε(t, x)vε(t, x) + qε(t, x)∂tvε(t, x)
  • dx = 0
Using the equations, integrating by parts and taking advantage of the fact that the system is in conservation form, we get        ∂tpε + ∂xxqε + 1 ε ∂xqε Df(uε) = 0 ∂tqε + pε − 1 ε qε = 0 Terminal conditions for pε and qε: pε(T, ·) = 1 ε ∇J(x, uε(T, ·)) , qε(T, ·) = 0 .
  • S. Bianchini, Comm. Pure Appl. Math., 2006
Necessary conditions for optimality
  • R
pε(x, 0)vε 0 (x) dx ≥ 0 ∀vε = ⇒ pε(x, 0) ≡ 0 Andrea Marson Approximation of optimal control problems for conservation laws
slide-81
SLIDE 81 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

Second order equations for q

Rewrite the system using only the q variables Andrea Marson Approximation of optimal control problems for conservation laws
slide-82
SLIDE 82 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

Second order equations for q

Rewrite the system using only the q variables Differentiate w.r.t. t ∂tqε + pε − 1 ε qε = 0 and use it in the first equation ∂tpε + ∂xxqε + 1 ε ∂xqε Df(uε) = 0 Andrea Marson Approximation of optimal control problems for conservation laws
slide-83
SLIDE 83 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

Second order equations for q

Rewrite the system using only the q variables Differentiate w.r.t. t ∂tqε + pε − 1 ε qε = 0 and use it in the first equation ∂tpε + ∂xxqε + 1 ε ∂xqε Df(uε) = 0 We get    ∂tqε + ∂xqε Df(uε) = ε
  • ∂ttqε − ∂xxqε
qε(T, ·) = 0 , ∂tqε(T, ·) = − 1 ε ∇J(x, uε(T, ·)) Andrea Marson Approximation of optimal control problems for conservation laws
slide-84
SLIDE 84 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

Second order equations for q

Rewrite the system using only the q variables Differentiate w.r.t. t ∂tqε + pε − 1 ε qε = 0 and use it in the first equation ∂tpε + ∂xxqε + 1 ε ∂xqε Df(uε) = 0 We get    ∂tqε + ∂xqε Df(uε) = ε
  • ∂ttqε − ∂xxqε
qε(T, ·) = 0 , ∂tqε(T, ·) = − 1 ε ∇J(x, uε(T, ·)) Necessary conditions for optimality ε ∂tqε(x, 0) − qε(x, 0) ≡ 0 Andrea Marson Approximation of optimal control problems for conservation laws
slide-85
SLIDE 85 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

We would like to pass to the limit as ε → 0...

Unfortunately, letting ε → 0, formally we get
  • ∂tq + ∂xq Df(u) = 0
q(T, ·) = −∇J(u(T, ·)) and necessary conditions for optimality write q(x, 0) ≡ 0 Andrea Marson Approximation of optimal control problems for conservation laws
slide-86
SLIDE 86 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

We would like to pass to the limit as ε → 0...

Unfortunately, letting ε → 0, formally we get
  • ∂tq + ∂xq Df(u) = 0
q(T, ·) = −∇J(u(T, ·)) and necessary conditions for optimality write q(x, 0) ≡ 0 One main problem: inconsistency of conditions at time t = T and at time t = 0 We can not bypass the obstructions due to the loss of regularity of the solutions to the non-relaxed problem Andrea Marson Approximation of optimal control problems for conservation laws
slide-87
SLIDE 87 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

Outline

1 Introduction Optimal control problem for conservation laws What is known 2 Another approach Γ convergence Approximation via vanishing viscosity Approximation via relaxation 3 Necessary conditions Linearized and adjoint equations An alternative attempt Andrea Marson Approximation of optimal control problems for conservation laws
slide-88
SLIDE 88 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

A recursive procedure (in progress)

Let εk ↓ 0, and assume to have uk = uk(t, x) ∈ W 1,1 uk(0, ·) = uk 0 ∈ Uεk ad =
  • u0 ∈ W 1,1(R) : u0W 1,1 ≤ δ , ∂xxu0L1 ≤ C/εk ,
∂xxxu0L1 ≤ C/ε2 k , TV ∂xxxu0 ≤ C/ε3 k
  • Andrea Marson
Approximation of optimal control problems for conservation laws
slide-89
SLIDE 89 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

A recursive procedure (in progress)

Let εk ↓ 0, and assume to have uk = uk(t, x) ∈ W 1,1 Uεk+1 ad =
  • u0 ∈ W 1,1(R) : u0W 1,1 ≤ δ , ∂xxu0L1 ≤ C/εk+1 ,
∂xxxu0L1 ≤ C/ε2 k+1 , TV ∂xxxu0 ≤ C/ε3 k+1
  • Then, we want to solve
min R J(x, uk+1(t, x)) dx : uk+1 ∈ Uk+1 ad
  • where uk+1 = uk+1(t, x) solves
  • ∂tuk+1 + Df(uk) ∂xuk+1 = ǫk+1
  • ∂xxuk+1 − ∂ttuk+1
linear equations uk+1(0, ·) = uk+1 , ∂tuk+1(0, x) = 0 Andrea Marson Approximation of optimal control problems for conservation laws
slide-90
SLIDE 90 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

