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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Control Problems for Hyperbolic Equations Fabio ANCONA University of Bologna , Italy 12th International Conference on

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems

Control Problems for Hyperbolic Equations

Fabio ANCONA University of Bologna , Italy 12th International Conference on Hyperbolic Problems University of Maryland, College Park, June 9–13, 2008

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 2

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems

Outline

1

Introduction General setting Physical Motivations Main problems

2

Controllability & Stabilizability Exact controllability Asymptotic stabilizability

3

Optimal control problems Generalized tangent vectors Linearized evolution equations

4

Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 3

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Outline

1

Introduction General setting Physical Motivations Main problems

2

Controllability & Stabilizability Exact controllability Asymptotic stabilizability

3

Optimal control problems Generalized tangent vectors Linearized evolution equations

4

Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 4

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

General setting

                     ∂t u + ∂x f(u) = h(x, u, z) , u(0, x) = ¯ u(x) , b.c. at x = ψ0(t) , x = ψ1(t) , with bdr data α0, α1 t ≥ 0 , ψ0(t) < x < ψ1(t) u = u(t, x) ∈ Rn conserved quantities f : Ω ⊆ Rn → Rn smooth flux h : R × Ω × Rm → Rn smooth source z = z(t, x) ∈ Z ⊂ Rm distributed control αj = αj(t) ∈ R

pj

boundary control

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 5

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

General setting

                     ∂t u + ∂x f(u) = h(x, u, z) , u(0, x) = ¯ u(x) , b.c. at x = ψ0(t) , x = ψ1(t) , with bdr data α0, α1 t ≥ 0 , ψ0(t) < x < ψ1(t) u = u(t, x) ∈ Rn conserved quantities f : Ω ⊆ Rn → Rn smooth flux h : R × Ω × Rm → Rn smooth source z = z(t, x) ∈ Z ⊂ Rm distributed control αj = αj(t) ∈ R

pj

boundary control

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 6

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

General setting

                     ∂t u + ∂x f(u) = h(x, u, z) , u(0, x) = ¯ u(x) , b.c. at x = ψ0(t) , x = ψ1(t) , with bdr data α0, α1 t ≥ 0 , ψ0(t) < x < ψ1(t) u = u(t, x) ∈ Rn conserved quantities f : Ω ⊆ Rn → Rn smooth flux h : R × Ω × Rm → Rn smooth source z = z(t, x) ∈ Z ⊂ Rm distributed control αj = αj(t) ∈ R

pj

boundary control

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 7

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

General setting

                     ∂t u + ∂x f(u) = h(x, u, z) , u(0, x) = ¯ u(x) , b.c. at x = ψ0(t) , x = ψ1(t) , with bdr data α0, α1 t ≥ 0 , ψ0(t) < x < ψ1(t) u = u(t, x) ∈ Rn conserved quantities f : Ω ⊆ Rn → Rn smooth flux h : R × Ω × Rm → Rn smooth source z = z(t, x) ∈ Z ⊂ Rm distributed control αj = αj(t) ∈ R

pj

boundary control

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 8

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

General setting

                     ∂t u + ∂x f(u) = h(x, u, z) , u(0, x) = ¯ u(x) , b.c. at x = ψ0(t) , x = ψ1(t) , with bdr data α0, α1 t ≥ 0 , ψ0(t) < x < ψ1(t) u = u(t, x) ∈ Rn conserved quantities f : Ω ⊆ Rn → Rn smooth flux h : R × Ω × Rm → Rn smooth source z = z(t, x) ∈ Z ⊂ Rm distributed control αj = αj(t) ∈ R

pj

boundary control

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 9

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

General setting

                     ∂t u + ∂x f(u) = h(x, u, z) , u(0, x) = ¯ u(x) , b.c. at x = ψ0(t) , x = ψ1(t) , with bdr data α0, α1 t ≥ 0 , ψ0(t) < x < ψ1(t) u = u(t, x) ∈ Rn conserved quantities f : Ω ⊆ Rn → Rn smooth flux h : R × Ω × Rm → Rn smooth source z = z(t, x) ∈ Z ⊂ Rm distributed control αj = αj(t) ∈ R

pj

boundary control

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 10

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

General setting

                     ∂t u + ∂x f(u) = h(x, u, z) , u(0, x) = ¯ u(x) , b.c. at x = ψ0(t) , x = ψ1(t) , with bdr data α0, α1 t ≥ 0 , ψ0(t) < x < ψ1(t) u = u(t, x) ∈ Rn conserved quantities f : Ω ⊆ Rn → Rn smooth flux h : R × Ω × Rm → Rn smooth source z = z(t, x) ∈ Z ⊂ Rm distributed control αj = αj(t) ∈ R

pj

boundary control

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

General Assumptions

Strictly Hyperbolic System ∂tu + ∂xf(u) = h(x, u, z) Df(u)ri(u) = λi(u)ri(u) i = 1, . . . , n λ1(u) < λ2(u) < · · · < λn(u) Weaker Formulation of B.C. Dirichlet b.c. not fulfilled pointwise

t

If λp(u) < ˙ ψ0(t) < λp+1(u) n − p cond’s at x = ψ0

λp+1 λn λp+2 x x = ψ0

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 12

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

General Assumptions

Strictly Hyperbolic System ∂tu + ∂xf(u) = h(x, u, z) Df(u)ri(u) = λi(u)ri(u) i = 1, . . . , n λ1(u) < λ2(u) < · · · < λn(u) Weaker Formulation of B.C. Dirichlet b.c. not fulfilled pointwise

t

If λp(u) < ˙ ψ0(t) < λp+1(u) n − p cond’s at x = ψ0

λp+1 λn λp+2 x x = ψ0

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 13

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Outline

1

Introduction General setting Physical Motivations Main problems

2

Controllability & Stabilizability Exact controllability Asymptotic stabilizability

3

Optimal control problems Generalized tangent vectors Linearized evolution equations

4

Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Isentropic gas dynamic (p-system)

Gas in a clinder with moving piston (in Lagrangian coord.)

  • ∂tv − ∂xu = 0

∂tu + ∂xp(v) = 0 x ∈]0, h[ v specific volume, u speed, p pressure

x gas x = h x = 0

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 15

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Isentropic gas dynamic (p-system)

Gas in a clinder with moving piston (in Lagrangian coord.)

  • ∂tv − ∂xu = 0

∂tu + ∂xp(v) = 0 x ∈]0, h[ v specific volume, u speed, p pressure

x gas x = h x = 0

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 16

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Stabilization problem for gas dynamic

a control acting only on speed u at x = h: u(t, h) = α(t). a reflection condition at x = 0: u(t, 0) = 0. Pb: given v(0, x) = ¯ v(x), u(0, x) = ¯ u(x) x ∈ ]0, h[ , Stabilize the system at an equilibrium (v, u) = (v∗, 0).

