Impulsive control of moving ensembles of interacting agents Maxim - PowerPoint PPT Presentation

Impulsive control of moving ensembles of interacting agents Maxim Staritsyn ∗ joint work with Nikolay Pogodaev ∗ CROWDS: Models and Control CIRM Marseille, France June 3–7, 2019 ∗ Matrosov Institute for System Dynamics and Control Theory, Irkutsk, Russia 1/30

Introduction

Impulsive systems formalize dynamic processes whose states can jump (change very fast) or “vibrate” quite rapidly. In the finite-dimensional case, such systems are frequently written down as generalized ODEs involving distributions or vector-valued measures. The option of impulsive control — actions of short duration but high “energy” (like hammering a nail or kicking a ball) — greatly expands the possibilities of the guide, and, for some tasks, is principally unavoidable. In this research, we translate some results of the impulsive control theory to the framework of multi-agent dynamical systems described by (nonlocal) continuity equations. 2/30

Allusions (beyond crowd dynamics) • Impulsive control of ensembles of non-interacting agents E.g., pendulums on an impulsively actuated cart, charged particles in an accelerator... • “Social networks” (groups of interacting agents) subject to an “agressive” media strategy 3/30

Problem statement

Optimal control problem The time evolution of the ensemble is modeled by a curve t �→ µ t in P 1 c ( R d ) being a (distributional) solution of � � � � ∂ t µ t + ∇ · µ t , u ( t ) = 0 , t ∈ [0 , T ] , µ 0 = ϑ, ( CE ) v µ t m � v [ µ, u ]( x ) = f 0 ( x ) + u i f i ( x ) + ( g ⋆ µ )( x ) , i =1 u i = u i ( t ) , i = 1 , . . . , m , — control inputs. Optimal control problem: � inf R d ℓ ( x ) d µ T ( x ) subject to ( CE ). ( P ) 4/30

Features and specifications � v [ µ, u ]( x ) = f 0 ( x ) + u i f i ( x ) + ( g ⋆ µ )( x ) , Assumptions: f i , i = 0 , m , and g are locally Lipschitz and satisfy the sublinear growth condition, ℓ ∈ C ( R d ; R ), µ 0 ∈ P 1 c ( R d ). • Controls depend on t only, i.e., the actuating force is common for all the agents; u = ( u 1 , . . . , u m ) ∈ U = U M . = � � � u ∈ L ∞ ([0 , T ]; R m ) � � � u � L 1 ([0 , T ]; R m ) ≤ M . � • The VF v is u -affine, while inputs u are not uniformly bounded ⇒ Ill-posedness of ( P ) 5/30

Ill-posedness ( P ) does not have a solution within U , since a minimizing sequence may converge to a measure. To have a well-posed model, we need to relax (extend) the set M of distributional solutions to ( CE ) under controls u ∈ U . A straightforward approach : embed U into ν ∈ C ∗ ([0 , T ]) : | ν | ([0 , T ]) ≤ M � � as usual: u �→ u L m . • Even in ODEs, this works only for scalar controls! • How to define the respective solution µ ( · ) [ ν ] of ( CE )? By analogy with ODEs, it should be a BV curve in P 1 c ( R d ). 6/30

Relaxation

Generalized states We relax ( CE ) in BV + ([0 , T ] , P 1 c ) — the set of right continuous arcs t �→ µ t s.t. Var [0 , T ] µ < ∞ . Here, card ( π ) − 1 Var [0 , T ] µ . � � = sup µ t i , µ t i +1 ) , W 1 π i =1 sup is taken over finite partitions π of [0 , T ], W 1 is the L 1 -Kantorovich distance. Definition [S.-2017] A function µ ( · ) ∈ BV + ([0 , T ]; P 1 c ) is called a generalized state of ( CE ) if ∃ { µ k ( · ) } ⊂ M converging to µ ( · ) at continuity points of µ ( · ) and at t = T ( µ k ⇁ µ ). 7/30

