Continuous dependence estimates for the ergodic problem with an application to homogenization

Transcription

1 Continuous dependence estimates for the ergodic problem with an application to homogenization Claudio Marchi Bayreuth, September 12 th, 2013 C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 1 / 21

2 Outline 1 Continuous dependence estimates for the ergodic problem The ergodic problem Motivations Known properties Main result 2 Rate of convergence for the homogenization problem The homogenization problem Motivations literature Main result C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 2 / 21

3 Continuous dependence estimates for the ergodic problem The ergodic problem The ergodic problem For the Hamilton-Jacobi-Bellman operator H(x, p, X) = max { tr (a(x, α)x) f (x, α) p l(x, α)}, (1) α A consider the ergodic problem: seek a pair (v, U) C(R n ) R s.t. H(x, Dv, D 2 v) = U in R n, v(0) = 0. (2) Aim: establish continuous dependence estimates for v; i.e. estimate of v 1 v 2 (in some cases, of v 1 v 2 C 2), where v 1 and v 2 are solutions to two equations (2) with different coefficients. C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 3 / 21

4 Continuous dependence estimates for the ergodic problem The ergodic problem The ergodic problem For the Hamilton-Jacobi-Bellman operator H(x, p, X) = max { tr (a(x, α)x) f (x, α) p l(x, α)}, (1) α A consider the ergodic problem: seek a pair (v, U) C(R n ) R s.t. H(x, Dv, D 2 v) = U in R n, v(0) = 0. (2) Aim: establish continuous dependence estimates for v; i.e. estimate of v 1 v 2 (in some cases, of v 1 v 2 C 2), where v 1 and v 2 are solutions to two equations (2) with different coefficients. C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 3 / 21

5 Continuous dependence estimates for the ergodic problem Motivations Motivations 1 dynamical systems in a torus [Cornfeld et al. 82], 2 the principal eigenvalue [Bensoussan-Frehse 92], 3 strong maximum principle [Alvarez-Bardi 10]. In Optimal Control theory 1 long-time behaviour of solution to parabolic equations, 2 homogenization or singular perturbation problems. C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 4 / 21

6 Continuous dependence estimates for the ergodic problem Motivations Motivations 1 dynamical systems in a torus [Cornfeld et al. 82], 2 the principal eigenvalue [Bensoussan-Frehse 92], 3 strong maximum principle [Alvarez-Bardi 10]. In Optimal Control theory 1 long-time behaviour of solution to parabolic equations, 2 homogenization or singular perturbation problems. C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 4 / 21

7 Continuous dependence estimates for the ergodic problem Known properties Properties of the ergodic problem Assume (A1) A is a compact metric space; (A2) uniform ellipticity: for ν > 0, a(x, α) νi, (x, α) R n A; (A3) periodicity: a, f and l are Z n -periodic in x; (A4) regularity: a, f and l are bdd, UC and Lipschitz in x uniformly in α: ϕ(x, α) K ϕ, ϕ(x, α) ϕ(y, α) K ϕ x y for every x, y R n, α A, for ϕ = a, f, l. C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 5 / 21

8 Continuous dependence estimates for the ergodic problem Known properties Introduce the stationary approximated problem for λ (0, 1) λv λ + H(x, Dv λ, D 2 v λ ) = 0 in R n, (3) and the evolutionary approximated problem t V +H(x, DV, D 2 V ) = 0 in (0, + ) R n, V (0, x) = 0 on R n. (4) Theorem 1 [Arisawa-Lions, 98] (i)!(v, U), with v periodic, s.t. the ergodic problem (2) is fulfilled; (ii) λ (0, 1),! periodic solution v λ C 2,θ (R n ) to (3); (iii)! x-periodic solution V to (4); (iv) as λ 0 +, λv λ and v λ v λ (0) uniformly converge to U and respectively to v; (v) as t +, V (t, x)/t and V (t, x) Ut uniformly converge to U and respectively to v + c 0 (for some c 0 R). C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 6 / 21

