On the Stackelberg strategies in control theory Enrique FERNÁNDEZ-CARA Dpto. E.D.A.N. - Univ. of Sevilla several joint works with F.D. ARARUNA Dpto. Matemática - UFPB - Brazil S. GUERRERO Lab. J.-L. Lions - UPMC - France M.C. SANTOS Dpto. Matemática - UFPE - Brazil Dedicated to Jean-Michel Coron in his 60th birthday E. Fernández-Cara Controllability of PDEs
Outline Background 1 Hierarchical control 2 The system and the controls. Meaning The Stackelberg-Nash strategy The main result. Idea of the proof Additional results and comments 3 E. Fernández-Cara Controllability of PDEs
Control issues The meaning of control CONTROL PROBLEMS What is usual: act to get good (or the best) results for � E ( U ) = F + . . . What is easier? Solving? Controlling? Two classical approaches: Optimal control Controllability E. Fernández-Cara Controllability of PDEs
Background Optimal control OPTIMAL CONTROL A general optimal control problem Minimize J ( v ) Subject to v ∈ V ad , y ∈ Y ad , ( v , y ) satisfies E ( y ) = F ( v ) + . . . ( S ) Main questions: ∃ , uniqueness/multiplicity, characterization, computation, . . . We could also consider similar bi-objective optimal control: "Minimize" J 1 ( v ) , J 2 ( v ) Subject to v ∈ V ad , . . . E. Fernández-Cara Controllability of PDEs
Background Controllability CONTROLLABILITY A null controllability problem Find ( v , y ) Such that v ∈ V ad , ( v , y ) satisfies ( ES ) , y ( T ) = 0 with y : [ 0 , T ] �→ H , E ( y ) ≡ y t + A ( y ) = F ( v ) + . . . ( ES ) Again many interesting questions: ∃ , uniqueness/multiplicity, characterization, computation, . . . A very rich subject for PDEs, see [Russell, J.-L. Lions, Coron, Zuazua, . . . ] E. Fernández-Cara Controllability of PDEs
Background Both viewpoints Question: How can we adopt both viewpoints together? Example: Optimal-control / controllability problem A simplified model for the autonomous car driving problem The system: x ( 0 ) = x 0 x = f ( x , u ) , ˙ Constraints: dist. ( x ( t ) , Z ( t )) ≥ ε ∀ t u ∈ U ad ( | u ( t ) | ≤ C ) u determines direction and speed Goals (prescribed x T and ˆ x ): x ( T ) = x T (or | x ( T ) − x T | ≤ ε . . . ) Minimize sup t | x ( t ) − ˆ x ( t ) | [Sontag, Sussman-Tang, . . . ] E. Fernández-Cara Controllability of PDEs
Optimal control + controllability Automatic driving Figure: The ICARE Project, INRIA, France. Autonomous car driving. Malis-Morin-Rives-Samson, 2004 The car in the street E. Fernández-Cara Controllability of PDEs
Optimal control + controllability Automatic driving Figure: Nissan ID. Autonomous car driving. 2015–2020 What is announced: • Nissan ID 1.0 (2015), highways and traffic jams (no lane change) • ID 2.0 (2018), overtaking and lane change • ID 3.0 (2020), complete autonomous driving in town http://reports.nissan-global.com/EN/?p=17295 E. Fernández-Cara Controllability of PDEs