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 17

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Stabilization problem for gas dynamic

a control acting only on speed u at x = h: u(t, h) = α(t). a reflection condition at x = 0: u(t, 0) = 0. Pb: given v(0, x) = ¯ v(x), u(0, x) = ¯ u(x) x ∈ ]0, h[ , Stabilize the system at an equilibrium (v, u) = (v∗, 0).

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 18

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Stabilization problem for gas dynamic

a control acting only on speed u at x = h: u(t, h) = α(t). a reflection condition at x = 0: u(t, 0) = 0. Pb: given v(0, x) = ¯ v(x), u(0, x) = ¯ u(x) x ∈ ]0, h[ , Stabilize the system at an equilibrium (v, u) = (v∗, 0).

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 19

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Stabilization problem for gas dynamic

a control acting only on speed u at x = h: u(t, h) = α(t). a reflection condition at x = 0: u(t, 0) = 0. Pb: given v(0, x) = ¯ v(x), u(0, x) = ¯ u(x) x ∈ ]0, h[ , Stabilize the system at an equilibrium (v, u) = (v∗, 0).

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Multicomponent chromatoghraphy

Separate two chemical species in a fluid by selective absorption on a solid medium

OUTLET S1 S2 x x = L x = 0 INLET

   ∂xc1 + ∂t

  • γc1

1+c1+c2

  • = 0

∂xc2 + ∂t

  • c2

1+c1+c2

  • = 0

x ∈]0, L[ ci concentration solute Si (γ ∈ ]0, 1])

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 21

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Multicomponent chromatoghraphy

Separate two chemical species in a fluid by selective absorption on a solid medium

OUTLET S1 S2 x x = L x = 0 INLET

   ∂xc1 + ∂t

  • γc1

1+c1+c2

  • = 0

∂xc2 + ∂t

  • c2

1+c1+c2

  • = 0

x ∈]0, L[ ci concentration solute Si (γ ∈ ]0, 1])

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Multicomponent chromatoghraphy

Temple system with GNL characteristic fields Ree, Aris & Amundson (1986, 1989) control concentration solute Si entering the tube at x = 0: ci(t, 0) = αi(t).

x t L T

ci = αi ATTAINABLE SET A(T) ci = ci

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 23

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Multicomponent chromatoghraphy

Temple system with GNL characteristic fields Ree, Aris & Amundson (1986, 1989) control concentration solute Si entering the tube at x = 0: ci(t, 0) = αi(t).

x t L T

ci = αi ATTAINABLE SET A(T) ci = ci

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 24

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Multicomponent chromatoghraphy

Temple system with GNL characteristic fields Ree, Aris & Amundson (1986, 1989) control concentration solute Si entering the tube at x = 0: ci(t, 0) = αi(t).

x t L T

ci = αi ATTAINABLE SET A(T) ci = ci

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 25

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Optimization problem for chromatography

Maximize separation of solutes at time T max

x, α

x (c1(T, ξ) − c2(T, ξ)) dξ+ + L

x

(c2(T, ξ) − c1(T, ξ)) dξ

                ∂xc1 + ∂t

  • γc1

1+c1+c2

  • = 0 ,

∂xc2 + ∂t

  • c2

1+c1+c2

  • = 0 ,

ci(0, x) = ¯ ci , ci(t, 0) = αi(t) . x ∈]0, L[ ,

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 26

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Outline

1

Introduction General setting Physical Motivations Main problems

2

Controllability & Stabilizability Exact controllability Asymptotic stabilizability

3

Optimal control problems Generalized tangent vectors Linearized evolution equations

4

Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 27

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Two Classes of Problems

  • 1. Controllability & Stabilizability
  • 2. Optimal control problems

(Mostly boundary controls will be considered)

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 28

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Two Classes of Problems

  • 1. Controllability & Stabilizability
  • 2. Optimal control problems

(Mostly boundary controls will be considered)

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 29

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Two Classes of Problems

  • 1. Controllability & Stabilizability
  • 2. Optimal control problems

(Mostly boundary controls will be considered)

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 30

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Boundary Controllability & Stabilizability

Boundary conditions (non characteristic boundary) bj(u(t, ψj(t))) = gj(αj(t)) (j = 0, 1) Given: initial datum u desired terminal profile Φ (e.g. a constant state Φ(x) ≡ u∗) Do exist: boundary controls αj at x = ψj so that solution uα(t, x) of corresponding IBVP satisfies:

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 31

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Boundary Controllability & Stabilizability

Boundary conditions (non characteristic boundary) bj(u(t, ψj(t))) = gj(αj(t)) (j = 0, 1) Given: initial datum u desired terminal profile Φ (e.g. a constant state Φ(x) ≡ u∗) Do exist: boundary controls αj at x = ψj so that solution uα(t, x) of corresponding IBVP satisfies:

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 32

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Boundary Controllability & Stabilizability

Boundary conditions (non characteristic boundary) bj(u(t, ψj(t))) = gj(αj(t)) (j = 0, 1) Given: initial datum u desired terminal profile Φ (e.g. a constant state Φ(x) ≡ u∗) Do exist: boundary controls αj at x = ψj so that solution uα(t, x) of corresponding IBVP satisfies:

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 33

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Boundary Controllability & Stabilizability

Boundary conditions (non characteristic boundary) bj(u(t, ψj(t))) = gj(αj(t)) (j = 0, 1) Given: initial datum u desired terminal profile Φ (e.g. a constant state Φ(x) ≡ u∗) Do exist: boundary controls αj at x = ψj so that solution uα(t, x) of corresponding IBVP satisfies:

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 34

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Boundary Controllability & Stabilizability

uα(T, ·) = Φ (finite time exact controllability)

b1(uα) = g1(α1)

x

t T uα = u x = ψ0 x = ψ1 b0(uα) = g0(α0)

  • r

lim

t→∞ uα(t, ·) = Φ ?

(asymptotic stabilizability)

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 35

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Boundary Controllability & Stabilizability

uα(T, ·) = Φ (finite time exact controllability)

b1(uα) = g1(α1)

x

t T uα = u x = ψ0 x = ψ1 b0(uα) = g0(α0)

  • r

lim

t→∞ uα(t, ·) = Φ ?

(asymptotic stabilizability)

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 36

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Boundary Controllability & Stabilizability

uα(T, ·) = Φ (finite time exact controllability)

b1(uα) = g1(α1)

x

t T uα = u x = ψ0 x = ψ1 b0(uα) = g0(α0)

  • r

lim

t→∞ uα(t, ·) = Φ ?

