Regularity of optimal control problems with super linear growth P - PowerPoint PPT Presentation

Regularity of optimal control problems with super linear growth P . Cardaliaguet Univ. Brest Toulon, May 18-21, 2008 Sixièmes Journées Franco-Chiliennes d’Optimisation P . Cardaliaguet (Univ. Brest) Regularity of optimal control problems 1 / 25

Statement of the main result Outline Statement of the main result 1 Sketch of the proof of the main result 2 Sketch of proof for the reverse Hölder inequality Lemma 3 P . Cardaliaguet (Univ. Brest) Regularity of optimal control problems 2 / 25

Statement of the main result Aim We investigate the regularity of the value function �� T � L ( x ( s ) , s , x ′ ( s )) ds + g ( x ( T )) u ( x , t ) = inf t where the infimum is taken over the x ( · ) ∈ W 1 , 1 ([ t , T ] , R N ) such that x ( t ) = x under the key assumption that L has a superlinear growth : ∃ p > 1, δ ∈ ( 0 , 1 ) , M > 0 with δ | ξ | p − M ≤ L ( x , s , ξ ) ≤ 1 δ | ξ | p + M P . Cardaliaguet (Univ. Brest) Regularity of optimal control problems 3 / 25

Statement of the main result Aim We investigate the regularity of the value function �� T � L ( x ( s ) , s , x ′ ( s )) ds + g ( x ( T )) u ( x , t ) = inf t where the infimum is taken over the x ( · ) ∈ W 1 , 1 ([ t , T ] , R N ) such that x ( t ) = x under the key assumption that L has a superlinear growth : ∃ p > 1, δ ∈ ( 0 , 1 ) , M > 0 with δ | ξ | p − M ≤ L ( x , s , ξ ) ≤ 1 δ | ξ | p + M P . Cardaliaguet (Univ. Brest) Regularity of optimal control problems 3 / 25

Statement of the main result Motivation Stochastic homogenization of Hamilton-Jacobi equations : t ( x , t ) + H ( x ǫ , t ǫ, Du ǫ ( x , t )) = 0 in R N × ( 0 , T ) u ǫ ( HJ ) (Souganidis (1999), Lions-Souganidis (2005), Schwab (2008)). Recall that our value function u is solution of − u t ( x , t ) + H ( x , t , Du ( x , t )) = 0 in R N × ( 0 , T ) ( HJ ) P . Cardaliaguet (Univ. Brest) Regularity of optimal control problems 4 / 25

Statement of the main result Known results Two types of results : Propagation of regularity : Lions (1985), Barles (1990), Rampazzo, Sartori (2000) If L and g are “sufficiently smooth", then u is Lipschitz continuous with a Lipschitz constant depending on the regularity of L and g . Weak point : not suitable for homogenization. P . Cardaliaguet (Univ. Brest) Regularity of optimal control problems 5 / 25

Statement of the main result Known results Two types of results : Propagation of regularity : Lions (1985), Barles (1990), Rampazzo, Sartori (2000) If L and g are “sufficiently smooth", then u is Lipschitz continuous with a Lipschitz constant depending on the regularity of L and g . Weak point : not suitable for homogenization. P . Cardaliaguet (Univ. Brest) Regularity of optimal control problems 5 / 25

Statement of the main result Interior regularity/Lipschitz regularity of optimal solutions : Clarke, Vinter (1985), Ambrosio, Ascenzi, Buttazzo (1989), Dal Maso, Frankowska (2003), Quincampoix, Zlateva (2006), Frankowska, Marchini (2005), Davini (2007). Key assumption : L = L ( x , x ′ ) has a superlinear growth. Consequences : - No Lavrentiev phenomenon. - Lipschitz continuity of optimal solutions implies Lipschitz continuity of the value function. Weak point : Cannot handle the time-dependant case. P . Cardaliaguet (Univ. Brest) Regularity of optimal control problems 6 / 25

Statement of the main result Interior regularity/Lipschitz regularity of optimal solutions : Clarke, Vinter (1985), Ambrosio, Ascenzi, Buttazzo (1989), Dal Maso, Frankowska (2003), Quincampoix, Zlateva (2006), Frankowska, Marchini (2005), Davini (2007). Key assumption : L = L ( x , x ′ ) has a superlinear growth. Consequences : - No Lavrentiev phenomenon. - Lipschitz continuity of optimal solutions implies Lipschitz continuity of the value function. Weak point : Cannot handle the time-dependant case. P . Cardaliaguet (Univ. Brest) Regularity of optimal control problems 6 / 25

Statement of the main result Difficulty for the time-dependent case If L = L ( x , t , ξ ) , one cannot expect u to be locally Lipschitz continuous because u Lipschitz continuous ⇒ optimal solutions are Lipschitz continuous. However Proposition (Lavrentiev (1926)) There are L and initial data ( x 0 , t 0 ) for which optimal solutions are not Lipschitz continuous. Manià’s Example (1934) : L ( x , t , ξ ) = ( t 3 − x ) 2 | ξ | 6 . P . Cardaliaguet (Univ. Brest) Regularity of optimal control problems 7 / 25

Statement of the main result Assumptions for the main result We suppose that L = L ( x , t , ξ ) is continuous, convex w.r.t. ξ , g is bounded by some M > 0, (Superlinear growth) there are p > 1 and δ > 0 such that δ | ξ | p − M ≤ L ( x , t , ξ ) ≤ 1 δ | ξ | p + M ∀ ( x , t ) ∈ R N × ( 0 , T ) P . Cardaliaguet (Univ. Brest) Regularity of optimal control problems 8 / 25