A recursive procedure (in progress)

Let εk ↓ 0, and assume to have uk = uk(t, x) ∈ W 1,1 Uεk+1 ad =
  • u0 ∈ W 1,1(R) : u0W 1,1 ≤ δ , ∂xxu0L1 ≤ C/εk+1 ,
∂xxxu0L1 ≤ C/ε2 k+1 , TV ∂xxxu0 ≤ C/ε3 k+1
  • Then, we want to solve
min R J(x, uk+1(t, x)) dx : uk+1 ∈ Uk+1 ad
  • where uk+1 = uk+1(t, x) solves
  • ∂tuk+1 + Df(uk) ∂xuk+1 = ǫk+1
  • ∂xxuk+1 − ∂ttuk+1
linear equations uk+1(0, ·) = uk+1 , ∂tuk+1(0, x) = 0 A solution does exist via the direct method of calculus of variation Andrea Marson Approximation of optimal control problems for conservation laws
slide-91
SLIDE 91 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

A recursive procedure (in progress)

Let εk ↓ 0, and assume to have uk = uk(t, x) ∈ W 1,1 Uεk+1 ad =
  • u0 ∈ W 1,1(R) : u0W 1,1 ≤ δ , ∂xxu0L1 ≤ C/εk+1 ,
∂xxxu0L1 ≤ C/ε2 k+1 , TV ∂xxxu0 ≤ C/ε3 k+1
  • Then, we want to solve
min R J(x, uk+1(t, x)) dx : uk+1 ∈ Uk+1 ad
  • where uk+1 = uk+1(t, x) solves
  • ∂tuk+1 + Df(uk) ∂xuk+1 = ǫk+1
  • ∂xxuk+1 − ∂ttuk+1
linear equations uk+1(0, ·) = uk+1 , ∂tuk+1(0, x) = 0 A solution does exist via the direct method of calculus of variation Does
  • uk
  • k≥1 converge to a minimum of the non-relaxed problem?
Andrea Marson Approximation of optimal control problems for conservation laws
slide-92
SLIDE 92 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

Towards a numerical algorithm (in progress)

By writing the linearized and adjoint equations... We look for uk+1 ∈ Uǫk+1 ad =
  • u0 ∈ W 1,1(R) : u0W 1,1 ≤ δ , ∂xxu0L1 ≤ C/εk+1 ,
∂xxxu0L1 ≤ C/ε2 k+1 , TV ∂xxxu0 ≤ C/ε3 k+1
  • such that the solution (uk+1, qk+1) =
  • uk+1(x, t), qk+1(x, t)
  • to the problem
                 ∂tuk+1 + Df(uk) ∂xuk+1 = ǫk+1
  • ∂xxuk+1 − ∂ttuk+1
linear equations for uk+1 ∂tqk+1 + ∂x
  • qk+1 Df(uk)
  • = ǫk+1
  • ∂ttqk+1 − ∂xxqk+1
adjoint equations for qk+1 uk+1(0, ·) = uk+1 , ∂tuk+1(0, x) = 0 initial conditions on uk+1 qk+1(T, ·) = 0 , ∂tqk+1(T, ·) = −∇J(x, uk+1(T, ·)) terminal conditions on qk+1 Andrea Marson Approximation of optimal control problems for conservation laws
slide-93
SLIDE 93 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

Towards a numerical algorithm (in progress)

By writing the linearized and adjoint equations... We look for uk+1 ∈ Uǫk+1 ad =
  • u0 ∈ W 1,1(R) : u0W 1,1 ≤ δ , ∂xxu0L1 ≤ C/εk+1 ,
∂xxxu0L1 ≤ C/ε2 k+1 , TV ∂xxxu0 ≤ C/ε3 k+1
  • such that the solution (uk+1, qk+1) =
  • uk+1(x, t), qk+1(x, t)
  • to the problem
                 ∂tuk+1 + Df(uk) ∂xuk+1 = ǫk+1
  • ∂xxuk+1 − ∂ttuk+1
linear equations for uk+1 ∂tqk+1 + ∂x
  • qk+1 Df(uk)
  • = ǫk+1
  • ∂ttqk+1 − ∂xxqk+1
adjoint equations for qk+1 uk+1(0, ·) = uk+1 , ∂tuk+1(0, x) = 0 initial conditions on uk+1 qk+1(T, ·) = 0 , ∂tqk+1(T, ·) = −∇J(x, uk+1(T, ·)) terminal conditions on qk+1 fulfills ǫk+1 ∂tqk+1(x, 0) − qk+1(x, 0) = 0 Andrea Marson Approximation of optimal control problems for conservation laws
slide-94
SLIDE 94 Introduction Another approach Necessary conditions Linearized and adjoint equations An alternative attempt

Thank you for your attention!

Andrea Marson Approximation of optimal control problems for conservation laws