(asymptotic stabilizability)

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 37

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Optimization problem

max

  • J (u, z, α) : z ∈ Z, α ∈ A
  • J (u, z, α) =

T +∞ L(x, u, z) dxdt + +∞ Φ

  • x, u(T, x)
  • dx+

+ T Ψ

  • u(t, 0), α(t)
  • dt

single boundary ψ0 ≡ 0 L, Φ, Ψ smooth A ⊂ L∞(0, T) admissible boundary controls at x = 0 Z ⊂ L1

loc(]0, +∞[×R) admissible distributed controls

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 38

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Optimization problem

max

  • J (u, z, α) : z ∈ Z, α ∈ A
  • J (u, z, α) =

T +∞ L(x, u, z) dxdt + +∞ Φ

  • x, u(T, x)
  • dx+

+ T Ψ

  • u(t, 0), α(t)
  • dt

single boundary ψ0 ≡ 0 L, Φ, Ψ smooth A ⊂ L∞(0, T) admissible boundary controls at x = 0 Z ⊂ L1

loc(]0, +∞[×R) admissible distributed controls

Fabio Ancona Control Problems for Hyperbolic Equations

slide-39
SLIDE 39

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Optimization problem

max

  • J (u, z, α) : z ∈ Z, α ∈ A
  • J (u, z, α) =

T +∞ L(x, u, z) dxdt + +∞ Φ

  • x, u(T, x)
  • dx+

+ T Ψ

  • u(t, 0), α(t)
  • dt

single boundary ψ0 ≡ 0 L, Φ, Ψ smooth A ⊂ L∞(0, T) admissible boundary controls at x = 0 Z ⊂ L1

loc(]0, +∞[×R) admissible distributed controls

Fabio Ancona Control Problems for Hyperbolic Equations

slide-40
SLIDE 40

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Optimization problem

max

  • J (u, z, α) : z ∈ Z, α ∈ A
  • J (u, z, α) =

T +∞ L(x, u, z) dxdt + +∞ Φ

  • x, u(T, x)
  • dx+

+ T Ψ

  • u(t, 0), α(t)
  • dt

single boundary ψ0 ≡ 0 L, Φ, Ψ smooth A ⊂ L∞(0, T) admissible boundary controls at x = 0 Z ⊂ L1

loc(]0, +∞[×R) admissible distributed controls

Fabio Ancona Control Problems for Hyperbolic Equations

slide-41
SLIDE 41

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems General setting Physical Motivations Main problems

Optimization problem

max

  • J (u, z, α) : z ∈ Z, α ∈ A
  • J (u, z, α) =

T +∞ L(x, u, z) dxdt + +∞ Φ

  • x, u(T, x)
  • dx+

+ T Ψ

  • u(t, 0), α(t)
  • dt

single boundary ψ0 ≡ 0 L, Φ, Ψ smooth A ⊂ L∞(0, T) admissible boundary controls at x = 0 Z ⊂ L1

loc(]0, +∞[×R) admissible distributed controls

Fabio Ancona Control Problems for Hyperbolic Equations

slide-42
SLIDE 42

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Exact controllability Asymptotic stabilizability

Outline

1

Introduction General setting Physical Motivations Main problems

2

Controllability & Stabilizability Exact controllability Asymptotic stabilizability

3

Optimal control problems Generalized tangent vectors Linearized evolution equations

4

Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Fabio Ancona Control Problems for Hyperbolic Equations

slide-43
SLIDE 43

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Exact controllability Asymptotic stabilizability

Finite time exact controllability to constant states u∗

  • 1. Quasilinear systems

∂t u + A(u) ∂x u = h(u) x ∈ ]a, b[ , with suff. small C1 initial data u (Cirinà, 1969; T.Li, B. Rao & co, 2002-2008; M.Gugat & G. Leugering, 2003)

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 44

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Exact controllability Asymptotic stabilizability

Finite time exact controllability to constant states u∗

  • 2. Nonlinear scalar convex con laws and GNL Temple

systems ∂t u + ∂x (f(u)) = 0 x ∈ ]a, b[ , with initial data u ∈ L∞ (L1) (discontinuous entropy weak solutions) (F .A., A.Marson, 1998; T. Horsin, 1998; F .A. & G.M. Coclite, 2005)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Exact controllability Asymptotic stabilizability

Finite time exact controllability to constant states u∗

  • 3. Isentropic gas dynamic (in Eulerian coord.)

   ∂tρ + ∂x(ρ u) = 0 ∂t(ρ u) + ∂x

  • ρ u2 + Kργ

= 0 with T.V.{bdr controls}≫ u∗ − u∞ (strong perturbation of the solution) (O. Glass, 2006)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Exact controllability Asymptotic stabilizability

NO exact controllability to constant states u∗

  • 4. Isentropic gas dynamic for a polytropic gas (in Eulerian

coord.)      ∂tρ + ∂x(ρ u) = 0 ∂tu + ∂x u2 2 + K γ − 1ργ−1

  • = 0

∃ initial datum so that corresponding sol. has dense set of discontinuities, whatever bdr controls are prescribed (A.Bressan & G.M.Coclite, 2002)

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 47

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Exact controllability Asymptotic stabilizability

Outline

1

Introduction General setting Physical Motivations Main problems

2

Controllability & Stabilizability Exact controllability Asymptotic stabilizability

3

Optimal control problems Generalized tangent vectors Linearized evolution equations

4

Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 48

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Exact controllability Asymptotic stabilizability

  • 1. Stabilizability with total control on both boundaries

Asymptotic stabilizability around a constant state with exponential rate (A.Bressan & G.M.Coclite, 2002)

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 49

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Exact controllability Asymptotic stabilizability

  • 2. Stabilizability with total control on single boundary

   b0(u(t, ψ0(t))) = 0 , b1(u(t, ψ1(t))) = g(α(t))

  • Assume Dg(α) has full rank

⇒ full control on waves entering the domain from x = ψ1

t λ2 λ1 λp b1(u) = g(α) x x = ψ0 x = ψ1

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 50

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Exact controllability Asymptotic stabilizability

  • 2. Stabilizability with total control on single boundary

   b0(u(t, ψ0(t))) = 0 , b1(u(t, ψ1(t))) = g(α(t))

  • Assume Dg(α) has full rank

⇒ full control on waves entering the domain from x = ψ1

t λ2 λ1 λp b1(u) = g(α) x x = ψ0 x = ψ1

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Exact controllability Asymptotic stabilizability

  • 2. Stabilizability with total control on single boundary
  • Assume p ≥ n − p and Db0(u) with maximum rank

rk

  • Db0 · r1(u), . . . , Db0 · rp(u)
  • = n − p

t λp λp+1 λ1 x = ψ0 x = ψ1 λn use control α acting x at x = ψ1 to generate first p components of u∗ to generate remaining n − p components of u∗ use reflections at x = ψ0 b1 = g(α)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Exact controllability Asymptotic stabilizability