9 Continuous dependence estimates for the ergodic problem Known properties Introduce the stationary approximated problem for λ (0, 1) λv λ + H(x, Dv λ, D 2 v λ ) = 0 in R n, (3) and the evolutionary approximated problem t V +H(x, DV, D 2 V ) = 0 in (0, + ) R n, V (0, x) = 0 on R n. (4) Theorem 1 [Arisawa-Lions, 98] (i)!(v, U), with v periodic, s.t. the ergodic problem (2) is fulfilled; (ii) λ (0, 1),! periodic solution v λ C 2,θ (R n ) to (3); (iii)! x-periodic solution V to (4); (iv) as λ 0 +, λv λ and v λ v λ (0) uniformly converge to U and respectively to v; (v) as t +, V (t, x)/t and V (t, x) Ut uniformly converge to U and respectively to v + c 0 (for some c 0 R). C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 6 / 21

10 Continuous dependence estimates for the ergodic problem Known properties Introduce the stationary approximated problem for λ (0, 1) λv λ + H(x, Dv λ, D 2 v λ ) = 0 in R n, (3) and the evolutionary approximated problem t V +H(x, DV, D 2 V ) = 0 in (0, + ) R n, V (0, x) = 0 on R n. (4) Theorem 1 [Arisawa-Lions, 98] (i)!(v, U), with v periodic, s.t. the ergodic problem (2) is fulfilled; (ii) λ (0, 1),! periodic solution v λ C 2,θ (R n ) to (3); (iii)! x-periodic solution V to (4); (iv) as λ 0 +, λv λ and v λ v λ (0) uniformly converge to U and respectively to v; (v) as t +, V (t, x)/t and V (t, x) Ut uniformly converge to U and respectively to v + c 0 (for some c 0 R). C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 6 / 21

11 Continuous dependence estimates for the ergodic problem Known properties Introduce the stationary approximated problem for λ (0, 1) λv λ + H(x, Dv λ, D 2 v λ ) = 0 in R n, (3) and the evolutionary approximated problem t V +H(x, DV, D 2 V ) = 0 in (0, + ) R n, V (0, x) = 0 on R n. (4) Theorem 1 [Arisawa-Lions, 98] (i)!(v, U), with v periodic, s.t. the ergodic problem (2) is fulfilled; (ii) λ (0, 1),! periodic solution v λ C 2,θ (R n ) to (3); (iii)! x-periodic solution V to (4); (iv) as λ 0 +, λv λ and v λ v λ (0) uniformly converge to U and respectively to v; (v) as t +, V (t, x)/t and V (t, x) Ut uniformly converge to U and respectively to v + c 0 (for some c 0 R). C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 6 / 21

12 Continuous dependence estimates for the ergodic problem Optimal control interpretation Known properties Consider the control system for s > 0 dx s = f (x s, α s )ds + 2σ(x s, α s )dw s, x 0 = x where (Ω, F, P) is a probability space endowed with a right continuous filtration (F t ) 0 t<+ and a p-dimensional Brownian motion W t. The control α is chosen in the set A of progressively measurable processes with value in A for minimizing one of the costs (E x denotes the expectation). + P(x, α) := E x l(x s, α s )e λs ds, t P(t, x, α) := E x l(x s, α s )ds 0 0 C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 7 / 21

13 Continuous dependence estimates for the ergodic problem Known properties Then, for a = σσ T, we have We deduce that (for some c 0 R) v λ (x) = inf P(x, α) α A V (t, x) = inf P(t, x, α). α A U = lim λ 0 + λ inf α A E x 1 = lim t + t [ v = lim λ 0 + λ = lim t + [ inf α A E x inf α A inf α A + 0 t P(t, x, α) Ut l(x s, α s )e λs ds l(x s, α s )ds 0 ] P(x, α) inf P(0, α) α A ] + c 0. C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 8 / 21

14 Continuous dependence estimates for the ergodic problem Main result Continuous dependence for the ergodic solution - 1 For i = 1, 2, consider { ) } tr (a i D 2 v i f i Dv i l i = U i in R n, v i (0) = 0. (Ei) max α A Theorem 2 [M., ppt] Under assumptions (A1)-(A4), we have v 1 v 2 CC l (max x,α a 1 a 2 + max x,α f 1 f 2 where C l := 1 + K l1 K l2 and C is independent of K li. ) + C max x,α l 1 l 2 C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 9 / 21

15 Continuous dependence estimates for the ergodic problem Main result Sketch of the proof Set w λ i := v λ i v λ i (0). By Theorem 1-(iv), it suffices to establish w λ 1 w λ 2 CC l (max x,α a 1 a 2 + max x,α f 1 f 2 ) + C max x,α l 1 l 2 (5) for every λ (0, 1). In order to prove this relation: (α) by the comparison principle, we get λ w λ 1 w λ 2 C 1 C l (max x,α a 1 a 2 +max x,α f 1 f 2 )+C max x,α l 1 l 2 ; (β) we proceed by contradiction; (γ) by the regularity of v λ i, we obtain a periodic function w, s.t. w = 1, w(0) = 0, H (Dw, D 2 w) 0 for some elliptic operator H with H (0, 0) = 0. (δ) contradiction to maximum principle. C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 10 / 21