Description of GSs: Time rescale t = ξ ( s ), where ξ : [0 , S ] �→ [0 , T ] is the inverse of � t m Ξ( t ) . � = t + | u i ( ς ) | d ς . 0 i =1 Define � �� � . 1 u ( t ) η s . � � = µ ξ ( s ) , α ( s ) , β ( s ) = � , . � � � � � 1 + � u ( t ) 1 + � u ( t ) � � � t = ξ ( s ) Then ( µ, u ) satisfies (CE) iff ( η, α, β ) satisfies the following reduced CE with a bounded vector field: � � � � ∂ s η s + ∇ · v ˆ η s , α ( s ) , β ( s ) η s = 0 , η 0 = ϑ, ( RCE ) m v [ η, α, β ]( x ) . � � � ˆ = α f 0 ( x ) + ( g ⋆ η )( x ) + f i ( x ) β i . (1) i =1 8/30

Relaxation of (RCE) Each u ∈ U produces a pair ( α, β ) satisfying � α d s = T , α ( s ) > 0 , α ( s ) + | β ( s ) | = 1 for a.e. s ∈ [0 , S ] . S We enlarge the set of admissible controls of (RCE) up to  �  α ≥ 0 , α + | β | ≤ 1 �   U .  �  ˆ ( α, β ) ∈ L ∞ ([0 , S ]; R 1+ m ) � S = . � � α d s = T  �    � 0 (!) In fact, triples ( η, α, β ), corresponding to “additional” controls, characterize GSs. 9/30

Discontinuous time change As ( α, β ) ∈ ˆ U , ξ = ξ [ α ] is not strictly monotone anymore, and ξ − 1 is undefined. We than shall operate with the pseudo-inverse  � � inf s ∈ [0 , S ] : ξ ( s ) > t , t ∈ [0 , T ) ,  ξ ← ( t ) = S , t = T ,  which is increasing, right continuous, and BV . 10/30

Theorem 1 Consider a sequence ( µ k , u k ) of control processes of ( CE ). Define controls ( α k , β k ), β k . = ( β k 1 , . . . , β k m ), by (8) with u = u k , and let η k be the associated solution of ( RCE ). Suppose that primitives F u k of u k converge to U ∈ BV + ([0 , T ]; R m ) at continuity points of U and at t = T . Then, 1. ∃ ( η k j , α k j , β k j ) ⊆ ( η k , α k , β k ) and ( η, α, β ) satisfying ( RCE ) � ξ ← ( t ) together with β ( s ) d s = U ( t ), t ∈ [0 , T ], s.t. 0 c ) × ˆ ( η k j , α k j , β k j ) → ( η, α, β ) in C ([0 , S ]; P 1 U , where ˆ U is equipped with topology σ ( L ∞ , L 1 ). 2. µ k j → η ξ ← at continuity points of ξ ← . = ξ ← [ α ] and at t = T . 11/30

Corollary Any GS is an arc of ( RCE ) up to a discontinuous time change s = ξ ← ( t ). Proposition For any ( α, β ) ∈ ˆ U , t �→ η ξ ← [ α ]( t ) [ α, β ] is a GS of ( CE ). THUS, ( RCE ) with ( α, β ) ∈ ˆ U , actually, describes M . e : GSs are indeed BV + ([0 , T ]; P 1 R´ esum´ c ). What about controls? The idea : Consider impulses as “fast motions” driven by the dominating part of the VF � v [ µ, u ]( x ) = f 0 ( x ) + u i f i ( x ) + ( g ⋆ µ )( x ) . 12/30

What generalized controls are Proposition Let µ ( · ) be a GS produced by { u k } ⊂ U , and ( F u k , F | u k | ) ⇁ ( U , V ). Assume that V sc = 0, and the set ∆ V . = { τ ∈ [0 , T ] : V ( τ ) − V ( τ − ) � = 0 } is naturally ordered. Then there exist • L ∞ -functions u τ : [0 , T τ . = V ( τ ) − V ( τ − )] �→ R m , τ ∈ ∆ V , � T τ m � | u τ u i ( ς ) d ς = U i ( τ ) − U i ( τ − ) , i | = 1; i = 1 , m ; 0 i =1 • AC -curves m τ : [0 , T τ ] �→ P 1 with the property m τ m τ 0 = µ τ − , T τ = µ τ , τ ∈ ∆ V , s.t. µ ( · ) satisfies the following measure continuity equation ( MCE ): 13/30