  • 2. Stabilizability with total control on single boundary
  • Assume p ≥ n − p and Db0(u) with maximum rank

rk

  • Db0 · r1(u), . . . , Db0 · rp(u)
  • = n − p

t λp λp+1 λ1 x = ψ0 x = ψ1 λn use control α acting x at x = ψ1 to generate first p components of u∗ to generate remaining n − p components of u∗ use reflections at x = ψ0 b1 = g(α)

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 53

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Exact controllability Asymptotic stabilizability

  • 2. Stabilizability with total control on single boundary
  • Assume p ≥ n − p and Db0(u) with maximum rank

rk

  • Db0 · r1(u), . . . , Db0 · rp(u)
  • = n − p

t λp λp+1 λ1 x = ψ0 x = ψ1 λn use control α acting x at x = ψ1 to generate first p components of u∗ to generate remaining n − p components of u∗ use reflections at x = ψ0 b1 = g(α)

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 54

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Exact controllability Asymptotic stabilizability

  • 2. Stabilizability with total control on single boundary
  • Nonlinear system ⇒ waves produced by bndr control

interact with each other generating new waves (2nd generation waves) ∃τ, bdr control α s.t. T.V.uα(τ, ·) = O(1) · |¯ u − u∗|2 uα(τ, ·) − u∗∞ = O(1) · |¯ u − u∗|2 ⇓ Asymptotic stabilization to equilibrium u∗ (b0(u∗) = 0) (F .A. & A.Marson, 2007)

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 55

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Exact controllability Asymptotic stabilizability

  • 2. Stabilizability with total control on single boundary
  • Nonlinear system ⇒ waves produced by bndr control

interact with each other generating new waves (2nd generation waves) ∃τ, bdr control α s.t. T.V.uα(τ, ·) = O(1) · |¯ u − u∗|2 uα(τ, ·) − u∗∞ = O(1) · |¯ u − u∗|2 ⇓ Asymptotic stabilization to equilibrium u∗ (b0(u∗) = 0) (F .A. & A.Marson, 2007)

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 56

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Exact controllability Asymptotic stabilizability

  • 2. Stabilizability with total control on single boundary
  • Nonlinear system ⇒ waves produced by bndr control

interact with each other generating new waves (2nd generation waves) ∃τ, bdr control α s.t. T.V.uα(τ, ·) = O(1) · |¯ u − u∗|2 uα(τ, ·) − u∗∞ = O(1) · |¯ u − u∗|2 ⇓ Asymptotic stabilization to equilibrium u∗ (b0(u∗) = 0) (F .A. & A.Marson, 2007)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Optimization problem

max

z∈Z, α∈A

T +∞ L(x, u, z) dxdt + +∞ Φ

  • x, u(T, x)
  • dx+

+ T Ψ

  • u(t, 0), α(t)
  • dt

u = uz,α(t, x) solution to (ψ0 ≡ 0):        ∂tu + ∂xf(u) = h(x, u, z) , u(0, x) = ¯ u(x) , b(u(t, 0)) = α(t)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Optimization problem

max

z∈Z, α∈A

T +∞ L(x, u, z) dxdt + +∞ Φ

  • x, u(T, x)
  • dx+

+ T Ψ

  • u(t, 0), α(t)
  • dt

u = uz,α(t, x) solution to (ψ0 ≡ 0):        ∂tu + ∂xf(u) = h(x, u, z) , u(0, x) = ¯ u(x) , b(u(t, 0)) = α(t)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Goals

  • 1. Establish existence of optimal solutions
  • 2. Seek necessary conditions for optimality of controls

z, α

  • 3. Provide algorithm to construct (almost) optimal solutions

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Goals

  • 1. Establish existence of optimal solutions
  • 2. Seek necessary conditions for optimality of controls

z, α

  • 3. Provide algorithm to construct (almost) optimal solutions

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Goals

  • 1. Establish existence of optimal solutions
  • 2. Seek necessary conditions for optimality of controls

z, α

  • 3. Provide algorithm to construct (almost) optimal solutions

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Main difficulties

Lack of regularity of sol’ns to cons. laws

x x x0 compression wave shock wave x0 u0(t, ·) u0

x → ∞

u t f ′(u0) u0(0, ·)

Non differentiability of input-to-trajectory map (z, α) → uz,α in any natural Banach space

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Main difficulties

Lack of regularity of sol’ns to cons. laws

x x x0 compression wave shock wave x0 u0(t, ·) u0

x → ∞

u t f ′(u0) u0(0, ·)

Non differentiability of input-to-trajectory map (z, α) → uz,α in any natural Banach space

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Non differentiability

∂tu + ∂x u2 2

  • = 0 ,

u(0, x) = ¯ uθ(x) . = (1 + θ)x · χ[0,1](x) (1)

  • Sol. to (1):

uθ(t, x) = (1 + θ)x 1 + (1 + θ)t · χ[0,√

1+(1+θ)t](x)

Notice: ¯ uθ is differentiable in L1 at θ = 0 lim

θ→0

¯ uθ − ¯ u0 − θ ¯ u0L1 θ = 0

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Non differentiability

∂tu + ∂x u2 2

  • = 0 ,

u(0, x) = ¯ uθ(x) . = (1 + θ)x · χ[0,1](x) (1)

  • Sol. to (1):

uθ(t, x) = (1 + θ)x 1 + (1 + θ)t · χ[0,√

1+(1+θ)t](x)

Notice: ¯ uθ is differentiable in L1 at θ = 0 lim

θ→0

¯ uθ − ¯ u0 − θ ¯ u0L1 θ = 0

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Non differentiability

∂tu + ∂x u2 2

  • = 0 ,

u(0, x) = ¯ uθ(x) . = (1 + θ)x · χ[0,1](x) (1)

  • Sol. to (1):

uθ(t, x) = (1 + θ)x 1 + (1 + θ)t · χ[0,√

1+(1+θ)t](x)

Notice: ¯ uθ is differentiable in L1 at θ = 0 lim

θ→0

¯ uθ − ¯ u0 − θ ¯ u0L1 θ = 0

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Non differentiability

The location of the jump in uθ(t, ·) depends on θ

u x u0(t, ·) uθ(t, ·) √ 1 + t

  • 1 + (1 + θ)t

≈ θ · (shift rate)

⇒ uθ(t, ·) is NOT diff. in L1 at θ = 0 for t > 0

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Non differentiability

The location of the jump in uθ(t, ·) depends on θ

u x u0(t, ·) uθ(t, ·) √ 1 + t

  • 1 + (1 + θ)t

≈ θ · (shift rate)

