Review of the Theory
Iterative Solution of the Optimal Control Problem
Example
Reference
An Example of Optimal Control of PDE
CIMPA Summer School on Inverse Problems and its Applications
Patricio Guzmán Meléndez
Universidad Técnica Federico Santa Marı́a
Lunes 11 de Enero de 2010
Patricio Guzmán Meléndez
An Example of Optimal Control of PDE
Universidad Técnica Federico Santa Marı́a
Review of the Theory
Iterative Solution of the Optimal Control Problem
Example
Reference
Outline
1
Review of the Theory
Example
2
Iterative Solution of the Optimal Control Problem
3
Example
A Dirichlet problem with Distributed Control
4
Reference
Patricio Guzmán Meléndez
An Example of Optimal Control of PDE
Universidad Técnica Federico Santa Marı́a
Review of the Theory
Iterative Solution of the Optimal Control Problem
Example
Reference
Statement of the Problem
Let V and H be two Hilbert spaces, and assume that V be dense in H with
continuous injection. Also let U and Z be Hilbert spaces.
State Equation
Ay(u) = f + Bu in V 0
Cost Functional
J(y, u) = ||Cy − zd ||2Z + (N u, u)U
Where A ∈ L(V, V 0 ) such that its bilinear form associated a(·, ·) is a coercive form on
V, B ∈ L(U , V 0 ), C ∈ L(V, Z) and N ∈ L(U , U ) such that (N u, u)U ≥ ν||u||2U .
Optimal Control Problem
Find u ∈ Uad ⊆ U such that J(u) =
Patricio Guzmán Meléndez
An Example of Optimal Control of PDE
ı́nf J(v)
v∈Uad
Universidad Técnica Federico Santa Marı́a
Review of the Theory
Iterative Solution of the Optimal Control Problem
Example
Reference
Theorem (Optimality System)
A necessary and sufficient condition for the existence of an optimal control u ∈ Uad is
that the following equations and inequalities hold:

y(u) ∈ V





p(u) ∈ V




 u∈U
ad
:
Ay(u) = f + Bu
:
A∗ p(u) = C ∗ ΛZ (Cy(u) − zd )
:
∗
Λ−1
U B p(u) + N u, v − u
U
≥ 0 ∀v ∈ Uad
If N is symmetric and positive definite, then the control u is unique. Also we have
that J 0 (u) = 2B ∗ p(u) + 2ΛU N u.
Patricio Guzmán Meléndez
An Example of Optimal Control of PDE
Universidad Técnica Federico Santa Marı́a
Review of the Theory
Iterative Solution of the Optimal Control Problem
Example
Reference
Theorem (Optimality System, Weak Form)
A necessary and sufficient condition for the existence of an optimal control u ∈ Uad is
that the following equations and inequalities hold:

y(u) ∈ V





p(u) ∈ V




 u∈U
ad
:
a(y(u), φ) = (f, φ) + b(u, φ) ∀φ ∈ V
:
a(ψ, p(u)) = (Cy(u) − zd , Cψ)Z ∀ψ ∈ V
:
∗
Λ−1
U B p(u) + N u, v − u
U
≥ 0 ∀v ∈ Uad
If N is symmetric and positive definite, then the control u is unique. Also we have
that J 0 (u) = 2B ∗ p(u) + 2ΛU N u.
Patricio Guzmán Meléndez
An Example of Optimal Control of PDE
Universidad Técnica Federico Santa Marı́a
Review of the Theory
Iterative Solution of the Optimal Control Problem
Example
Reference
Example
A Dirichlet problem with Distributed Control
For instance consider V = H01 (Ω) and H = L2 (Ω) = U = Z. Let N = νI with v > 0,
A = −4, B = I and C = I. In this example the cost funcional is given by:
J(u) = ||y(u) − zd ||2 + ν||u||2
The optimality system is as follows:

y(u) ∈ H01 (Ω)






p(u) ∈ H01 (Ω)





