Problem formulationPontryagin stochastic maximum principle

Plan of the proof

Pontryagin’s maximum principle for optimalcontrol of SPDEs

AbdulRahman Al-Hussein

Department of Mathematics, College of Science, Qassim University,P. O. Box 6644, Buraydah 51452, Saudi Arabia

E-mail: alhusseinqu@hotmail.com

Workshop on Stochastic Control Problems forFBSDEs and Applications

Marrakesh, December 13-18, 2010

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

StatementAssumptionsThe adjoint equation

Outline

1 Problem formulationStatementAssumptionsThe adjoint equation

2 Pontryagin stochastic maximum principle

3 Plan of the proofEstimatesVariational inequality

ProofTwo more lemmasProof of main result

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

StatementAssumptionsThe adjoint equation

Consider the SPDE on a sep. Hilbert space K :dX u(·)(t) = (A(t) X u(·)(t) + F (X u(·)(t),u(t)) )dt + G(X u(·)(t))dM(t),X u(·)(0) = x .

(1)

M is a continuous martingale, << M >>t =∫ t

0 Q(s) ds,for a predictable process Q(·) s.t. Q(t) ∈ L1(K ) is symmetric,positive definite, Q(t) ≤ Q, where Q ∈ L1(K ) (positive definite).

F : K ×O → K (O is a sep. Hilbert space).

G : K → L2(K0,K ), K0 = Q−1/2(K ).

A(t , ω) is a predictable unbounded linear operator on K .

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

StatementAssumptionsThe adjoint equation

. We shall derive a stochastic maximum principle for this controlproblem by using the adjoint equation (BSPDE).

u(·) : [0,T ]× Ω→ O is admissible if u(·) ∈ L2F (0,T ;O) and

u(t) ∈ U a.e., a.s. ( U is a nonempty convex subset of O ) .

The set of admissible controls Uad .

L2F (0,T ; E) := ψ : [0,T ]× Ω→ E ,predictable,

E [∫ T

0 |ψ(t)|2Edt ] <∞ .

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

StatementAssumptionsThe adjoint equation

. We shall derive a stochastic maximum principle for this controlproblem by using the adjoint equation (BSPDE).

u(·) : [0,T ]× Ω→ O is admissible if u(·) ∈ L2F (0,T ;O) and

u(t) ∈ U a.e., a.s. ( U is a nonempty convex subset of O ) .

The set of admissible controls Uad .

L2F (0,T ; E) := ψ : [0,T ]× Ω→ E ,predictable,

E [∫ T

0 |ψ(t)|2Edt ] <∞ .

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

StatementAssumptionsThe adjoint equation

Definethe cost functional:

J(u(·)) := E [

∫ T

0g(X u(·)(t), u(t) ) dt + φ(X u(·)(T )) ],

J∗ := infJ(u(·)) : u(·) ∈ Uad.

The control problem for this SPDE (1) is to find a control u∗(·) and thecorresponding solution X u∗(·) of (1) s.t.

J∗ = J(u∗(·)).

u∗(·) is an optimal control.

X u∗(·) (or briefly X ∗) is an optimal solution.

The pair (X ∗ ,u∗(·)) is an optimal pair.

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

StatementAssumptionsThe adjoint equation

Definethe cost functional:

J(u(·)) := E [

∫ T

0g(X u(·)(t), u(t) ) dt + φ(X u(·)(T )) ],

J∗ := infJ(u(·)) : u(·) ∈ Uad.

The control problem for this SPDE (1) is to find a control u∗(·) and thecorresponding solution X u∗(·) of (1) s.t.

J∗ = J(u∗(·)).

u∗(·) is an optimal control.

X u∗(·) (or briefly X ∗) is an optimal solution.

The pair (X ∗ ,u∗(·)) is an optimal pair.

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

StatementAssumptionsThe adjoint equation

(H1) F ,G,g, φ are C1 w.r.t. x , F is C1 w.r.t. u,the derivatives Fx , Fu, Gx ,gx are uniformly bounded.

Also |φx |K ≤ C1 (1 + |x |K ), some constant C1 > 0.

(H2) gx satisfies Lipschitz condition with respect to u uniformly inx .