⇒ uθ(t, ·) is NOT diff. in L1 at θ = 0 for t > 0

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Non differentiability

lim

θ→0

uθ(t, ·) − u0(t, ·) θ yields a measure µt with a nonzero singular part located at the point of jump x(t) = √ 1 + t

  • f u0(t, ·)

(µt)s = ∆u0(t, x(t))

  • size of the jump

· d dθ

  • 1 + (1 + θ)t
  • θ=0
  • shift rate

· δx(t) = t 2(1 + t) · δx(t)

  • ∆u0(t, x(t)) = u0(t, x(t)−) − u0(t, x(t)+) =

1 √ 1 + t

  • Fabio Ancona

Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Non differentiability

lim

θ→0

uθ(t, ·) − u0(t, ·) θ yields a measure µt with a nonzero singular part located at the point of jump x(t) = √ 1 + t

  • f u0(t, ·)

(µt)s = ∆u0(t, x(t))

  • size of the jump

· d dθ

  • 1 + (1 + θ)t
  • θ=0
  • shift rate

· δx(t) = t 2(1 + t) · δx(t)

  • ∆u0(t, x(t)) = u0(t, x(t)−) − u0(t, x(t)+) =

1 √ 1 + t

  • Fabio Ancona

Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Non differentiability

lim

θ→0

uθ(t, ·) − u0(t, ·) θ yields a measure µt with a nonzero singular part located at the point of jump x(t) = √ 1 + t

  • f u0(t, ·)

(µt)s = ∆u0(t, x(t))

  • size of the jump

· d dθ

  • 1 + (1 + θ)t
  • θ=0
  • shift rate

· δx(t) = t 2(1 + t) · δx(t)

  • ∆u0(t, x(t)) = u0(t, x(t)−) − u0(t, x(t)+) =

1 √ 1 + t

  • Fabio Ancona

Control Problems for Hyperbolic Equations

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SLIDE 72

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Outline

1

Introduction General setting Physical Motivations Main problems

2

Controllability & Stabilizability Exact controllability Asymptotic stabilizability

3

Optimal control problems Generalized tangent vectors Linearized evolution equations

4

Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 73

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Generalized tangent vectors

A generalized tangent vector generated by a family of solutions

  • uθ

, with uθ(t) − u0(t) θ ⇀ µt, is an element (v, ξ) ∈ L1(R) × R♯ jumps in u v (vertical displacement) takes into account of the absolutely continuous part of µt ξ (horizontal displacement) takes into account of the singular part of µt (no Cantor part in µt) (A.Bressan & A.Marson, 1995)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Generalized tangent vectors

A generalized tangent vector generated by a family of solutions

  • uθ

, with uθ(t) − u0(t) θ ⇀ µt, is an element (v, ξ) ∈ L1(R) × R♯ jumps in u v (vertical displacement) takes into account of the absolutely continuous part of µt ξ (horizontal displacement) takes into account of the singular part of µt (no Cantor part in µt) (A.Bressan & A.Marson, 1995)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Generalized tangent vectors

A generalized tangent vector generated by a family of solutions

  • uθ

, with uθ(t) − u0(t) θ ⇀ µt, is an element (v, ξ) ∈ L1(R) × R♯ jumps in u v (vertical displacement) takes into account of the absolutely continuous part of µt ξ (horizontal displacement) takes into account of the singular part of µt (no Cantor part in µt) (A.Bressan & A.Marson, 1995)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Generalized tangent vectors

A generalized tangent vector generated by a family of solutions

  • uθ

, with uθ(t) − u0(t) θ ⇀ µt, is an element (v, ξ) ∈ L1(R) × R♯ jumps in u v (vertical displacement) takes into account of the absolutely continuous part of µt ξ (horizontal displacement) takes into account of the singular part of µt (no Cantor part in µt) (A.Bressan & A.Marson, 1995)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Vertical displacement

u0(t, ·) ≈ θv u x ≈ θv uθ(t, ·)

v(t, x) = lim

θ→0

uθ(t, x) − u0(t, x) θ

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Vertical displacement

u0(t, ·) ≈ θv u x ≈ θv uθ(t, ·)

v(t, x) = lim

θ→0

uθ(t, x) − u0(t, x) θ

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Horizontal displacement

uθ(t, ·) u x u0(t, ·) ≈ θξα ≈ θξβ xθ

α

xθ

β

x0

β

x0

α

ξα(t) = lim

θ→0

xθ

α(t) − x0 α(t)

θ rates of horizontal displacement of locations xθ

1(t) < · · · > xθ N(t)

  • f jumps in uθ(t, ·)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Horizontal displacement

uθ(t, ·) u x u0(t, ·) ≈ θξα ≈ θξβ xθ

α

xθ

β

x0

β

x0

α

ξα(t) = lim

θ→0

xθ

α(t) − x0 α(t)

θ rates of horizontal displacement of locations xθ

1(t) < · · · > xθ N(t)

  • f jumps in uθ(t, ·)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Admissible variations

x t (τθ, ηθ) (τ0, η0)

  • ξγ

xβ xθ

β

ξα ξβ xθ

α

[xα

uθ(t) ≈ u0(t) + θv(t) +

  • ξα<0

∆u0(t, xα(t)) · χ[x0(t)+θξα(t),x0(t)] +

  • ξα>0

∆u0(t, xα(t)) · χ[x0(t),x0(t)+θξα(t)]

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Admissible variations

x t (τθ, ηθ) (τ0, η0)

  • ξγ

xβ xθ

β

ξα ξβ xθ

α

[xα

uθ(t) ≈ u0(t) + θv(t) +

  • ξα<0

∆u0(t, xα(t)) · χ[x0(t)+θξα(t),x0(t)] +

  • ξα>0

∆u0(t, xα(t)) · χ[x0(t),x0(t)+θξα(t)]

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Admissible variations

x t (τθ, ηθ) (τ0, η0)

  • ξγ

xβ xθ

β

ξα ξβ xθ

α

[xα

uθ(t) ≈ u0(t) + θv(t) +

  • ξα<0

∆u0(t, xα(t)) · χ[x0(t)+θξα(t),x0(t)] +

  • ξα>0

∆u0(t, xα(t)) · χ[x0(t),x0(t)+θξα(t)]

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Admissible variations

x t (τθ, ηθ) (τ0, η0)

  • ξγ

xβ xθ

β

ξα ξβ xθ

α

[xα

uθ(t) ≈ u0(t) + θv(t) +

  • ξα<0

∆u0(t, xα(t)) · χ[x0(t)+θξα(t),x0(t)] +

  • ξα>0

∆u0(t, xα(t)) · χ[x0(t),x0(t)+θξα(t)]

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Outline

1

Introduction General setting Physical Motivations Main problems

2

Controllability & Stabilizability Exact controllability Asymptotic stabilizability