 u∈U
ad
:
−4y(u) = f + u in Ω
:
−4p(u) = y(u) − zd in Ω
Z
[p(u) + N u] (v − u) dΩ ≥ 0 ∀v ∈ Uad
:
Ω
Patricio Guzmán Meléndez
An Example of Optimal Control of PDE
Universidad Técnica Federico Santa Marı́a
Review of the Theory
Iterative Solution of the Optimal Control Problem
Example
Reference
Sketch of the Iterative Method
We will use the following steepest descent iterative method: uk+1 = uk − τ k J 0 (uk )
where J 0 represents the descent direction and τ k the acceleration parameter.
A method for the search of an optimal control can therefore be devised in terms of the
following iterative algorithm:
1
Find the expression of the adjoint equation and the derivative J 0 .
2
Provide an initial guess u0 of the control and then solve the state equation in y.
3
Being known the state variable and given a target function zd solve the adjoint
equation.
4
Evaluate J y J 0 . If the chosen stopping test is fulfilled then exit.
5
In the other case evaluate τ k and compute the new control.
6
Repeat (1) to (5) til the chosen stopping test is fulfilled.
Given a tolerance (TOL), the following stopping test can be use:
||z k − zd ||Z ≤ T OL , ||J 0 (uk )||U 0 ≤ T OL , ||J 0 (uk )||U 0 ≤ T OL||J 0 (u0 )||U 0
Patricio Guzmán Meléndez
An Example of Optimal Control of PDE
Universidad Técnica Federico Santa Marı́a
Review of the Theory
Iterative Solution of the Optimal Control Problem
Example
Reference
A Dirichlet problem with Distributed Control
Example
Let Ω = (0, 1). Consider V = H01 (Ω) and H = L2 (Ω) = U = Z. Finally let N = νI
with v > 0 and A = −4.
1
2
3
State Equation
−4y = f + u
y
= 0
Adjoint Equation
−4p = y − zd
p
= 0
, in Ω
, in ∂Ω
, in Ω
, in ∂Ω
Cost Functional
Z 1
Z
(y − zd )2 dx + ν
J(y, u) =
0
1
u2 dx ;
0
For this example we will use f = 1 and zd =
Patricio Guzmán Meléndez
An Example of Optimal Control of PDE
1 0
J (p, u) = p + νu
2
x
1−x
, 0 ≤ x ≤ 1/2
, 1/2 ≤ x ≤ 1
Universidad Técnica Federico Santa Marı́a
Review of the Theory
Iterative Solution of the Optimal Control Problem
Example
Reference
A Dirichlet problem with Distributed Control
Description of the Algorithm
The iterative problem is described by the following steps:
−4y k = 1 + uk , in Ω
(A)
yk
= 0
, in ∂Ω
(B)
(C)
−4pk
pk
=
=
y k − zd
0
, in Ω
, in ∂Ω




τk
=
ν||uk ||2 + ||y k − zd ||2
||2νuk + 2pk ||2



uk+1
=
uk − τ k 2(νuk + pk )
A finite element scheme is implemented in the steps (A) and (B) for each k, using:
n
o
Vh = vh ∈ C 0 (Ω) / vh |[xi ,xi+1 ] ∈ P 1 , i ∈ {0, . . . , N } , vh (0) = vh (1) = 0
with h = 10−2 and 0 = x0 ≤ x1 ≤ . . . ≤ xN = 1 a grid of Ω.
Patricio Guzmán Meléndez
An Example of Optimal Control of PDE
Universidad Técnica Federico Santa Marı́a
Review of the Theory
Iterative Solution of the Optimal Control Problem
Example
Reference
A Dirichlet problem with Distributed Control
Description of the Algorithm
In the step (C) the functional J(y, u) is minimized using the conjugate gradient
method with an acceleration parameter tk .
It was used u0 = 0, τ 0 = τ = 10 and the following stopping test:
||J 0 (uk )|| < T OL||J 0 (u0 )|| , T OL = 10−3
Observations:
X
1 y k (x) =
αkj φj (x) −→ Approximation of the Optimal State in the step k.
h
j
2
pkh (x)
=
X
3
ukh (x) =
X
βjk φj (x) −→ Approximation of the Adjoint State in the step k.
j
γjk φj (x) −→ Approximación of the Optimal Control in the step k.
j
Patricio Guzmán Meléndez
An Example of Optimal Control of PDE
Universidad Técnica Federico Santa Marı́a
Review of the Theory
Iterative Solution of the Optimal Control Problem
Example
Reference
A Dirichlet problem with Distributed Control
Results
ν
10−2
Patricio Guzmán Meléndez
An Example of Optimal Control of PDE
ite
15
J
0,020255
Universidad Técnica Federico Santa Marı́a
Review of the Theory
Iterative Solution of the Optimal Control Problem
Example
Reference
A Dirichlet problem with Distributed Control
Results
ν
10−3
Patricio Guzmán Meléndez
An Example of Optimal Control of PDE
ite
133
J
0,004669
Universidad Técnica Federico Santa Marı́a
Review of the Theory
Iterative Solution of the Optimal Control Problem
Example
Reference
A Dirichlet problem with Distributed Control
Results
ν
10−4
Patricio Guzmán Meléndez
An Example of Optimal Control of PDE
ite
668
J
0,001080
Universidad Técnica Federico Santa Marı́a
Review of the Theory
Iterative Solution of the Optimal Control Problem
Example
Reference
Reference
Numerical Models for Differential Problems.
Alfio Quarteroni
Springer-Verlag Italia, Milan 2009.
Patricio Guzmán Meléndez
An Example of Optimal Control of PDE
Universidad Técnica Federico Santa Marı́a