(H3) A(t , ω) is a predictable linear operator on K , belongs toL(V ; V ′) ( (V ,K ,V ′) is a Gelfand triple ),

1 2⟨A(t , ω) y , y

⟩+ α |y |2

V≤ λ |y |2 a.e. t ∈ [0,T ] , a.s. ∀ y ∈ V ,

for some α, λ > 0.

2 ∃ C2 ≥ 0 s.t. |A(t , ω) y |V ′ ≤ C2 |y |V ∀ (t , ω), ∀ y ∈ V .

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

StatementAssumptionsThe adjoint equation

Define the Hamiltonian:

H : [0,T ]× Ω× K ×O × K × L2(K )→ R,

H(t , x ,u, y , z) := −g(x ,u) +⟨F (x ,u) , y

⟩+⟨G(x)Q1/2(t) , z

⟩2 .

The (adjoint) BSPDE:

−dY u(·)(t) = [ A∗(t) Y u(·)(t) +∇xH(X u(·)(t),u(t),Y u(·)(t),Z u(·)(t)Q1/2(t)) ]dt−Z u(·)(t)dM(t)− dNu(·)(t), 0 ≤ t ≤ T ,

Y u(·)(T ) = −∇φ(X u(·)(T )),

A∗(t) is the adjoint operator of A(t).

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

StatementAssumptionsThe adjoint equation

Define the Hamiltonian:

H : [0,T ]× Ω× K ×O × K × L2(K )→ R,

H(t , x ,u, y , z) := −g(x ,u) +⟨F (x ,u) , y

⟩+⟨G(x)Q1/2(t) , z

⟩2 .

The (adjoint) BSPDE:

−dY u(·)(t) = [ A∗(t) Y u(·)(t) +∇xH(X u(·)(t),u(t),Y u(·)(t),Z u(·)(t)Q1/2(t)) ]dt−Z u(·)(t)dM(t)− dNu(·)(t), 0 ≤ t ≤ T ,

Y u(·)(T ) = −∇φ(X u(·)(T )),

A∗(t) is the adjoint operator of A(t).

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

StatementAssumptionsThe adjoint equation

M2,c[0,T ](K ) the space of continuous square integrable

martingales in K .

Two elements M and N ofM2,c[0,T ](K ) are very strongly orthogonal

(VSO) ifE [M(τ)⊗ N(τ)] = E [M(0)⊗ N(0)],

for all [0,T ] - valued stopping times τ.

In fact: M and N are VSO⇔ << M,N >> = 0.

Λ2(K ;P,M) the space of integrands w.r.t. M s.t.

Φ(t , ω)Q1/2(t , ω) ∈ L2(K ), for every h ∈ K

the K - valued process Φ Q1/2(h) is predictable,

E [∫ T

0 ||(Φ Q1/2)(t)||22 dt ] <∞.

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

StatementAssumptionsThe adjoint equation

M2,c[0,T ](K ) the space of continuous square integrable

martingales in K .

Two elements M and N ofM2,c[0,T ](K ) are very strongly orthogonal

(VSO) ifE [M(τ)⊗ N(τ)] = E [M(0)⊗ N(0)],

for all [0,T ] - valued stopping times τ.

In fact: M and N are VSO⇔ << M,N >> = 0.

Λ2(K ;P,M) the space of integrands w.r.t. M s.t.

Φ(t , ω)Q1/2(t , ω) ∈ L2(K ), for every h ∈ K

the K - valued process Φ Q1/2(h) is predictable,

E [∫ T

0 ||(Φ Q1/2)(t)||22 dt ] <∞.

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

StatementAssumptionsThe adjoint equation

M2,c[0,T ](K ) the space of continuous square integrable

martingales in K .

Two elements M and N ofM2,c[0,T ](K ) are very strongly orthogonal

(VSO) ifE [M(τ)⊗ N(τ)] = E [M(0)⊗ N(0)],

for all [0,T ] - valued stopping times τ.

In fact: M and N are VSO⇔ << M,N >> = 0.

Λ2(K ;P,M) the space of integrands w.r.t. M s.t.

Φ(t , ω)Q1/2(t , ω) ∈ L2(K ), for every h ∈ K

the K - valued process Φ Q1/2(h) is predictable,

E [∫ T

0 ||(Φ Q1/2)(t)||22 dt ] <∞.