3

Optimal control problems Generalized tangent vectors Linearized evolution equations

4

Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 86

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Evolution of generalized tangent vectors

If uθ(¯ t, ·) generates a generalized tangent vector discontinuities of u0 interact at most two at the time uθ is piecewise Lipschitz with uniform in θ Lipschitz constant outside the discontinuities Then uθ(t, ·) generates a generalized tangent vector

  • v(t, ·), ξ(t)
  • for t > ¯

t (A.Bressan & A.Marson, 1995)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Evolution of generalized tangent vectors

If uθ(¯ t, ·) generates a generalized tangent vector discontinuities of u0 interact at most two at the time uθ is piecewise Lipschitz with uniform in θ Lipschitz constant outside the discontinuities Then uθ(t, ·) generates a generalized tangent vector

  • v(t, ·), ξ(t)
  • for t > ¯

t (A.Bressan & A.Marson, 1995)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Evolution of generalized tangent vectors

If uθ(¯ t, ·) generates a generalized tangent vector discontinuities of u0 interact at most two at the time uθ is piecewise Lipschitz with uniform in θ Lipschitz constant outside the discontinuities Then uθ(t, ·) generates a generalized tangent vector

  • v(t, ·), ξ(t)
  • for t > ¯

t (A.Bressan & A.Marson, 1995)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Evolution of generalized tangent vectors

If uθ(¯ t, ·) generates a generalized tangent vector discontinuities of u0 interact at most two at the time uθ is piecewise Lipschitz with uniform in θ Lipschitz constant outside the discontinuities Then uθ(t, ·) generates a generalized tangent vector

  • v(t, ·), ξ(t)
  • for t > ¯

t (A.Bressan & A.Marson, 1995)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Evolution of generalized tangent vectors

If uθ(¯ t, ·) generates a generalized tangent vector discontinuities of u0 interact at most two at the time uθ is piecewise Lipschitz with uniform in θ Lipschitz constant outside the discontinuities Then uθ(t, ·) generates a generalized tangent vector

  • v(t, ·), ξ(t)
  • for t > ¯

t (A.Bressan & A.Marson, 1995)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Evolution of generalized tangent vectors

If uθ(¯ t, ·) generates a generalized tangent vector discontinuities of u0 interact at most two at the time uθ is piecewise Lipschitz with uniform in θ Lipschitz constant outside the discontinuities Then uθ(t, ·) generates a generalized tangent vector

  • v(t, ·), ξ(t)
  • for t > ¯

t (A.Bressan & A.Marson, 1995)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Evolution of generalized tangent vectors

Moreover v(t, x) is a broad solution of ∂tv + Df(u)∂xv +

  • D2f(u) · v
  • ∂xu = Duh(x, u, z) · v

ξα(t) satisfies an ODE along the α-th discontinuity x = xα(t) explicit restarting conditions at the interaction of two discontinuities (A.Bressan & A.Marson, 1995)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Evolution of generalized tangent vectors

Moreover v(t, x) is a broad solution of ∂tv + Df(u)∂xv +

  • D2f(u) · v
  • ∂xu = Duh(x, u, z) · v

ξα(t) satisfies an ODE along the α-th discontinuity x = xα(t) explicit restarting conditions at the interaction of two discontinuities (A.Bressan & A.Marson, 1995)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Evolution of generalized tangent vectors

Moreover v(t, x) is a broad solution of ∂tv + Df(u)∂xv +

  • D2f(u) · v
  • ∂xu = Duh(x, u, z) · v

ξα(t) satisfies an ODE along the α-th discontinuity x = xα(t) explicit restarting conditions at the interaction of two discontinuities (A.Bressan & A.Marson, 1995)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Evolution of generalized tangent vectors

Moreover v(t, x) is a broad solution of ∂tv + Df(u)∂xv +

  • D2f(u) · v
  • ∂xu = Duh(x, u, z) · v

ξα(t) satisfies an ODE along the α-th discontinuity x = xα(t) explicit restarting conditions at the interaction of two discontinuities (A.Bressan & A.Marson, 1995)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Evolution of generalized tangent vectors

Moreover v(t, x) is a broad solution of ∂tv + Df(u)∂xv +

  • D2f(u) · v
  • ∂xu = Duh(x, u, z) · v

ξα(t) satisfies an ODE along the α-th discontinuity x = xα(t) explicit restarting conditions at the interaction of two discontinuities (A.Bressan & A.Marson, 1995)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Necessary conditions for optimality

Necessary conditions for optimality obtained by means of generalized cotangent vectors (v∗, ξ∗) satisfying

  • v∗(t, x) · v(t, x) dx +
  • j

ξ∗

j (t)ξj(t) = const

backward transported along trajectories of ∂tu + ∂xf(u) = h(x, u, z) (A. Bressan, A. Marson, 1995; A. Bressan, W. Shen, 2007)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Necessary conditions for optimality

Necessary conditions for optimality obtained by means of generalized cotangent vectors (v∗, ξ∗) satisfying

  • v∗(t, x) · v(t, x) dx +
  • j

ξ∗

j (t)ξj(t) = const

backward transported along trajectories of ∂tu + ∂xf(u) = h(x, u, z) (A. Bressan, A. Marson, 1995; A. Bressan, W. Shen, 2007)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Necessary conditions for optimality

Necessary conditions for optimality obtained by means of generalized cotangent vectors (v∗, ξ∗) satisfying

  • v∗(t, x) · v(t, x) dx +
  • j

ξ∗

j (t)ξj(t) = const

backward transported along trajectories of ∂tu + ∂xf(u) = h(x, u, z) (A. Bressan, A. Marson, 1995; A. Bressan, W. Shen, 2007)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Goal

Extend variational calculus on generalized tangent and cotangent vectors to first order variations uθ that do not satisfy structural stability assumption on wave structure of reference solution u0 uniform Lipschitzianity assumption on continuous part of reference solution u0

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Goal

Extend variational calculus on generalized tangent and cotangent vectors to first order variations uθ that do not satisfy structural stability assumption on wave structure of reference solution u0 uniform Lipschitzianity assumption on continuous part of reference solution u0

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Goal

Extend variational calculus on generalized tangent and cotangent vectors to first order variations uθ that do not satisfy structural stability assumption on wave structure of reference solution u0 uniform Lipschitzianity assumption on continuous part of reference solution u0

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Shock interactions

the discontinuities of u0 interact at most two at time

t x ξα

  • ξγ

(τ 0, η0) (τ θ, ηθ)

u0 uθ

ξβ

Stability of outgoing wave structure ⇒ existence of outgoing tangent vectors

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Shock interactions

the discontinuities of u0 interact at most two at time

t x ξα

  • ξγ

(τ 0, η0) (τ θ, ηθ)

u0 uθ

ξβ

Stability of outgoing wave structure ⇒ existence of outgoing tangent vectors

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Shock interactions

If more than two discontinuities interact at the time...

uθ

t x

u0

...instability of outgoing wave structure Existence of outgoing tangent vectors?

