Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system Controllability implies ergodicity Armen Shirikyan Department of Mathematics University of Cergy–Pontoise Evolution Equations: Long-time behaviour and Control Université Savoie Mont-Blanc 15 June 2015 1 / 23
Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system Outline Introduction Preliminaries Controllability Stationary measures and mixing Randomly forced differential equations Mixing in the total variation metric Main result Differential equations on a compact manifold Mixing in the Kantorovich–Wasserstein metric Navier–Stokes system with a random boundary force Main result Open problems 2 / 23
Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system Control problem and its stochastic counterpart Let X be a compact metric space, let E be separable Hilbert space, and let S : X × E → X be a continuous mapping. Let us consider the discrete-time control system u k = S ( u k − 1 , ζ k ) , k ≥ 1 , (1) supplemented with the initial condition u 0 = u . (2) Here u ∈ X is an initial state, { u k } ⊂ X is the trajectory of the system, and { ζ k } ⊂ E are control functions at our disposal. We say that (1) is globally approximately controllable if for any u , ˆ u ∈ X and any ε > 0 one can find an integer m ≥ 1 and a finite sequence ζ 1 , . . . , ζ k ∈ E such that d ( u m , ˆ u ) < ε. (3) 3 / 23
Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system Control problem and its stochastic counterpart Let X be a compact metric space, let E be separable Hilbert space, and let S : X × E → X be a continuous mapping. Let us consider the discrete-time control system u k = S ( u k − 1 , ζ k ) , k ≥ 1 , (1) supplemented with the initial condition u 0 = u . (2) Here u ∈ X is an initial state, { u k } ⊂ X is the trajectory of the system, and { ζ k } ⊂ E are control functions at our disposal. We say that (1) is globally approximately controllable if for any u , ˆ u ∈ X and any ε > 0 one can find an integer m ≥ 1 and a finite sequence ζ 1 , . . . , ζ k ∈ E such that d ( u m , ˆ u ) < ε. (3) 3 / 23
Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system Control problem and its stochastic counterpart Let us take a probability measure ℓ on E and consider the stochastic system u k = S ( u k − 1 , η k ) , k ≥ 1 , (4) where { η k } is a sequence of random variables in E distributed according to the law ℓ . In this case, for any initial state u , the trajectory of (4), (2) is a stochastic process in X . Very often, one is interested in the asymptotic behaviour of the law of u k as k → + ∞ . For instance, an interesting problem is to find a stationary measure µ , so that D ( u ) = µ = ⇒ D ( u k ) = µ for any k ≥ 0 , (5) and to study the asymptotic stability of µ as k → ∞ . GOAL: To investigate the relationship between controllability properties of (1) and the long-time behaviour of trajectories for (4) . 4 / 23
Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system Control problem and its stochastic counterpart Let us take a probability measure ℓ on E and consider the stochastic system u k = S ( u k − 1 , η k ) , k ≥ 1 , (4) where { η k } is a sequence of random variables in E distributed according to the law ℓ . In this case, for any initial state u , the trajectory of (4), (2) is a stochastic process in X . Very often, one is interested in the asymptotic behaviour of the law of u k as k → + ∞ . For instance, an interesting problem is to find a stationary measure µ , so that D ( u ) = µ = ⇒ D ( u k ) = µ for any k ≥ 0 , (5) and to study the asymptotic stability of µ as k → ∞ . GOAL: To investigate the relationship between controllability properties of (1) and the long-time behaviour of trajectories for (4) . 4 / 23
Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system Control problem and its stochastic counterpart Let us take a probability measure ℓ on E and consider the stochastic system u k = S ( u k − 1 , η k ) , k ≥ 1 , (4) where { η k } is a sequence of random variables in E distributed according to the law ℓ . In this case, for any initial state u , the trajectory of (4), (2) is a stochastic process in X . Very often, one is interested in the asymptotic behaviour of the law of u k as k → + ∞ . For instance, an interesting problem is to find a stationary measure µ , so that D ( u ) = µ = ⇒ D ( u k ) = µ for any k ≥ 0 , (5) and to study the asymptotic stability of µ as k → ∞ . GOAL: To investigate the relationship between controllability properties of (1) and the long-time behaviour of trajectories for (4) . 4 / 23
Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system Approximate and solid controllability Let K ⊂ E be a closed subset and let ˆ u ∈ X . We say that (1) is globally approximately controllable to ˆ u with a K -valued control if for any ε > 0 there is m ≥ 1 such that, given an initial point u ∈ X , one can find ζ 1 , . . . , ζ m ∈ K satisfying the inequality � S m ( u ; ζ 1 , . . . , ζ m ) , ˆ � d u ≤ ε, (6) where S k ( u ; ζ 1 , . . . , ζ k ) stands for the trajectory of (1), (2). We say that (1) is solidly controllable from u 0 if there is a compact set Q ⊂ K , a non-degenerate ball B ⊂ X , and ε > 0 such that, for any continuous mapping Φ : Q → X satisfying � � Φ ( ζ ) , S ( u 0 , ζ ) ≤ ε, sup d (7) ζ ∈ Q we have Φ ( Q ) ⊃ B . 5 / 23
Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system Approximate and solid controllability Let K ⊂ E be a closed subset and let ˆ u ∈ X . We say that (1) is globally approximately controllable to ˆ u with a K -valued control if for any ε > 0 there is m ≥ 1 such that, given an initial point u ∈ X , one can find ζ 1 , . . . , ζ m ∈ K satisfying the inequality � S m ( u ; ζ 1 , . . . , ζ m ) , ˆ � d u ≤ ε, (6) where S k ( u ; ζ 1 , . . . , ζ k ) stands for the trajectory of (1), (2). We say that (1) is solidly controllable from u 0 if there is a compact set Q ⊂ K , a non-degenerate ball B ⊂ X , and ε > 0 such that, for any continuous mapping Φ : Q → X satisfying � � Φ ( ζ ) , S ( u 0 , ζ ) ≤ ε, sup d (7) ζ ∈ Q we have Φ ( Q ) ⊃ B . 5 / 23
Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system Local stabilisability Let D δ = { ( u , u ′ ) ∈ X × X : d ( u , u ′ ) ≤ δ } . We say that (1) is locally stabilisable if for any R > 0 there is a finite-dimensional subspace E ⊂ E , positive numbers C , δ , α ≤ 1, and q < 1, and a continuous mapping ( u , u ′ , η ) �→ η ′ , Φ : D δ × B E ( R ) → E , which is continuously differentiable in η and satisfies the following inequalities for any ( u , u ′ ) ∈ D δ : S ( u , ζ ) , S ( u ′ , ζ + Φ ( u , u ′ , ζ )) ≤ q d ( u , u ′ ) , � � sup d (8) ζ ∈ B E ( R ) � Φ ( u , u ′ , ζ ) � E + � D ζ Φ ( u , u ′ , ζ ) � L ( E ) ≤ C d ( u , u ′ ) α . (9) � � sup ζ ∈ B E ( R ) 6 / 23
Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system Metrics on the space of probability measures Let us denote by P ( X ) the space of probability measures on X , let C ( X ) be the space of continuous functions f : X → R , and let L ( X ) be the subspace of f ∈ C ( X ) satisfying | f ( u ) − f ( v ) | � f � L := sup | f ( u ) | + sup < ∞ d ( u , v ) u ∈ X u � = v Total variation metric: � µ 1 − µ 2 � var = sup | µ 1 (Γ) − µ 2 (Γ) | Γ ∈B ( X ) = 1 sup | ( f , µ 1 ) − ( f , µ 2 ) | . 2 � f � ∞ ≤ 1 Kantorovich–Wasserstein metric: � µ 1 − µ 2 � ∗ L = sup | ( f , µ 1 ) − ( f , µ 2 ) | . � f � L ≤ 1 7 / 23
Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system Discrete-time Markov processes The stochastic system (4), (2) defines a discrete-time Markov process in X , provided that { η k } is an i.i.d. sequence in E . We denote by P k ( u , Γ) its transition function, P k ( u , Γ) = P u { u k ∈ Γ } , and by P k : C ( X ) → C ( X ) and P ∗ k : P ( X ) → P ( X ) the corresponding Markov operators: � � ( P ∗ ( P k f )( u ) = P k ( u , d v ) f ( v ) , k µ )(Γ) = P k ( u , Γ) µ ( d u ) . X X Let us recall that ( P ∗ ( P k f )( u ) = E u f ( u k ) , k µ )(Γ) = P µ { u k ∈ Γ } . 8 / 23
Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system Exponential mixing Definition A measure µ ∈ P ( X ) is said to be stationary for (4) if P ∗ P ∗ 1 µ = µ = ⇒ k µ = µ for all k ≥ 1 . A stationary measure µ for (4) is said to be exponentially mixing in the total variation metric if there are positive numbers C and γ such that � P ∗ k λ − µ � var ≤ Ce − γ k for all k ≥ 0, λ ∈ P ( X ) . (10) Similarly, a stationary measure µ is said to be exponentially mixing in the Kantorovich–Wasserstein metric if there are positive numbers C and γ such that (10) holds with � · � var replaced by � · � ∗ L . 9 / 23
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