Continuous dependence estimates for large time behavior for Bellman-Isaacs equations and applications to the ergodic problem
aa r X i v : . [ m a t h . A P ] J u l Continuous dependence estimates for large time behavior forBellman-Isaacs equations and applications to the ergodicproblem ∗ Claudio Marchi † July 28 th , 2010 Abstract
This paper concerns continuous dependence estimates for Hamilton-Jacobi-Bellman-Isaacs operators (briefly, HJBI). For the parabolic Cauchy problem, we establish suchan estimate in the whole space [0 , + ∞ ) × R n . Moreover, under some periodicity andellipticity assumptions, we obtain a similar estimate for the ergodic constant associ-ated to the HJBI operator. An interesting byproduct of the latter result will be thelocal uniform convergence for some classes of singular perturbation problems. MSC 2000:
Keywords:
Continuous dependence estimates, parabolic Hamilton-Jacobi equations, vis-cosity solutions, ergodic problems, differential games, singular perturbations.
We consider the following Cauchy problem (cid:26) ∂ t u + H ( x, Du, D u ) = 0 in (0 , + ∞ ) × R n u (0 , x ) = 0 on R n (1.1)for the Hamilton-Jacobi-Bellman-Isaacs (briefly, HJBI) operator H ( x, p, X ) = min β ∈ B max α ∈ A {− tr ( a ( x, α, β ) X ) + f ( x, α, β ) · p + ℓ ( x, α, β ) } (1.2)where ∂ t ≡ ∂/∂t , Du and D u stand respectively for the gradient and for the Hessianmatrix of the real-valued function u = u ( t, x ).For instance, equations of this kind naturally arise in zero-sum two-persons stochasticdifferential games: consider the control system for s > dx s = f ( x s , α s , β s ) + √ σ ( x s , α s , β s ) dW s , x = x (1.3) ∗ Work partially supported by the INDAM-GNAMPA project “Fenomeni di propagazione di fronti eproblemi di omogeneizzazione”. † Dip. di Matematica Pura ed Applicata, Universit`a di Padova, via Trieste 63, 35121 Padova, Italy( [email protected] ). , F , P ) is a probability space endowed with a continuous right filtration ( F t ) ≤ t< + ∞ and a p -adapted Brownian motion W t . The control law α (respectively, β ) belongs to theset A (resp., B ) of progressively measurable processes which take value in the compact set A (resp., B ). The two controls α and β are chosen respectively by the first and the secondplayer whose purpose are opposite: the former wants to minimize the cost functional P ( t, x, α, β ) := E x Z t ℓ ( x s , α s , β s ) ds (1.4)(here, E x denotes the expectation) while the latter wants to maximize it. It is well known(see [21]) that the lower value function u ( t, x ) := inf α ∈ Γ sup β ∈B P ( t, x, α [ β ] , β )is a viscosity solution to problem (1.1)-(1.2) with a = T σσ where Γ stands for the set ofadmissible strategies of the first player (namely, nonanticipating maps α : B → A ; see [21]).This paper is devoted to two main purposes. The former purpose is to establish acontinuous dependence estimate for problem (1.1)-(1.2) in the whole space [0 , + ∞ ) × R n ; inother words, we want to provide an estimate of sup R n | u ( · , t ) − v ( · , t ) | for every t ∈ [0 , + ∞ ),where u and v are solutions to two problems (1.1)-(1.2) having different coefficients. Thelatter purpose of this paper is to establish a continuous dependence estimate for the ergodic constant associated to HJBI operators H as in (1.2) (see Section 3 for the precisedefinition and main properties). An interesting byproduct of this estimate is the localuniform convergence for some classes of singular perturbation problems (see Section 3.1).The continuous dependence estimates for fully nonlinear equations have been widelystudied in literature, starting from the paper by Souganidis [32] for first-order equations.In fact, such estimates play a crucial role in many contexts as error estimates for approx-imation schemes (see [9, 18] and references therein), regularity results (for instance, see[8, 12, 26]) and rate of convergence for vanishing viscosity methods (see [14, 22, 26, 27] andreferences therein). In particular, let us recall that Cockburn, Gripenberg and Londen [14]tackled up the continuous dependence estimate for quasi-linear second-order equationswith Neumann boundary conditions, while Grinpenberg [22] addressed