Statement of the main result Main result Theorem There are θ > p and, for any τ > 0 , K τ > 0 such that � | x 0 − x 1 | ( θ − p ) / ( θ − 1 ) + | t 0 − t 1 | ( θ − p ) /θ � | u ( x 0 , t 0 ) − u ( x 1 , t 1 ) | ≤ K τ for any x 0 , x 1 ∈ R N , for any t 0 , t 1 ∈ [ 0 , T − τ ] , where θ = θ ( M , δ, p , T ) and K τ = K ( τ, M , δ, q , T ) . P . Cardaliaguet (Univ. Brest) Regularity of optimal control problems 9 / 25

Sketch of the proof of the main result Outline Statement of the main result 1 Sketch of the proof of the main result 2 Sketch of proof for the reverse Hölder inequality Lemma 3 P . Cardaliaguet (Univ. Brest) Regularity of optimal control problems 10 / 25

Sketch of the proof of the main result Main steps Three main steps : Step 1 : a strange inequality satisfied by optimal solutions, Step 2 : use of the reverse Hölder inequality to get regularity of the optimal solutions, Step 3 : this regularity gives the Hölder continuity of u . P . Cardaliaguet (Univ. Brest) Regularity of optimal control problems 11 / 25

Sketch of the proof of the main result Step 1 : a strange inequality Lemma There is A = A ( M , δ, T ) ≥ 1 such that, for any optimal solution ¯ x starting from x 0 at time t 0 , � p � t 0 + h � t 0 + h � 1 1 ( α ( s )) p ds ≤ A α ( s ) ds ∀ h ∈ [ 0 , T − t 0 ] h h t 0 t 0 where α ( s ) = | ¯ x ′ ( s ) | + 1 Idea of proof : test the optimality of ¯ x against ¯ x ( t 0 + h ) − x 0 � ( t − t 0 ) + x 0 if t ∈ [ t 0 , t 0 + h ] ˜ x ( t ) = h ¯ x ( t ) otherwise P . Cardaliaguet (Univ. Brest) Regularity of optimal control problems 12 / 25

Sketch of the proof of the main result Remarks on the inequality If α ∈ L p ( 0 , 1 ) for some p > 1, we have from the Hölder inequality : � p � h �� h � h � � 1 ≤ 1 h ( 1 − 1 / q ) p = 1 | α | p | α | p | α | ∀ h ∈ [ 0 , 1 ] h p h h 0 0 0 So the inequality satisfied by α ( s ) = | ¯ x ′ ( s ) | + 1 : � p � t 0 + h � t 0 + h � 1 1 ( α ( s )) p ds ≤ A α ( s ) ds ∀ h ∈ [ 0 , T − t 0 ] h h t 0 t 0 is a reverse Hölder inequality. P . Cardaliaguet (Univ. Brest) Regularity of optimal control problems 13 / 25

Sketch of the proof of the main result Remarks on the inequality If α ∈ L p ( 0 , 1 ) for some p > 1, we have from the Hölder inequality : � p � h �� h � h � � 1 ≤ 1 h ( 1 − 1 / q ) p = 1 | α | p | α | p | α | ∀ h ∈ [ 0 , 1 ] h p h h 0 0 0 So the inequality satisfied by α ( s ) = | ¯ x ′ ( s ) | + 1 : � p � t 0 + h � t 0 + h � 1 1 ( α ( s )) p ds ≤ A α ( s ) ds ∀ h ∈ [ 0 , T − t 0 ] h h t 0 t 0 is a reverse Hölder inequality. P . Cardaliaguet (Univ. Brest) Regularity of optimal control problems 13 / 25

Sketch of the proof of the main result Step 2 : use of the reverse Hölder inequality Fix A > 1 and p > 1. Lemma (reverse Hölder inequality) There are θ = θ ( A , p ) > p and C = C ( A , p ) > 0 such that, for any α ∈ L p ( 0 , 1 ) satisfying � p � h � h � 1 1 | α ( s ) | p ds ≤ A | α ( s ) | ds ∀ h ∈ [ 0 , 1 ] , h h 0 0 one has � h | α ( s ) | ds ≤ C � α � L p h 1 − 1 /θ ∀ h ∈ [ 0 , 1 ] . 0 Remark : This result is a weak form of Gehring’s reverse Hölder inequality. P . Cardaliaguet (Univ. Brest) Regularity of optimal control problems 14 / 25

Sketch of the proof of the main result Step 2 (end) : Regularity of optimal solutions Corollary There are θ > p and C such that, for any x 0 ∈ R N and any t 0 < T, if ¯ x is optimal for the initial position x 0 at time t 0 , then � t 0 + h x ′ ( s ) | ds ≤ C ( T − t 0 ) 1 /θ − 1 / p h 1 − 1 /θ | ¯ ∀ h ∈ [ t 0 , T ] t 0 Remark : this proves that optimal solutions are 1 /θ − Hölder continuous. P . Cardaliaguet (Univ. Brest) Regularity of optimal control problems 15 / 25

Sketch of the proof of the main result Step 3 : Use of the regularity of optimal solutions Corollary (Space regularity) Let x 0 , x 1 ∈ R N , t 0 < T. Then u ( x 1 , t 0 ) − u ( x 0 , t 0 ) ≤ K 1 ( T − t 0 ) − ( p − 1 )( θ − p ) / ( p ( θ − 1 )) | x 1 − x 0 | ( θ − p ) / ( θ − 1 ) where K 1 = K 1 ( M , p , T , δ ) . Proof : Let ¯ x be optimal for x 0 and consider ¯ x ( t 0 + h ) − x 1 � ( t − t 0 ) + x 0 if t ∈ [ t 0 , t 0 + h ] ˜ x h ( t ) = h ¯ x ( t ) otherwise which is admissible for x 1 . Use the regularity of ¯ x and optimize w.r.t. h . P . Cardaliaguet (Univ. Brest) Regularity of optimal control problems 16 / 25