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

StatementAssumptionsThe adjoint equation

DefinitionA solution of a BSPDE: −dY (t) =

(A(t)Y (t) + f (t ,Y (t),Z (t)Q1/2(t))

)dt

−Z (t)dM(t)− dN(t), 0 ≤ t ≤ T ,Y (T ) = ξ,

is (Y ,Z ,N) ∈ L2F (0,T ; V )× Λ2(K ;P,M)×M2,c

[0,T ](K ) s.t. ∀ t ∈ [0,T ] :

Y (t) = ξ +

∫ T

t

(A(s) Y (s) + f (s,Y (s),Z (s)Q1/2(s))

)ds

−∫ T

tZ (s)dM(s)−

∫ T

tdN(s),

N(0) = 0, N is VSO to M.

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

StatementAssumptionsThe adjoint equation

. The following theorem gives the unique solution to the adjointequation.

Theorem 1 (Existence & uniqueness of the solution of the adjointBSPDE)

Assume (H1)–(H3). There exists a unique solution (Y u(·),Z u(·),Nu(·) )

to the adjoint BSPDE in L2F (0,T ; V )× Λ2(K ;P,M)×M2,c

[0,T ](K ).

The proof of this theorem can be found in:[ Al-Hussein, A., Backward stochastic partial differential equations driven byinfinite dimensional martingales and applications, Stochastics, Vol. 81, No. 6,2009, 601-626 ].

Denote the solution of the (adjoint) BSPDE corresponding to u∗(·) by(Y ∗,Z ∗,N∗).

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

StatementAssumptionsThe adjoint equation

. The following theorem gives the unique solution to the adjointequation.

Theorem 1 (Existence & uniqueness of the solution of the adjointBSPDE)

Assume (H1)–(H3). There exists a unique solution (Y u(·),Z u(·),Nu(·) )

to the adjoint BSPDE in L2F (0,T ; V )× Λ2(K ;P,M)×M2,c

[0,T ](K ).

The proof of this theorem can be found in:[ Al-Hussein, A., Backward stochastic partial differential equations driven byinfinite dimensional martingales and applications, Stochastics, Vol. 81, No. 6,2009, 601-626 ].

Denote the solution of the (adjoint) BSPDE corresponding to u∗(·) by(Y ∗,Z ∗,N∗).

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

StatementAssumptionsThe adjoint equation

. The following theorem gives the unique solution to the adjointequation.

Theorem 1 (Existence & uniqueness of the solution of the adjointBSPDE)

Assume (H1)–(H3). There exists a unique solution (Y u(·),Z u(·),Nu(·) )

to the adjoint BSPDE in L2F (0,T ; V )× Λ2(K ;P,M)×M2,c

[0,T ](K ).

The proof of this theorem can be found in:[ Al-Hussein, A., Backward stochastic partial differential equations driven byinfinite dimensional martingales and applications, Stochastics, Vol. 81, No. 6,2009, 601-626 ].

Denote the solution of the (adjoint) BSPDE corresponding to u∗(·) by(Y ∗,Z ∗,N∗).

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

Outline

1 Problem formulationStatementAssumptionsThe adjoint equation

2 Pontryagin stochastic maximum principle

3 Plan of the proofEstimatesVariational inequality

ProofTwo more lemmasProof of main result

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

Theorem 2 (Pontryagin stochastic maximum principle)

Suppose (H1)–(H3). Assume (X ∗,u∗(·)) is an optimal pair for ourcontrol problem.

Then there exists a unique solution (Y ∗,Z ∗,N∗) to the correspondingBSPDE s.t. the following inequality holds:

⟨∇uH(t ,X ∗(t),u∗(t),Y ∗(t),Z ∗(t)Q1/2(t)),u − u∗(t)

⟩O ≤ 0,

∀ u ∈ U, a.e. t ∈ [0,T ], a.s.

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

EstimatesVariational inequalityTwo more lemmasProof of main result

Outline

1 Problem formulationStatementAssumptionsThe adjoint equation

2 Pontryagin stochastic maximum principle

3 Plan of the proofEstimatesVariational inequality

ProofTwo more lemmasProof of main result

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

EstimatesVariational inequalityTwo more lemmasProof of main result

. Let u∗(·) be an optimal control and X ∗ be the corresponding

solution of SPDE (1). Let u(·) be s.t. u∗(·) + u(·) ∈ Uad .