16 Continuous dependence estimates for the ergodic problem Main result Continuous dependence for the ergodic solution - 2 For i = 1, 2, consider H i (x, Dv i, D 2 v i ) = U i in R n, v i (0) = 0. Theorem 3 [M., ppt] Assume (A1)-(A4) and that, for some θ (0, 1), H i C 1,θ loc. Then, for some C R and K R n R n S n ( ) v 1 v 2 C 2 (R n ) C max a 1 a 2 + max f 1 f 2 + max l 1 l 2 x,α x,α x,α +[H 1 H 2 ] 1,K where [H 1 H 2 ] 1,K is the Lipschitz constant of (H 1 H 2 ) on the set K. C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 11 / 21

17 Rate of convergence for the homogenization problem RATE OF CONVERGENCE IN HOMOGENIZATION PROBLEM C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 12 / 21

18 Rate of convergence for the homogenization problem The homogenization problem The homogenization problem We consider the homogenization problem ( u ε + H x, x ε, Duε, D 2 u ε) = 0 in R n (6) where H(x, y, p, X) = max { tr (a(x, y, α)x) f (x, y, α) p l(x, y, α)}. α A Aim Investigate how fast, as ε 0, u ε converges to the solution u to the effective equation (which needs to be suitably defined) ( ) u + H x, Du, D 2 u = 0 in R n. (7) C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 13 / 21

19 Rate of convergence for the homogenization problem The homogenization problem The homogenization problem We consider the homogenization problem ( u ε + H x, x ε, Duε, D 2 u ε) = 0 in R n (6) where H(x, y, p, X) = max { tr (a(x, y, α)x) f (x, y, α) p l(x, y, α)}. α A Aim Investigate how fast, as ε 0, u ε converges to the solution u to the effective equation (which needs to be suitably defined) ( ) u + H x, Du, D 2 u = 0 in R n. (7) C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 13 / 21

20 Rate of convergence for the homogenization problem The homogenization problem Assume (A1) A is a compact metric space; (A2) uniform ellipticity: for ν > 0, a(x, y, α) νi, (x, y, α); (A3) periodicity: a, f and l are Z n -periodic in y; (A4) regularity: a, f and l are bdd, UC and Lipschitz in (x, y) uniformly in α: for ϕ = a, f, l, there holds ϕ K and ϕ(x 1, y 1, α) ϕ(x 2, y 2, α) K ( x 1 x 2 + y 1 y 2 ) for every x 1, x 2, y 1, y 2 R n, α A. C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 14 / 21

21 Rate of convergence for the homogenization problem Motivations Motivation: a stochastic optimal control problem Consider the dynamics in a medium with microscopic heterogeneities ( dx s = f x s, x ) s ε, α s ds + ( 2σ x s, x ) s ε, α s dw s, x 0 = x (s > 0) where (Ω, F, P) is a probability space endowed with a right continuous filtration (F t ) 0 t<+ and a p-dimensional Brownian motion W t. The control α is chosen in the set A of progressively measurable processes with value in A for minimizing the cost P(x, α) = E x + 0 ( l x s, x ) s ε, α s e s ds. (E x denotes the expectation). The value function u ε (x) := inf α A P(x, α) solves (6) with a = σσ T. The purpose of homogenization is to investigate what happens when the size of heterogeneities becomes smaller and smaller. C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 15 / 21

22 Rate of convergence for the homogenization problem literature Literature Evans, 89-92; Alvarez-Bardi, 01 H(x, p, X) is the ergodic constant for H(x,, p, X + ) i.e.:! a y-periodic sol v = v(y; x, p, X) to the cell problem H(x, y, p, X + D 2 yyv) = H(x, p, X) in R n, v(0; x, p, X) = 0; u C 2,θ, for some θ (0, 1]; u ε converges to u locally uniformly in R n. This feature is formally motivated by the expansion u ε (x) = u(x) + εu 1 (x, x/ε) + ε 2 u 2 (x, x/ε) (8) C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 16 / 21