� T m � � � � ˙ U ac � � 0 = ∂ t ϕ ( t , x )+ f 0 ( x )+ i ( t ) f i ( x ) · ∇ ϕ ( t , x ) d µ t ( x ) d t R d 0 i =1 � T τ � m � � � � � ∂ ς ϕ τ ( ς, x )+ u τ · ∇ ϕ τ ( ς, x ) d m τ � + i ( ς ) f i ( x ) ς ( x ) d ς R d 0 τ ∈ ∆ V i =1 for all collections Φ = ( ϕ, { ϕ τ } τ ∈ ∆ V ), ϕ : (0 , T ) × R d �→ R , ϕ τ : [0 , T τ ] × R d �→ R , τ ∈ ∆ V , s.t. • ϕ is r.c. in t for all x ∈ R d , and C ∞ on each ( τ j , τ j +1 ) × R d ; c • ϕ τ , τ ∈ ∆ V , are C ∞ on (0 , T τ ) × R d , and c • ϕ ( τ − , x ) = ϕ τ (0 , x ) and ϕ ( τ, x ) = ϕ τ ( T τ , x ), for all τ ∈ ∆ V and x ∈ R d . 14/30

Limit CE and the case of commutative VFs R´ esum´ e : The actual input of the relaxed CE is the total collection ( U , V , { u τ } ). Jumps of µ ( · ) are represented through a local PDE ( limit continuity equation ): � � R d ϕ τ ( T τ , x ) d µ τ ( x ) − R d ϕ τ (0 , x ) d µ τ − ( x ) = � T τ � m � � � ∂ ς ϕ τ ( ς, x ) + � u τ · ∇ ϕ τ ( ς, x ) d m τ � i ( ς ) f i ( x ) ς ( x ) d ς 0 R d i =1 ⇒ If f i commute , then (by the Frobenius theorem) t �→ µ t is independent of { u τ } and is completely defined by U (i.e. such GSs do not depend on their approximations by AC -curves). 15/30

Relaxed optimal control problem Optimal impulsive control problem ( P ): �� � � µ ( · ) ∈ M � Minimize ℓ d µ T , Reduced problem (ˆ P ): �� � � η ( · ) satisfies ( RCE ) � Minimize ℓ d η S . Proposition 1. inf( P ) = inf( ˆ P ). 2. Assume that ℓ is Lipschitz continuous. Then, problem ( ˆ P ) has a solution (and therefore, so does ( P )). 16/30

Necessary optimality condition

Pontryagin’s Maximum Principle for (RP), cf. [Bonnet, Rossi, 2017] Theorem 2 Assume that g ∈ C 1 ( R d ; R d ), ∇ g ( − x ) = −∇ g ( x ) for all x ∈ R d , and ℓ ∈ C 1 ( R d ; R ). Let (¯ α, ¯ β ) ∈ ˆ U be optimal for (ˆ P ). Then ∃ λ ∈ R and an arc γ : [0 , S ] �→ P 1 c ( R 2 d ) of the Hamiltonian system  � � � α ( s ) , ¯ � � ∂ s γ s + ∇ ( y , p ) H γ s , ¯ β ( s ) γ s = 0 ,  � � π 1 π 2 π 1 ♯ γ 0 = ϑ, ♯ γ S = ( −∇ ℓ ) ♯ ♯ γ S ,  s.t. the following maximum condition holds for L 1 -a.a. s ∈ [0 , S ]: α ( s ) , ¯ H 1 ( γ s , λ ) , H 0 ( γ s ) � � � � H γ s , λ, ¯ β ( s ) = max . 17/30