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Shock interactions

If more than two discontinuities interact at the time...

uθ

t x

u0

...instability of outgoing wave structure Existence of outgoing tangent vectors?

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 107

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Shock interactions

If more than two discontinuities interact at the time...

uθ

t x

u0

...instability of outgoing wave structure Existence of outgoing tangent vectors?

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Shock generation

uθ is piecewise Lipschitz with uniform in θ Lipschitz constant outside the discontinuities ⇒ no gradient catastrophe in u0

x x x0 compression wave shock wave x0 u0(t, ·) u0

x → ∞

u t f ′(u0) u0(0, ·)

⇒ no new discontinuities in u0

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Shock generation

uθ is piecewise Lipschitz with uniform in θ Lipschitz constant outside the discontinuities ⇒ no gradient catastrophe in u0

x x x0 compression wave shock wave x0 u0(t, ·) u0

x → ∞

u t f ′(u0) u0(0, ·)

⇒ no new discontinuities in u0

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Generalized tangent vectors Linearized evolution equations

Shock generation

uθ is piecewise Lipschitz with uniform in θ Lipschitz constant outside the discontinuities ⇒ no gradient catastrophe in u0

x x x0 compression wave shock wave x0 u0(t, ·) u0

x → ∞

u t f ′(u0) u0(0, ·)

⇒ no new discontinuities in u0

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Outline

1

Introduction General setting Physical Motivations Main problems

2

Controllability & Stabilizability Exact controllability Asymptotic stabilizability

3

Optimal control problems Generalized tangent vectors Linearized evolution equations

4

Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 112

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

A first step ... towards the goal

Provide necessary conditions for optimality of piecewise Lipschitz solutions with finite number of discontinuities, that may contain compression waves Extend variational calculus on generalized tangent and cotangent vectors for a particular class of hyperbolic systems (Temple systems) Derive a Pontryagin type maximum principle for optimal solutions of such systems (F .A., A. Marson, in preparation, 2008)

Fabio Ancona Control Problems for Hyperbolic Equations

slide-113
SLIDE 113

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

A first step ... towards the goal

Provide necessary conditions for optimality of piecewise Lipschitz solutions with finite number of discontinuities, that may contain compression waves Extend variational calculus on generalized tangent and cotangent vectors for a particular class of hyperbolic systems (Temple systems) Derive a Pontryagin type maximum principle for optimal solutions of such systems (F .A., A. Marson, in preparation, 2008)

Fabio Ancona Control Problems for Hyperbolic Equations

slide-114
SLIDE 114

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

A first step ... towards the goal

Provide necessary conditions for optimality of piecewise Lipschitz solutions with finite number of discontinuities, that may contain compression waves Extend variational calculus on generalized tangent and cotangent vectors for a particular class of hyperbolic systems (Temple systems) Derive a Pontryagin type maximum principle for optimal solutions of such systems (F .A., A. Marson, in preparation, 2008)

Fabio Ancona Control Problems for Hyperbolic Equations

slide-115
SLIDE 115

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

A first step ... towards the goal

Provide necessary conditions for optimality of piecewise Lipschitz solutions with finite number of discontinuities, that may contain compression waves Extend variational calculus on generalized tangent and cotangent vectors for a particular class of hyperbolic systems (Temple systems) Derive a Pontryagin type maximum principle for optimal solutions of such systems (F .A., A. Marson, in preparation, 2008)

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

What a Temple system is

Exists a system of coordinates w = (w1, . . . , wn) consisting of Riemann invariants so that ∂twi + λi(w)∂xwi = h(x, w, z) , i = 1, . . . n and the level sets

  • u : wi(u) = const
  • ,

i = 1, . . . n are hyperplanes ⇒ Hugoniot curves ≡ integral curves of characteristic fields and are straight lines. Models: chromatography, traffic flow

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

What a Temple system is

Exists a system of coordinates w = (w1, . . . , wn) consisting of Riemann invariants so that ∂twi + λi(w)∂xwi = h(x, w, z) , i = 1, . . . n and the level sets

  • u : wi(u) = const
  • ,

i = 1, . . . n are hyperplanes ⇒ Hugoniot curves ≡ integral curves of characteristic fields and are straight lines. Models: chromatography, traffic flow

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Stability of wave structure at interactions...

t x uθ u0

...even in the presence of three or more interacting discontinuities (No wave of new families emerges at the interaction) ⇒ ∃ outgoing tangent vectors

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 119

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Stability of wave structure at interactions...

t x uθ u0

...even in the presence of three or more interacting discontinuities (No wave of new families emerges at the interaction) ⇒ ∃ outgoing tangent vectors

Fabio Ancona Control Problems for Hyperbolic Equations

slide-120
SLIDE 120

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Stability of wave structure at interactions...

t x uθ u0

...even in the presence of three or more interacting discontinuities (No wave of new families emerges at the interaction) ⇒ ∃ outgoing tangent vectors

Fabio Ancona Control Problems for Hyperbolic Equations

slide-121
SLIDE 121

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Outline

1

Introduction General setting Physical Motivations Main problems

2

Controllability & Stabilizability Exact controllability Asymptotic stabilizability

3

Optimal control problems Generalized tangent vectors Linearized evolution equations

4

Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

A PDE for first order variations

Key point: consider a perturbation uθ that generates a generalized tangent vector (v, ξ) on the domain [0, T] × R. Then the limit Radon measure uθ(t) − u0(t) θ ⇀ µt = µac + µs (µs =

α ∆αu0 ξα δxα)

is a (measure valued) solution of µt +

  • Df(u0)µac

x

  • +
  • α
  • ∆αu0 ξα λkα(u0,−

α

, u0,+

α

) δxα

  • x = 0

(λkα(u0,−

α

, u0,+

α

) is shock speed of jump ∆αu0)

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 123

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

if a new shock of u0 is generated at ¯ t, apply divergence theorem for measure valued solutions to obtain µ(¯ t, ·), relying on µ(t, ·) for t < ¯ t in time intervals where no new shock is generated evolution of µ is determined by the linearized equation for generalized tangent vectors and the corresponding ODE along discontinuities of u0

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 124

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

if a new shock of u0 is generated at ¯ t, apply divergence theorem for measure valued solutions to obtain µ(¯ t, ·), relying on µ(t, ·) for t < ¯ t in time intervals where no new shock is generated evolution of µ is determined by the linearized equation for generalized tangent vectors and the corresponding ODE along discontinuities of u0