Hierarchical control The system and the controls. Meaning Another way to connect optimal control and controllability: HIERARCHICAL CONTROL (Stackelberg) The main ideas in the context of Navier-Stokes: Two controls - one leader, one follower y t +( y · ∇ ) y − ∆ y + ∇ p = f 1 O + v 1 ω , ( x , t ) ∈ Ω × ( 0 , T ) ∇ · y = 0 , ( x , t ) ∈ Ω × ( 0 , T ) y = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , T ) y ( x , 0 ) = y 0 ( x ) , x ∈ Ω Different domains O , ω Two objectives: Get y ≈ y d in O d × ( 0 , T ) , with reasonable effort: �� �� | y − y d | 2 + µ | v | 2 Minimize α O d × ( 0 , T ) ω × ( 0 , T ) An optimal control problem Get y ( T ) = 0 - A null controllability problem Before explaining what to do . . . let us complicate the situation! E. Fernández-Cara Controllability of PDEs
Hierarchical control The system and the controls. Meaning BEYOND: A MORE COMPLEX CONTROL PROBLEM, NAVIER-STOKES (Stackelberg-Nash, Stackelberg-Pareto, . . . ) Three controls: one leader, two followers y t +( y · ∇ ) y − ∆ y + ∇ p = f 1 O + v 1 1 O 1 + v 2 1 O 2 , ( x , t ) ∈ Ω × ( 0 , T ) ∇ · y = 0 , ( x , t ) ∈ Ω × ( 0 , T ) y = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , T ) y ( x , 0 ) = y 0 ( x ) , x ∈ Ω Different domains O , O i , ( i = 1 , 2 ) Three objectives: “Simultaneously”, y ≈ y i , d in O i , d × ( 0 , T ) , i = 1 , 2, reasonable effort: �� �� | y − y i , d | 2 + µ | v i | 2 , Minimize α i i = 1 , 2 O i , d × ( 0 , T ) O i × ( 0 , T ) Bi-objective optimal control - The task of the followers In practice, an equilibrium ( v 1 ( f ) , v 2 ( f )) for each f ? Get y ( T ) = 0 Null controllability - The task of the leader Can we find f such that y ( T ) = 0? E. Fernández-Cara Controllability of PDEs
Hierarchical control The system and the controls. Meaning y t +( y · ∇ ) y − ∆ y + ∇ p = f 1 O + v 1 1 O 1 + v 2 1 O 2 , ( x , t ) ∈ Ω × ( 0 , T ) ∇ · y = 0 , ( x , t ) ∈ Ω × ( 0 , T ) y = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , T ) y ( x , 0 ) = y 0 ( x ) , x ∈ Ω Many applications: Heating: Controlling temperatures Heat sources at different locations - Heat PDE (linear, semilinear, etc.) Tumor growth: Controlling tumor cell densities Radiotherapy strategies - Reaction-diffusion PDEs bilinear control Fluid mechanics: Controlling fluid velocity fields Several mechanical actions - Stokes, Navier-Stokes or similar Finances: Controlling the price of an option Agents at different stock prices, etc. - Backwards in time heat-like PDE Degenerate coefficients Contributions: Lions, Díaz-Lions, Glowinski-Periaux-Ramos, Guillén, . . . Optimal control + AC E. Fernández-Cara Controllability of PDEs