23 Rate of convergence for the homogenization problem literature Literature Evans, 89-92; Alvarez-Bardi, 01 H(x, p, X) is the ergodic constant for H(x,, p, X + ) i.e.:! a y-periodic sol v = v(y; x, p, X) to the cell problem H(x, y, p, X + D 2 yyv) = H(x, p, X) in R n, v(0; x, p, X) = 0; u C 2,θ, for some θ (0, 1]; u ε converges to u locally uniformly in R n. This feature is formally motivated by the expansion u ε (x) = u(x) + εu 1 (x, x/ε) + ε 2 u 2 (x, x/ε) (8) C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 16 / 21

24 Rate of convergence for the homogenization problem literature Camilli-M., 09 u ε converges to u uniformly in R n ; u ε u Cε 2θ 2+θ. Remark. The maximal rate is 2/3. C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 17 / 21

25 Rate of convergence for the homogenization problem Main result An improvement for the rate of convergence Theorem [M., ppt] u ε u Cε θ. Remark. The maximal rate is 1 as 1 in the formal expansion (8), 2 in the regular case for linear problem (see: [Bensoussan-JL Lions-Papanicolaou, 78] or [Jikov-Kozlov-Oleinik, 94]). C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 18 / 21

26 Rate of convergence for the homogenization problem Sketch of the proof Main result Fix ε; for each γ > 0, introduce ϕ(x) := u ε (x) u(x) ε 2 v ( x ε ; [u](x) ) γ 2 x 2 where v(y; [u](x)) := v(y; x, Du(x), D 2 u(x)). By u C 2,θ and Theorem 2, ϕ C 0 (R n ). Let ˆx be its maximum point. For c := Cε θ, consider ( x ) ϕ(x) := u ε (x) u(x) ε 2 v ε ; [u](ˆx) γ 2 x 2 c x ˆx 2 and denote by x its maximum point in B(ˆx, ε). Main estimate u ε ( x) u(ˆx) C[ε θ + γ 1/2 ]. C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 19 / 21

27 Rate of convergence for the homogenization problem Sketch of the proof Main result Fix ε; for each γ > 0, introduce ϕ(x) := u ε (x) u(x) ε 2 v ( x ε ; [u](x) ) γ 2 x 2 where v(y; [u](x)) := v(y; x, Du(x), D 2 u(x)). By u C 2,θ and Theorem 2, ϕ C 0 (R n ). Let ˆx be its maximum point. For c := Cε θ, consider ( x ) ϕ(x) := u ε (x) u(x) ε 2 v ε ; [u](ˆx) γ 2 x 2 c x ˆx 2 and denote by x its maximum point in B(ˆx, ε). Main estimate u ε ( x) u(ˆx) C[ε θ + γ 1/2 ]. C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 19 / 21

28 Rate of convergence for the homogenization problem Sketch of the proof Main result Fix ε; for each γ > 0, introduce ϕ(x) := u ε (x) u(x) ε 2 v ( x ε ; [u](x) ) γ 2 x 2 where v(y; [u](x)) := v(y; x, Du(x), D 2 u(x)). By u C 2,θ and Theorem 2, ϕ C 0 (R n ). Let ˆx be its maximum point. For c := Cε θ, consider ( x ) ϕ(x) := u ε (x) u(x) ε 2 v ε ; [u](ˆx) γ 2 x 2 c x ˆx 2 and denote by x its maximum point in B(ˆx, ε). Main estimate u ε ( x) u(ˆx) C[ε θ + γ 1/2 ]. C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 19 / 21

29 Rate of convergence for the homogenization problem Main result The relation ϕ( x) ϕ(ˆx) = ϕ(ˆx) ϕ(x) yields u ε (x) u(x) C[ε θ + γ 1/2 ] + γ 2 x 2. As γ 0 +, we get u ε (x) u(x) Cε θ. C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 20 / 21

30 Rate of convergence for the homogenization problem Main result Example: diffusion independent of α and y Consider the homogenization problem ( u ε tr a(x)d 2 u ε) ( + F x, x ε, Duε) = 0 in R n with a C 1,1 and F(x, y, p) = max α A { f (x, y, α) p l(x, y, α)}. In this case, the effective equation is ( ) u tr a(x)d 2 u + F(x, y, Du) dy = 0 in R n. (0,1] n Corollary u ε u Cε. C. Marchi (Università di Padova) Continuous dependence Bayreuth, 12/9/13 21 / 21