For 0 ≤ ε ≤ 1 consider the variational control:

uε(t) = u∗(t) + εu(t), t ∈ [0,T ].

The convexity of U ⇒ uε(·) ∈ Uad .

. Get the corresponding Xε of (1).

. Let p be the solution of the linear equation: dp(t) = (A(t)p(t) + Fx (X ∗(t),u∗(t))p(t))dt+ Fu(X ∗(t),u∗(t))u(t)dt + Gx (X ∗(t))p(t)dM(t),

p(0) = 0.

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

EstimatesVariational inequalityTwo more lemmasProof of main result

. Let u∗(·) be an optimal control and X ∗ be the corresponding

solution of SPDE (1). Let u(·) be s.t. u∗(·) + u(·) ∈ Uad .

For 0 ≤ ε ≤ 1 consider the variational control:

uε(t) = u∗(t) + εu(t), t ∈ [0,T ].

The convexity of U ⇒ uε(·) ∈ Uad .

. Get the corresponding Xε of (1).

. Let p be the solution of the linear equation: dp(t) = (A(t)p(t) + Fx (X ∗(t),u∗(t))p(t))dt+ Fu(X ∗(t),u∗(t))u(t)dt + Gx (X ∗(t))p(t)dM(t),

p(0) = 0.

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

EstimatesVariational inequalityTwo more lemmasProof of main result

. We obtain:

Lemma 1Assume (H1), (H3). Let

ηε(t) =Xε(t)− X ∗(t)

ε− p(t), t ∈ [0,T ].

Then:

1 supt∈[0,T ]

E [ |p(t)|2 ] <∞,

2 supt∈[0,T ]

E [ |Xε(t)− X ∗(t)|2 ] = O(ε2),

3 limε→0+

supt∈[0,T ]

E [ |ηε(t)|2 ] = 0.

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

EstimatesVariational inequalityTwo more lemmasProof of main result

Lemma 2 (Variational inequality)

Suppose (H1)–(H3). Then ∀ ε > 0,

J(uε(·))− J(u∗(·)) = ε E [φx (X ∗(T )) p(T ) ]

+ ε E [

∫ T

0gx (X ∗(s),u∗(s)) p(s) ds ]

+ E [

∫ T

0

(g(X ∗(s),uε(s))− g(X ∗(s),u∗(s))

)ds ]

+ o(ε).

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

EstimatesVariational inequalityTwo more lemmasProof of main result

Sketch proof of Lemma 2

Note

J(uε(·))− J(u∗(·)) = I1(ε) + I2(ε),

where

I1(ε) = E [φ(Xε(T ))− φ(X ∗(T )) ],

I2(ε) = E [

∫ T

0

(g(Xε(s),uε(s))− g(X ∗(s),u∗(s))

)ds ].

Then apply Lemma 1 (2), (3).

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

EstimatesVariational inequalityTwo more lemmasProof of main result

Lemma 3If (H1)–(H3) hold, then

− ε E⟨

Y ∗(T ),p(T )⟩

+ ε E [

∫ T

0gx (X ∗(s),u∗(s)) p(s) ds ]

+ E [

∫ T

0(−δεH(s) +

⟨Y ∗(s) , δεF (s)

⟩) ds ] ≥ o(ε),

where

δεF (s) = F (X ∗(s),uε(s))− F (X ∗(s),u∗(s)),

δεH(s) = H(X ∗(s),uε(s),Y ∗(s),Z ∗(s)Q1/2(s))

−H(X ∗(s),u∗(s),Y ∗(s),Z ∗(s)Q1/2(s)).

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

EstimatesVariational inequalityTwo more lemmasProof of main result

Proof of Lemma 3

. Since u∗(·) is an optimal control, J(uε(·))− J(u∗(·)) ≥ 0.

. Next apply Lemma 2 and the definition of the Hamiltonian H.

Next need the following relation:

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

EstimatesVariational inequalityTwo more lemmasProof of main result

Proof of Lemma 3

. Since u∗(·) is an optimal control, J(uε(·))− J(u∗(·)) ≥ 0.

. Next apply Lemma 2 and the definition of the Hamiltonian H.

