Thuong Nguyen. SADCO Internal Review Metting

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1 Asymptotic behavior of singularly perturbed control system: non-periodic setting Thuong Nguyen (Joint work with A. Siconolfi) SADCO Internal Review Metting Rome, Nov 10-12, 2014 Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

2 Overview 1 Introduction 2 Periodic setting: Results of Alvarez & Bardi 2003, Non-periodic setting: Some new results Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

3 1. Introduction Singularly perturbed control system: X (s) = f ( X (s), Y (s), α(s) ), s > 0 Y (s) = 1 ε g( X (s), Y (s), α(s) ), s > 0 X (0) = x, Y (0) = y. (S ε ) Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

4 1. Introduction Singularly perturbed control system: X (s) = f ( X (s), Y (s), α(s) ), s > 0 Y (s) = 1 ε g( X (s), Y (s), α(s) ), s > 0 X (0) = x, Y (0) = y. (S ε ) α : [0, + ) A measurable function (control), A R d be a compact subset; f : R n R m A R n and g : R n R m A R m are the dynamics; (x, y) R n R m initial position, ε > 0 small parameter; Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

5 trajectory: ( X ε (s), X ε (s) ) ( X ε (s; x, y, α), X ε (s; x, y, α) ), X ε ( ) is slow trajectory, Y ε ( ) is fast trajectory. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

6 trajectory: ( X ε (s), X ε (s) ) ( X ε (s; x, y, α), X ε (s; x, y, α) ), X ε ( ) is slow trajectory, Y ε ( ) is fast trajectory. Value function: for (x, y) R n R m, t > 0 v ε (x, y, t) := inf α { t l ( X ε (s), Y ε (s), α(s) ) ds+h ( X ε (t) )} 0 l : R n R m A R is running cost; h : R n R is final cost (or terminal cost). Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

7 Approaches for singular perturbation problem Goal. Asymptotic analysis when ε goes to zero? Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

8 Approaches for singular perturbation problem Goal. Asymptotic analysis when ε goes to zero? Dynamical system approach: Asymptotic behavior of trajectory ( X ε ( ), Y ε ( ) ), as ε 0. Order reduction method: for ODE [Levinson & Tichonov]; for control system [Kokotovíc, Khalil, O Reilly, Bensoussan, Dontchev, Zolezzi, Veliov,...]; Averaging method (limit occupational measures): Artstein, Gaitsgory, Leizarowitz and orthers. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

9 PDE approach: [Alvarez & Bardi, 2001, 2003, 2010] Asymptotic behavior of value function v ε, as ε 0? Under suitable assumptions, v ε is the unique viscosity solution of HJB equation {v t ε + H ( x, y, D x v ε, D y v ) ε = 0 in R n R m (0, + ), ε v ε (x, y, 0) = h(x) in R n R m, where H(x, y, p, q) = max a A { p f (x, y, a) q g(x, y, a) l(x, y, a) }. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

10 PDE approach aims at characterizing limit of v ε as viscosity (sub-super) solutions of appropriate effective HJB equation { v t + H(x, Dv) = 0 in R n (0, + ), v(x, 0) = h(x) in R n. (1) In this talk, we will follow the PDE approach. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

11 2. Periodic setting: [Alvarez & Bardi 2003, 2010] Standing assumptions: Standard assumptions on datum: f, g, l, h bounded uniformly continuous; f, g Lipschitz continuous in state variables, uniformly in control variable; Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

12 2. Periodic setting: [Alvarez & Bardi 2003, 2010] Standing assumptions: Standard assumptions on datum: f, g, l, h bounded uniformly continuous; f, g Lipschitz continuous in state variables, uniformly in control variable; Periodicity: f, g, l are Z m -periodic in the fast variable; Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

13 2. Periodic setting: [Alvarez & Bardi 2003, 2010] Standing assumptions: Standard assumptions on datum: f, g, l, h bounded uniformly continuous; f, g Lipschitz continuous in state variables, uniformly in control variable; Periodicity: f, g, l are Z m -periodic in the fast variable; Bounded time controllability on the fast system { Y (τ) = g ( x, Y (τ), α(τ) ), τ > 0 (FS) Y (0) = y, x is fixed in R n. for fixed x R n, for any y, z R m, there exist T > 0 and α A such that Y y (τ; α, x) = z for some τ T. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

14 Main results: periodic setting Cell problem: For each (x 0, p 0 ) R n R n, consider the family of equations, for a constant λ R, H ( x 0, y, p 0, Du(y) ) = λ in R m. (CP) Find λ R such that (CP) has periodic sub- and supersolutions. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