the case of theDirichlet boundary data for the same equations. Afterwards, Jakobsen and Karlsen [26]extended their results to more general classes of equations (see also [27] for elliptic prob-lems). Furthermore, Jakobsen and Georgelin [25] extended the previous results to problemswith more general boundary conditions and domains.The first main purpose of this paper is to establish a continuous dependence estimatefor problem (1.1)-(1.2) with the following three features: the estimate holds in the wholespace [0 , + ∞ ) × R n , the dependence on the L ∞ -distance between the coefficients is explicit,the constants can be explicitly characterized.As one may expect, it turns out that our estimate increases linearly with t . A similarestimate could be obtained by an easy application of the Comparison Principle providedthat some bound on the C -norm of the solution is available. Unfortunately, this is notthe case for operators in form (1.2). In fact, our approach is based on the ComparisonPrinciple techniques for viscosity solutions (see [16]): doubling the variables and adding apenalization term. Let us observe that this approach does not require the non-degeneracyof the operator H ; actually, we shall also apply our result to some degenerate problems.2n the ergodic problem for the operator H , we seek a pair ( v, U ), with v ∈ C ( R n )and U ∈ R (that is, v is a real-valued function while U is a constant) which, in viscositysense, satisfy H ( x, Dv, D v ) = U in R n . (1.5)This problem has been widely studied in literature, especially in connection with homoge-nization or singular perturbation problems (see [2, 3, 6, 13, 20, 30] and references therein),with long-time behavior of solutions to parabolic equations (for instance, see [4, 5, 7]) andwith dynamical systems in a torus (see [1, 15, 31]).It is well known (see [3, 4]) that, under some periodicity and non-degeneracy as-sumptions, there exists exactly one value U ∈ R (called ergodic constant ) such that equa-tion (1.5) admits at least one bounded solution (which is periodic and unique up to aconstant). In [20], Evans obtained a continuous dependence estimate for the ergodic con-stant for operators which are Lipschitz continuous in the variable x uniformly with respectto ( p, X ). Afterwards, Alvarez and Bardi [3], extended his result to operators H as in (1.2)provided that the dispersion matrix is left unchanged (see also [3, Section 6] for possiblydegenerate equations).The second main purpose of this paper is to obtain a continuous dependence estimatefor the ergodic constant of HJBI operators (1.2) only under the periodicity and the non-degeneracy assumption namely, we shall also consider equations with different dispersionmatrices. An interesting byproduct will be the local uniform convergence for some classesof singular perturbation problems for HJBI operators.In conclusion, the aim of this paper is threefold: a continuous dependence esti-mate for problem (1.1)-(1.2) in the whole space [0 , + ∞ ) × R n , a similar estimate for theergodic constant for HJBI operators (1.2) and a local uniform convergence for singularperturbation problems.This paper is organized as follows: in the rest of this section, we provide somenotations and list the standing assumptions. Section 2 contains the continuous dependenceestimate for the parabolic Cauchy problem (1.1) and its application to some degenerateproblems as well. Section 3 concerns the continuous dependence estimate for the ergodicconstant in (1.5); section 3.1 is devoted to illustrate how to derive the local uniformconvergence for singular perturbation problems. Notations:
We define M n,p and S n respectively as the set of n × p real matrices andthe set of n × n symmetric matrices. The latter is endowed with the Euclidean norm andwith the usual order, namely: for X = ( X ij ) i,j =1 ,...,n ∈ S n , k X k := ( P ni,j =1 X ij ) / and,for X, Y ∈ S n , we shall write X ≥ Y , if X − Y is a semi-definite positive matrix.For every positive t , we set Q t := [0 , t ) × R n and Q ∞ := [0 , + ∞ ) × R n .For every real-valued function h , we set k h k ∞ := ess sup | h ( y ) | ; for γ ∈ (0 , γ -H¨older norm: | h | γ := sup y = x | h ( y ) − h ( x ) || y − x | γ . Moreover, J , + h ( ξ ) and J , − h ( ξ ) standrespectively for the second-order superjet and subjet of h at the point ξ (see [16] for theprecise definition and main properties). A real function ω is said a modulus of continuity whenever it is a nonnegative continuous non-decreasing real function on [0 , + ∞ ) with ω (0) = 0. Standing assumptions:
For the operator H in (1.2), the following assumptions will holdthroughout this paper 3 A A and B are two compact metric spaces.( A a = T σσ . The functions σ , f and ℓ are bounded continuous functions in R n × A × B with value respectively in M n,p , R n , and R ; namely, for some C >
0, there holds: k σ k ∞ , k f k ∞ , k ℓ k ∞ ≤ C .( A
3) The drift vectors f and the dispersion matrix σ are Lipschitz continuous in x uni-formly in ( α, β ), namely: for some positive constant C φ , every φ = σ, f satisfies | φ ( x, α, β ) − φ ( y, α, β ) | ≤ C φ | x − y | ∀ x, y ∈ R n , ∀ ( α, β ) ∈ A × B. The running cost ℓ is uniformly continuous in x uniformly in ( α, β ), namely: thereexists a modulus of continuity ω such that | ℓ ( x, α, β ) − ℓ ( y, α, β ) | ≤ ω ( | x − y | ) ∀ x, y ∈ R n , ∀ ( α, β ) ∈ A × B. For i = 1 ,
2, consider the parabolic Cauchy problems ( ∂ t u i + min β ∈ B max α ∈ A (cid:8) − tr (cid:0) a i ( x, α, β ) D u i (cid:1) + f i ( x, α, β ) · Du i + ℓ i ( x, α, β ) (cid:9) = 0 in Q ∞ u i (0 , x ) = 0 on R n . (Pi)where the coefficients fulfill our standing assumptions ( A A k u ( t, · ) − u ( t, · ) k ∞ for every t ∈ [0 , + ∞ ). Insection 2.1 we shall apply this estimate to some degenerate problems. Remark 2.1
By standard viscosity theory (for instance, see [16]), assumptions ( A )-( A )guarantee that the Comparison Principle applies to problem (Pi) ; whence, by the Perronmethod, one can easily deduce that (Pi) admits exactly one solution u i ∈ C ( Q ∞ ) with | u i ( t, x ) | ≤ tC, ∀ ( t, x ) ∈ Q ∞ (2.1) where C is the constant introduced in assumption ( A ). Theorem 2.1
Let u i be the unique solution to problem (Pi) which satisfies the bound (2.1) ( i = 1 , ). Furthermore, let us assume that, for some γ ∈ (0 , , u i ( t, · ) is γ -H¨oldercontinuous uniformly in t , namely: for some C H > , there holds | u i ( t, · ) | γ ≤ C H , ∀ t ∈ [0 , + ∞ ) , i = 1 , . (2.2) Then, there exist a positive constant M such that, for every ( x, t ) ∈ Q ∞ , there holds | u ( t, x ) − u ( t, x ) | ≤ tM (cid:20) max x,α,β k σ − σ k γ + max x,α,β | f − f | γ/ + max x,α,β | ℓ − ℓ | + ω (cid:18) C (max x,α,β k σ − σ k + max x,α,β | f − f | / ) (cid:19)(cid:21) . roof of Theorem 2.1 We shall argue using some techniques introduced in [14, 26].We fix t > η ∈ (1 , + ∞ ) and ε ∈ (0 , E t := [0 , t ) × R n × R n and s t := sup E t (cid:26) u ( τ, x ) − u ( τ, y ) − (cid:18) η | x − y | + ε | x | + | y | ) + εt − τ (cid:19)(cid:27) . (2.3)Our purpose is to establish an upper bound for s t . To this end, without any loss ofgenerality, we can assume s t >
0. For δ ∈ (0 , ψ ( τ, x, y ) := u ( τ, x ) − u ( τ, y ) − δs t τt − (cid:18) η | x − y | + ε | x | + | y | ) + εt − τ (cid:19) . (2.4)Let us observe that definition (2.3) entailssup E t ψ ≥ sup E t (cid:26) u ( τ, x ) − u ( τ, y ) − (cid:18) η | x − y | + ε | x | + | y | ) + εt − τ (cid:19)(cid:27) − sup τ ∈ [0 ,t ) (cid:26) δs t τt (cid:27) ≥ (1 − δ ) s t > . (2.5)Since the functions u and u are bounded in Q t and since the function ψ tends to −∞ as τ → t − , we deduce that there exists a point ( τ , x , y ) ∈ E t where the function ψ attainsits global maximum, namely ψ ( τ , x , y ) = sup E t ψ ≥ C := (2 C H ) / (2 − γ ) , there holds | x − y | ≤ Cη − / (2 − γ ) , ε (cid:0) | x | + | y | (cid:1) ≤ Ct (2.7)where C H , γ and C are the constants introduced respectively in assumption (2.2) and ( A C is independent of t . Actually, in order to prove theformer estimate, we observe that inequality ψ ( τ , x , x ) + ψ ( τ , y , y ) ≤ ψ ( τ , x , y )and assumption (2.2) give η | x − y | ≤ [ u ( τ , x ) − u ( τ , y )] + [ u ( τ , x ) − u ( τ , y )] ≤ C H | x − y | γ . Let us now prove the latter estimate in (2.7): by estimates (2.1) and (2.6), we infer ε (cid:0) | x | + | y | (cid:1) ≤ u ( τ , x ) − u ( τ , y ) ≤ Ct.