Next need the following relation:

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

EstimatesVariational inequalityTwo more lemmasProof of main result

Lemma 4

E⟨

Y ∗(T ),p(T )⟩

= E [

∫ T

0gx (X ∗(s),u∗(s)) p(s) ds ]

+ E [

∫ T

0

⟨Y ∗(s),Fu(X ∗(s),u∗(s))u(s)

⟩ds ].

Proof of Lemma 4

. Use Ito’s formula together with⟨∇xH(X ∗(t),u∗(t),Y ∗(t),Z ∗(t)Q1/2(t)),p(t)

⟩= − gx (X ∗(t),u∗(t)) p(t) +

⟨Fx (X ∗(t),u∗(t)) p(t) ,Y ∗(t)

⟩+⟨

G(X ∗(t))Q1/2(t) ,Z ∗(t)Q1/2(t)⟩

2 a.s. ∀ t ∈ [0,T ].

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

EstimatesVariational inequalityTwo more lemmasProof of main result

Lemma 4

E⟨

Y ∗(T ),p(T )⟩

= E [

∫ T

0gx (X ∗(s),u∗(s)) p(s) ds ]

+ E [

∫ T

0

⟨Y ∗(s),Fu(X ∗(s),u∗(s))u(s)

⟩ds ].

Proof of Lemma 4

. Use Ito’s formula together with⟨∇xH(X ∗(t),u∗(t),Y ∗(t),Z ∗(t)Q1/2(t)),p(t)

⟩= − gx (X ∗(t),u∗(t)) p(t) +

⟨Fx (X ∗(t),u∗(t)) p(t) ,Y ∗(t)

⟩+⟨

G(X ∗(t))Q1/2(t) ,Z ∗(t)Q1/2(t)⟩

2 a.s. ∀ t ∈ [0,T ].

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

EstimatesVariational inequalityTwo more lemmasProof of main result

Proof of main resultConsider the adjoint equation. From Lemma 3, Lemma 4 get

E [

∫ T

0

⟨Y ∗(s), δεF (s)− εFu(X ∗(s),u∗(s))u(s)

⟩ds ]

− E [

∫ T

0δεH(s)ds ] ≥ o(ε).

The continuity, boundedness of Fu in (H1) and the DCT give

1εE [

∫ T

0

⟨Y ∗(s), δεF (s)− ε Fu(X ∗(s),u∗(s)) u(s)

⟩ds ]

= E[ ∫ T

0

⟨Y ∗(s),

∫ 1

0

(Fu(X ∗(s),u∗(s) + θ(uε(s)− u∗(s)))

−Fu(X ∗(s),u∗(s)))

u(s) dθ⟩

ds]→ 0,

as ε→ 0+.

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

EstimatesVariational inequalityTwo more lemmasProof of main result

Proof of main resultConsider the adjoint equation. From Lemma 3, Lemma 4 get

E [

∫ T

0

⟨Y ∗(s), δεF (s)− εFu(X ∗(s),u∗(s))u(s)

⟩ds ]

− E [

∫ T

0δεH(s)ds ] ≥ o(ε).

The continuity, boundedness of Fu in (H1) and the DCT give

1εE [

∫ T

0

⟨Y ∗(s), δεF (s)− ε Fu(X ∗(s),u∗(s)) u(s)

⟩ds ]

= E[ ∫ T

0

⟨Y ∗(s),

∫ 1

0

(Fu(X ∗(s),u∗(s) + θ(uε(s)− u∗(s)))

−Fu(X ∗(s),u∗(s)))

u(s) dθ⟩

ds]→ 0,

as ε→ 0+.

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

EstimatesVariational inequalityTwo more lemmasProof of main result

continued proof

In particular

E [

∫ T

0

⟨Y ∗(s), δεF (s)− εFu(X ∗(s),u∗(s)) u(s)

⟩ds ] = o(ε).

∴ − E [

∫ T

0δεH(s) ds ] ≥ o(ε).

∴ by dividing this inequality by ε and letting ε→ 0+ :

E [

∫ T

0∇uH(t ,X ∗(t),u∗(t),Y ∗(t),Z ∗(t)Q1/2(t)),u(t)

⟩dt ] ≤ 0.

The theorem then follows.

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs

Problem formulationPontryagin stochastic maximum principle

Plan of the proof

EstimatesVariational inequalityTwo more lemmasProof of main result

Thank you

AbdulRahman Al-Hussein Pontryagin’s maximum principle for optimal control of SPDEs