15 Main results: periodic setting Cell problem: For each (x 0, p 0 ) R n R n, consider the family of equations, for a constant λ R, H ( x 0, y, p 0, Du(y) ) = λ in R m. (CP) Find λ R such that (CP) has periodic sub- and supersolutions. Theorem 1 [Alvarez & Bardi 2010] For each (x 0, p 0 ) R n R n, there exists a unique real value c 0 = c 0 (x 0, p 0 ) such that the cell problem (CP), with λ = c 0, admits a periodic subsolution and a periodic supersolution. The effective Hamiltonian H is then defined by setting H(x 0, p 0 ) = c 0. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

16 Remark 1. (Correctors) Periodic subsolution and periodic supersolution of cell problem play the roles of correctors allowing to adapt Evans s perturbed test function method. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

17 Remark 1. (Correctors) Periodic subsolution and periodic supersolution of cell problem play the roles of correctors allowing to adapt Evans s perturbed test function method. Remark 2. (Connection to ergodic control problem) lim δu w(y, t; x 0, p 0 ) δ(y; x 0, p 0 ) = lim δ 0 t + t = H(x 0, p 0 ) uniformly in y R m. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

18 where u δ (y; x 0, p 0 ) is the unique solution to the equation δu δ + H ( x 0, y, p 0, Du δ (y) ) = 0 in R m, u δ periodic, w(y, t; x 0, p 0 ) is the unique solution to the problem { w t + H(x 0, y, p 0, D y w) = 0, (y, t) R m (0, + ), w(y, 0) = 0, w periodic in y, y R m. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

19 Weak semilimits Under standing assumptions, { v ε}, ε > 0, is equibounded in R n R m [0, + ), hence we can define upper semilimit and lower semilimit of v ε, as ε 0, respectively, by v (x, t) := v (x, t) := lim sup ε 0, (x,t ) (x,t) lim inf ε 0, (x,t ) (x,t) sup v ε (x, y, t ), y R m inf v ε (x, y, t ). y R m Note that, v (x, t) and v (x, t) are, respectively, bounded upper semicontinuous and bounded lower semicontinuous in R n [0, + ). Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

20 Convergence result Theorem 2 [Alvarez & Bardi 2010] The weak semilimits v and v are, respectively, subsolution and supersolution of the effective problem { v t + H(x, Dv) = 0 in R n (0, + ) (HJ) v(x, 0) = h(x) in R n. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

21 Convergence result Theorem 2 [Alvarez & Bardi 2010] The weak semilimits v and v are, respectively, subsolution and supersolution of the effective problem { v t + H(x, Dv) = 0 in R n (0, + ) (HJ) v(x, 0) = h(x) in R n. In addition, if (HJ) satisfies comparison principle, then v ε locally uniformly converges on R n R m [0, + ), as ε 0, to the unique solution v = v = v of (HJ). Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

22 Key ideas and methods (periodic case) Viscosity solution theory; Periodic homogenization [Lions, Papanicolau, Varadhan 1986]; Perturbed test function method [Evans 1989, 1992]. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

23 3. Non-periodic setting Standard assumptions on datum: f, g, h bounded uniformly continuous; f, g Lipschitz continuous in state variables, uniformly in control variable; l is uniformly continuous; Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

24 3. Non-periodic setting Standard assumptions on datum: f, g, h bounded uniformly continuous; f, g Lipschitz continuous in state variables, uniformly in control variable; l is uniformly continuous; Coercivity: l is coercive in the fast variable, i.e., min l(x, y, a) +, as y +, uniformly in x; a A Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

25 3. Non-periodic setting Standard assumptions on datum: f, g, h bounded uniformly continuous; f, g Lipschitz continuous in state variables, uniformly in control variable; l is uniformly continuous; Coercivity: l is coercive in the fast variable, i.e., min l(x, y, a) +, as y +, uniformly in x; a A Local bounded time controllability on the fast system: Given compact subsets C R n, D R m. For fixed x C, for any y, z D, there exist T = T (C, D) > 0 and α A such that Y y (τ; α, x) = z for some τ T. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

26 Critical value For each (x 0, p 0 ) R n R n, consider the family of equations, for a constant λ R, H ( x 0, y, p 0, Du(y) ) = λ in R m. (2) Remark. There is no hope to have periodic or even bounded sub- and supersolutions to (2) for some distinguished value λ. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

27 Theorem 1 For each (x 0, p 0 ) R n R m, there exits a unique real value c 0 = c 0 (x 0, p 0 ) such that the equation H ( x 0, y, p 0, Du(y) ) = c 0 in R m, admits a bounded subsolution and a locally bounded coercive supersolution. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

28 Theorem 1 For each (x 0, p 0 ) R n R m, there exits a unique real value c 0 = c 0 (x 0, p 0 ) such that the equation H ( x 0, y, p 0, Du(y) ) = c 0 in R m, admits a bounded subsolution and a locally bounded coercive supersolution. Key ideas and methods (non-periodic case) Approximation by coercive, convex Hamiltonians; Some tools from Weak KAM theory [Fathi & Siconolfi 2005]. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