Hence, the proof of estimates (2.7) is accomplished.We introduce the test function φ ( τ, x, y ) := δs t τt + η | x − y | + ε | x | + | y | ) + εt − τ (2.8)5nd we invoke [16, Theorem 8.3]: for every ν >
0, there exist values a, b ∈ R and matrices X, Y ∈ S n such that( a, D x φ ( τ , x , y ) , X ) ∈ J , + u ( τ , x ) , ( b, D y φ ( τ , x , y ) , Y ) ∈ J , − u ( τ , y ) , (2.9) a − b = ∂ τ φ ( τ , x , y ) ≡ δs t t + ε ( t − τ ) (2.10) (cid:18) X − Y (cid:19) ≤ Θ + ν Θ , (2.11)where Θ := η (cid:18) I − I − I I (cid:19) + ε (cid:18) I I (cid:19) . From the last inequality, one can deduce that,for every ( α, β ) ∈ A × B , there holdstr ( a ( x , α, β ) X ) − tr ( a ( y , α, β ) Y ) ≤ η k σ ( x , α, β ) − σ ( y , α, β ) k + 2 εC + ν tr (cid:0) ΣΘ (cid:1) (2.12)with Σ := (cid:18) T σ ( x , α, β ) σ ( x , α, β ) T σ ( x , α, β ) σ ( y , α, β ) T σ ( y , α, β ) σ ( x , α, β ) T σ ( y , α, β ) σ ( y , α, β ) (cid:19) . In order to prove this inequality, we shall use the arguments by Ishii [23]. Multiplyingrelation (2.11) by matrix Σ (which is symmetric and nonnegative definite) and evaluatingthe trace, we obtaintr (cid:0) T σ ( x , α, β ) σ ( x , α, β ) X − T σ ( y , α, β ) σ ( y , α, β ) Y (cid:1) ≤ η tr (cid:2) T ( σ ( x , α, β ) − σ ( y , α, β )) ( σ ( x , α, β ) − σ ( y , α, β )) (cid:3) + ε tr (cid:0) T σ ( x , α, β ) σ ( x , α, β ) (cid:1) + ε tr (cid:0) T σ ( y , α, β ) σ ( y , α, β ) (cid:1) + ν tr (cid:0) ΣΘ (cid:1) ;therefore, by assumption ( A u is a subsolution to problem (Pi) with i = 1, the former relation in (2.9)yields0 ≥ a + min β ∈ B max α ∈ A {− tr ( a ( x , α, β ) X ) + f ( x , α, β ) · D x φ + ℓ ( x , α, β ) }≥ b + min β ∈ B max α ∈ A {− tr ( a ( y , α, β ) Y ) + f ( x , α, β ) · ( η ( x − y ) + εx ) + ℓ ( x , α, β ) }− η max α,β k σ ( x , α, β ) − σ ( y , α, β ) k − εC − ν max α,β tr (cid:0) ΣΘ (cid:1) + δs t t + ε ( t − τ ) where the last inequality is due to the definition of φ (2.8) and to relations (2.10) and (2.12).Since u is a supersolution to equation (Pi) with i = 2, by assumption ( A s t : δs t t + ε ( t − τ ) ≤ η max α,β k σ ( x , α, β ) − σ ( y , α, β ) k + 2 εC + ν max α,β tr (cid:0) ΣΘ (cid:1) + εC ( | x | + | y | ) + η | x − y | max α,β | f ( x , α, β ) − f ( y , α, β ) | + max α,β | ℓ ( x , α, β ) − ℓ ( y , α, β ) | . Owing to the definition of s t in (2.3), we deduce u ( τ, x ) − u ( τ, y ) − η | x − y | − ε (cid:0) | x | + | y | (cid:1) ≤ s t + εt − τ ∀ ( τ, x, y ) ∈ E t . τ, x ) ∈ Q t , we infer u ( τ, x ) − u ( τ, x ) ≤ tδ (cid:20) η max α,β k σ ( x , α, β ) − σ ( y , α, β ) k + 2 εC + ν max α,β tr (cid:0) ΣΘ (cid:1) + η | x − y | max α,β | f ( x , α, β ) − f ( y , α, β ) | + εC ( | x | + | y | )+ max α,β | ℓ ( x , α, β ) − ℓ ( y , α, β ) | (cid:21) + εt − τ + ε | x | . By the regularity of the coefficients (see assumption ( A C :=2 C σ C + 2 + C f C + C , we have u ( τ, x ) − u ( τ, x ) ≤ tδ (cid:20) η (cid:18) C σ | x − y | + max x,α,β k σ − σ k (cid:19) + ηC f | x − y | + η | x − y | max x,α,β | f − f | + ω ( | x − y | ) + max x,α,β | ℓ − ℓ | (cid:21) + ε (cid:20) t − τ + Ctδ ( | x | + | y | + 2 C ) + | x | (cid:21) + ν max α,β tr (cid:0) ΣΘ (cid:1) ≤ tδ ˜ C (cid:20) η − γ/ (2 − γ ) + η max x,α,β k σ − σ k + η (1 − γ ) / (2 − γ ) max x,α,β | f − f | (cid:21) + tδ (cid:20) max x,α,β | ℓ − ℓ | + ω ( Cη − / (2 − γ ) ) (cid:21) + ε (cid:20) t − τ + Ctδ ( | x | + | y | + 2 C ) + | x | (cid:21) + ν max α,β tr (cid:0) ΣΘ (cid:1) . Letting ν → + and afterwards ε → + , by estimate 2.7, we infer u ( τ, x ) − u ( τ, x ) ≤ tδ ˜ C (cid:20) η − γ/ (2 − γ ) + η max x,α,β k σ − σ k + η (1 − γ ) / (2 − γ ) max x,α,β | f − f | (cid:21) + tδ (cid:20) max x,α,β | ℓ − ℓ | + ω ( Cη − / (2 − γ ) ) (cid:21) Letting δ → − and afterwards τ → t − , by the continuity of the functions u and u , forevery x ∈ R n , we deduce u ( t, x ) − u ( t, x ) ≤ t ˜ C (cid:20) η − γ/ (2 − γ ) + η max x,α,β k σ − σ k + η (1 − γ ) / (2 − γ ) max x,α,β | f − f | (cid:21) + t (cid:20) max x,α,β | ℓ − ℓ | + ω ( Cη − / (2 − γ ) ) (cid:21) Since η belongs to [1 , + ∞ ), we infer u ( t, x ) − u ( t, x ) ≤ t ˜ C (cid:20) η − γ/ (2 − γ ) + η (cid:18) max x,α,β k σ − σ k + max x,α,β | f − f | (cid:19)(cid:21) + t (cid:20) max x,α,β | ℓ − ℓ | + ω ( Cη − / (2 − γ ) ) (cid:21) r ∈ (0 , h ( s ) := rs + s − γ/ (2 − γ ) in [1 , + ∞ ) is lessor equal to 2 r γ/ (this value is attained in s = r − (2 − γ ) / ). Therefore, choosing η =[max x,α,β k σ − σ k + max x,α,β | f − f | ] − (2 − γ ) / , we conclude u ( t, x ) − u ( t, x ) ≤ t ˜ C (cid:18) max x,α,β k σ − σ k + max x,α,β | f − f | (cid:19) γ/ + t (cid:20) max x,α,β | ℓ − ℓ | + ω (cid:18) C (max x,α,β k σ − σ k + max x,α,β | f − f | ) / (cid:19)(cid:21) ≤ t ˜ C (cid:20) max x,α,β k σ − σ k γ + max x,α,β | f − f | γ/ (cid:21) + t (cid:20) max x,α,β | ℓ − ℓ | + ω (cid:18) C (max x,α,β k σ − σ k + max x,α,β | f − f | / ) (cid:19)(cid:21) for every x ∈ R n . Owing to the arbitrariness of the value t , one side of the inequality inour statement is completely proved. Being similar, the proof of the other one is omitted. ✷ This Section is devoted to illustrate an application of Theorem 2.1 to some classes ofparabolic Cauchy problems for degenerate HJBI operators.