Hierarchical control The system and the controls. Meaning TOO DIFFICULT - A SIMPLIFIED PROBLEM Again three controls: one leader, two followers y t − y xx = f 1 O + v 1 1 O 1 + v 2 1 O 2 , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , T ) y ( 0 , t ) = y ( 1 , t ) = 0 , t ∈ ( 0 , T ) ( H ) y ( x , 0 ) = y 0 ( x ) , x ∈ ( 0 , 1 ) Different intervals O , O i Again three objectives: Simultaneously, y ≈ y i , d in O i , d × ( 0 , T ) , i = 1 , 2, reasonable effort: �� �� | y − y i , d | 2 + µ | v i | 2 , Minimize α i i = 1 , 2 O i , d × ( 0 , T ) O i × ( 0 , T ) Bi-objective optimal control - Followers’ task Get y ( T ) = 0 Null controllability - Leader’s task What can we do? E. Fernández-Cara Controllability of PDEs
Hierarchical control The Stackelberg-Nash strategy THE STACKELBERG-NASH STRATEGY Step 1: f is fixed �� �� | y − y i , d | 2 + µ | v i | 2 , i = 1 , 2 J i ( v 1 , v 2 ) := α i O i , d × ( 0 , T ) O i × ( 0 , T ) Find a Nash equilibrium ( v 1 ( f ) , v 2 ( f )) with v i ( f ) ∈ L 2 ( O i × ( 0 , T )) : ∀ v 1 ∈ L 2 ( O 1 × ( 0 , T )) J 1 ( v 1 ( f ) , v 2 ( f )) ≤ J 1 ( v 1 , v 2 ( f )) ∀ v 2 ∈ L 2 ( O 2 × ( 0 , T )) J 2 ( v 1 ( f ) , v 2 ( f )) ≤ J 2 ( v 1 ( f ) , v 2 ) Equivalent to: y t − y xx = f 1 O − 1 µφ 1 1 O 1 − 1 µφ 2 1 O 2 − φ i , t − φ i , xx = α i ( y − y i , d ) 1 O i , i = 1 , 2 ( HN ) φ i ( 0 , t ) = φ i ( 1 , t ) = 0 , y ( 0 , t ) = y ( 1 , t ) = 0 , t ∈ ( 0 , T ) y ( x , 0 ) = y 0 ( x ) , φ i ( x , T ) = 0 , x ∈ ( 0 , 1 ) Then: v i ( f ) = − 1 µ φ i | O i × ( 0 , T ) (Pontryagin) ∃ ( v 1 ( f ) , v 2 ( f )) ? Uniqueness? E. Fernández-Cara Controllability of PDEs
Hierarchical control The Stackelberg-Nash strategy THE STACKELBERG-NASH STRATEGY Step 2: Find f such that y t − y xx = f 1 O − 1 µφ 1 1 O 1 − 1 µφ 2 1 O 2 − φ i , t − φ i , xx = α i ( y − y i , d ) 1 O i , i = 1 , 2 ( HSN ) 1 φ i ( 0 , t ) = φ i ( 1 , t ) = 0 , y ( 0 , t ) = y ( 1 , t ) = 0 , t ∈ ( 0 , T ) y ( x , 0 ) = y 0 ( x ) , φ i ( x , T ) = 0 , x ∈ ( 0 , 1 ) y ( x , T ) = 0 , x ∈ ( 0 , 1 ) ( HSN ) 2 with � f � L 2 ( O× ( 0 , T )) ≤ C � y 0 � L 2 For instance, for y i . d ≡ 0, equivalent to: → R ( M ) , with Ly 0 := y ( · , T ) , Mf := y ( · , T ) . . . R ( L ) ֒ ∀ ψ T ∈ L 2 ( 0 , 1 ) In turn, equivalent to: � L ∗ ψ T � ≤ � M ∗ ψ T � (classical, functional analysis; [Russell, 1973]) E. Fernández-Cara Controllability of PDEs
Hierarchical control The result. Idea of the proof Theorem Assume: O 1 , d = O 2 , d , O i , d ∩ O � = ∅ , large µ ρ 2 | y i , d | 2 dx dt < + ∞ , i = 1 , 2 , then: ρ such that, if � � ∃ ˆ O d × ( 0 , T ) ˆ ∀ y 0 ∈ L 2 (Ω) ∃ null controls f ∈ L 2 ( O × ( 0 , T )) & Nash pairs ( v 1 ( f ) , v 2 ( f )) Idea of the proof: 1 - Large µ ⇒ ∀ f ∈ L 2 ( O × ( 0 , T )) ∃ ! Nash equilibrium ( v 1 ( f ) , v 2 ( f )) y t − y xx = f 1 O − 1 µ φ 1 1 O 1 − 1 µ φ 2 1 O 2 − φ i , t − φ i , xx = α i ( y − y i , d ) 1 O i , i = 1 , 2 φ i ( 0 , t ) = φ i ( 1 , t ) = 0 , y ( 0 , t ) = y ( 1 , t ) = 0 , t ∈ ( 0 , T ) y ( x , 0 ) = y 0 ( x ) , φ i ( x , T ) = 0 , x ∈ ( 0 , 1 ) v i ( f ) = − 1 µφ i | O i × ( 0 , T ) E. Fernández-Cara Controllability of PDEs
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