29 For each (x 0, p 0 ) R n R n, define H(x 0, p 0 ) := inf { λ R : (2) has a bounded subsolution } We let H 0 (y, q) := H(x 0, y, p 0, q). To approximate H 0 by a sequence of coercive, convex Hamiltonians H k, k N, and consider a sequence of equations H k (y, Du) = λ in R m. (3) Define the critical value for H k c k := inf { λ R : (3) has a subsolution }. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

30 Claim 1. The c k is finite for any k, and make up a non-increasing bounded sequence, and hence has a limit c 0 := lim c k. k + Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

31 Claim 1. The c k is finite for any k, and make up a non-increasing bounded sequence, and hence has a limit c 0 := Consider the critical equation lim c k. k + H k (y, Du) = c k in R m. (HJ k ) Claim 2. There exist a sequence of equibounded subsolutions u k and a sequence of locally equibounded, equicoercive solutions w k to (HJ k ). (Using semidistances S k and Aubry set A k for H k ) Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

32 Claim 3. We set u(y) = lim sup # u k (y) := sup { lim sup u k (y k ) : y k y }, k + w(y) = lim inf # w k (y) := inf { lim inf w k(y k ) : y k y }, k + then u and w are, respectively, bounded subsolution and locally bounded coercive supersolution to the equation H 0 (y, Du) = c 0 in R m. (HJ 0 ) (Using stability property of viscosity solution) Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

33 Claim 3. We set u(y) = lim sup # u k (y) := sup { lim sup u k (y k ) : y k y }, k + w(y) = lim inf # w k (y) := inf { lim inf w k(y k ) : y k y }, k + then u and w are, respectively, bounded subsolution and locally bounded coercive supersolution to the equation H 0 (y, Du) = c 0 in R m. (HJ 0 ) (Using stability property of viscosity solution) Claim 4. c 0 = H(x 0, p 0 ). Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

34 Weak semilimits The value functions v ε, ε > 0, is locally equibounded, therefore we can define the weak sup-semilimit and weak inf-semilimit of v ε, respectively, by lim sup # v ε (x, y, t) = sup { lim sup v ε (x ε, y ε, t ε ) : (x ε, y ε, t ε ) (x, y, t) } ε 0 lim inf # v ε (x, y, t) = inf { lim inf ε 0 v ε (x ε, y ε, t ε ) : (x ε, y ε, t ε ) (x, y, t) }. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

35 Weak semilimits The value functions v ε, ε > 0, is locally equibounded, therefore we can define the weak sup-semilimit and weak inf-semilimit of v ε, respectively, by lim sup # v ε (x, y, t) = sup { lim sup v ε (x ε, y ε, t ε ) : (x ε, y ε, t ε ) (x, y, t) } ε 0 lim inf # v ε (x, y, t) = inf { lim inf ε 0 v ε (x ε, y ε, t ε ) : (x ε, y ε, t ε ) (x, y, t) }. Claim. v := lim sup # v ε and v := lim inf # v ε are, respectively, u.s.c and l.s.c, and moreover they are independent of the fast variable y. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

36 Convergence result Theorem 2 The weak sup-semilimit v and weak inf-semilimit v are, respectively, subsolution and supersolution of the effective Cauchy problem { v t + H(x, Dv) = 0 in R n (0, + ) (4) v(x, 0) = h(x) in R n. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

37 Convergence result Theorem 2 The weak sup-semilimit v and weak inf-semilimit v are, respectively, subsolution and supersolution of the effective Cauchy problem { v t + H(x, Dv) = 0 in R n (0, + ) (4) v(x, 0) = h(x) in R n. In addition, if (4) satisfies comparison principle, v ε locally uniformly converges on R n R m [0, + ), as ε 0, to the solution v = v = v of (4). Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

38 Work is in progress... Existence of continuous viscosity solution to the critical equation H ( x 0, y, p 0, Du(y) ) = H(x 0, p 0 ) in R m. Connection to ergodic control problem. Representation formula for H and the explicit form of limit optimal control problem via limit occupational measures. Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

39 References O. Alvarez, M. Bardi, Viscosity solutions methods for singular perturbations in deterministic and stochastic control. SIAM J. Control Optim. 40 (2001), O. Alvarez, M. Bardi, Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result. Arch. Ration. Mech. Anal. 170 (2003), O. Alvarez, M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations. Mem. Amer. Math. Soc. 204, no. 960 (2010). Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

40 L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), no. 3-4, L. C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), no. 3-4, A. Fathi, A. Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonian, Calc. Var. 22 (2005), P. L. Lions, G. Papanicolau and S. R. S. Varadhan, Homogeneization of Hamilton-Jacobi equations, Unpublished, Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26

41 Thank you for your attention! Thuong Nguyen (Roma Sapienza) Asymptotic of SPCS: non-periodic setting November 10-12, / 26