Fabio Ancona Control Problems for Hyperbolic Equations

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SLIDE 125

Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Outline

1

Introduction General setting Physical Motivations Main problems

2

Controllability & Stabilizability Exact controllability Asymptotic stabilizability

3

Optimal control problems Generalized tangent vectors Linearized evolution equations

4

Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

The Maximum Principle

Assume ( z, w) = (optimal control–optimal trajectory) be a solution to the optimal control problem

  • w with a finite number of discontinuities

cotangent vector (v∗(t, x), ξ∗(t)) be a backward solution of ∂tv∗ + ∂xv∗ · Λ( w) + v∗ DΛ( w) · ∂x( w) = = −v∗Dw h(x, w, z) − DwL(x, w, z) , Λ( w) = diag

  • λi(

w)

  • v∗(T, x) = DwΦ
  • x,

w(T, x)

  • ξ∗

α(T) = ∆Φ

  • xα,

w(T, xα)

  • + backward ODEs along the jumps for ξ∗

α

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

The Maximum Principle

Assume ( z, w) = (optimal control–optimal trajectory) be a solution to the optimal control problem

  • w with a finite number of discontinuities

cotangent vector (v∗(t, x), ξ∗(t)) be a backward solution of ∂tv∗ + ∂xv∗ · Λ( w) + v∗ DΛ( w) · ∂x( w) = = −v∗Dw h(x, w, z) − DwL(x, w, z) , Λ( w) = diag

  • λi(

w)

  • v∗(T, x) = DwΦ
  • x,

w(T, x)

  • ξ∗

α(T) = ∆Φ

  • xα,

w(T, xα)

  • + backward ODEs along the jumps for ξ∗

α

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

The Maximum Principle

Assume ( z, w) = (optimal control–optimal trajectory) be a solution to the optimal control problem

  • w with a finite number of discontinuities

cotangent vector (v∗(t, x), ξ∗(t)) be a backward solution of ∂tv∗ + ∂xv∗ · Λ( w) + v∗ DΛ( w) · ∂x( w) = = −v∗Dw h(x, w, z) − DwL(x, w, z) , Λ( w) = diag

  • λi(

w)

  • v∗(T, x) = DwΦ
  • x,

w(T, x)

  • ξ∗

α(T) = ∆Φ

  • xα,

w(T, xα)

  • + backward ODEs along the jumps for ξ∗

α

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

The Maximum Principle

Assume ( z, w) = (optimal control–optimal trajectory) be a solution to the optimal control problem

  • w with a finite number of discontinuities

cotangent vector (v∗(t, x), ξ∗(t)) be a backward solution of ∂tv∗ + ∂xv∗ · Λ( w) + v∗ DΛ( w) · ∂x( w) = = −v∗Dw h(x, w, z) − DwL(x, w, z) , Λ( w) = diag

  • λi(

w)

  • v∗(T, x) = DwΦ
  • x,

w(T, x)

  • ξ∗

α(T) = ∆Φ

  • xα,

w(T, xα)

  • + backward ODEs along the jumps for ξ∗

α

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

The Maximum Principle

Assume ( z, w) = (optimal control–optimal trajectory) be a solution to the optimal control problem

  • w with a finite number of discontinuities

cotangent vector (v∗(t, x), ξ∗(t)) be a backward solution of ∂tv∗ + ∂xv∗ · Λ( w) + v∗ DΛ( w) · ∂x( w) = = −v∗Dw h(x, w, z) − DwL(x, w, z) , Λ( w) = diag

  • λi(

w)

  • v∗(T, x) = DwΦ
  • x,

w(T, x)

  • ξ∗

α(T) = ∆Φ

  • xα,

w(T, xα)

  • + backward ODEs along the jumps for ξ∗

α

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

The Maximum Principle

Assume ( z, w) = (optimal control–optimal trajectory) be a solution to the optimal control problem

  • w with a finite number of discontinuities

cotangent vector (v∗(t, x), ξ∗(t)) be a backward solution of ∂tv∗ + ∂xv∗ · Λ( w) + v∗ DΛ( w) · ∂x( w) = = −v∗Dw h(x, w, z) − DwL(x, w, z) , Λ( w) = diag

  • λi(

w)

  • v∗(T, x) = DwΦ
  • x,

w(T, x)

  • ξ∗

α(T) = ∆Φ

  • xα,

w(T, xα)

  • + backward ODEs along the jumps for ξ∗

α

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

The Maximum Principle

Assume ( z, w) = (optimal control–optimal trajectory) be a solution to the optimal control problem

  • w with a finite number of discontinuities

cotangent vector (v∗(t, x), ξ∗(t)) be a backward solution of ∂tv∗ + ∂xv∗ · Λ( w) + v∗ DΛ( w) · ∂x( w) = = −v∗Dw h(x, w, z) − DwL(x, w, z) , Λ( w) = diag

  • λi(

w)

  • v∗(T, x) = DwΦ
  • x,

w(T, x)

  • ξ∗

α(T) = ∆Φ

  • xα,

w(T, xα)

  • + backward ODEs along the jumps for ξ∗

α

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

The Maximum Principle

Then at every point of continuity of w(t, x) and v∗(t, x) there holds v∗(t, x) · h(x, w, z) + L(x, w, z) = = max

z∈Z

  • v∗(t, x) · h(x,

w, z) + L(x, w, z)

  • Fabio Ancona

Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Future directions

Consider feedback controls z = z(u) which yield regular solutions of balance law ∂tu + ∂xf(u) = h(u, z) Study the optimization problem within a class of (more regular) approximate solutions, e.g.      ∂tuε + ∂xf(uε) = h(x, uε, z) + ε ∂2

xuε

uε(0, x) = u(x) , uε(t, 0) = g(α(t)) ε → 0+

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Future directions

Consider feedback controls z = z(u) which yield regular solutions of balance law ∂tu + ∂xf(u) = h(u, z) Study the optimization problem within a class of (more regular) approximate solutions, e.g.      ∂tuε + ∂xf(uε) = h(x, uε, z) + ε ∂2

xuε

uε(0, x) = u(x) , uε(t, 0) = g(α(t)) ε → 0+

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Future directions

Consider feedback controls z = z(u) which yield regular solutions of balance law ∂tu + ∂xf(u) = h(u, z) Study the optimization problem within a class of (more regular) approximate solutions, e.g.      ∂tuε + ∂xf(uε) = h(x, uε, z) + ε ∂2

xuε

uε(0, x) = u(x) , uε(t, 0) = g(α(t)) ε → 0+

Fabio Ancona Control Problems for Hyperbolic Equations

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Introduction Controllability & Stabilizability Optimal control problems Pontryagin Maximum Principle for Temple systems Temple systems Evolution of first order variations Pontryagin Maximum Principle

Thank you for your attention!!

Fabio Ancona Control Problems for Hyperbolic Equations