Corollary 2.1
Assume that, besides our standing assumptions, for some ν > , thereholds min β ∈ B max α ∈ A {− tr ( a i ( x, α, β ) X ) + f i ( x, α, β ) · p + ℓ i ( x, α, β ) } ≥ ν | p | − C (2.13) for every ( x, p, X ) ∈ R n × R n × S n ( i = 1 , ). Then, there exists M > such that, forevery ( t, x ) ∈ Q ∞ , there holds | u ( t, x ) − u ( t, x ) | ≤ tM (cid:20) max x,α,β k σ − σ k + max x,α,β | f − f | / + max x,α,β | ℓ − ℓ | + ω (cid:18) C (max x,α,β k σ − σ k + max x,α,β | f − f | / ) (cid:19)(cid:21) , where u and u are respectively the solution to (Pi) with i = 1 and i = 2 . Remark 2.2
Relation (2.13) is fulfilled provided that there exists A ′ i ⊂ A such that σ i ( x, α, β ) = 0 ∀ α ∈ A ′ i , B (0 , ν ) ⊂ conv { f i ( x, α, β ) | α ∈ A ′ i } for every x ∈ R n , β ∈ B (here, B (0 , ν ) stands for the ball centered in with radius ν whileconv A is the convex hull of A ⊂ R n ). Proof of Corollary 2.1
A straightforward application of Theorem 2.1 yields thestatement provided that the functions u and u satisfy condition (2.2) with γ = 1. Letus prove this property by using some arguments of [4, Theorem II.1]. Assume that thereholds | u i ( t, x ) − u i ( t + h, x ) | ≤ Ch, ∀ ( t, x ) ∈ Q ∞ , h > , i = 1 , . (2.14)8n this case, relations (2.13) and (2.14) guarantee in viscosity sense C ≥ H ( x, Du i ( · , t ) , D u i ( · , t )) ≥ ν | Du i ( · , t ) | − C, in R n for all t ∈ [0 , + ∞ ), i = 1 ,
2. In particular, we have: | Du i | ≤ Cν − , which amounts to(2.2) with γ = 1.In conclusion, let us prove inequality (2.14). By estimate (2.1), we infer that thefunctions u i ( t + h, x ) ± Ch are respectively a super and a subsolution to (Pi). Applyingthe Comparison Principle, we accomplish the proof of estimate (2.14). ✷ This section is devoted to provide a continuous dependence estimate for the ergodic con-stant associated to the HJBI operator (1.2). Let us recall that, in the ergodic problem,we seek a constant U such that the equation H ( x, Dv, D v ) = U in R n (3.1)admits at least one solution v . For δ >
0, let us also introduce the approximated equation δw δ + H (cid:0) x, Dw δ , D w δ (cid:1) = 0 in R n . (3.2)Beside our standing assumptions, throughout this section, the operator H also fulfills( A
4) Periodicity: the functions σ , f and ℓ are Z n -periodic in x ;( A
5) Non-degeneracy: there exists a positive constant ν such that: a ( x, α, β ) ≥ νI, ∀ ( x, α, β ) ∈ R n × A × B. For later use, in the following Proposition, we shall collect several known propertiesof the ergodic problem.
Proposition 3.1
Under assumptions ( A )-( A ), the following properties hold: i ) There exists exactly one constant U such that equation (3.1) admits a bounded con-tinuous (and periodic) solution v . Moreover, v is unique up to an additive constant. ii ) Let u be the solution to the Cauchy problem (1.1) ; then, as t → + ∞ , u ( t, x ) /t converges to the ergodic constant U of equation (3.1) uniformly in x . iii ) The approximated equation (3.2) admits exactly one bounded continuous solution w δ : δ k w δ k ∞ ≤ max x,α,β | ℓ | . Moreover, as δ → + , δw δ and ( w δ − w δ (0)) convergerespectively to the ergodic constant U and to the solution v of (3.1) with v (0) = 0 . iv ) There exist two constants κ ∈ (0 , and K > , both depending only on the parame-ters of our assumptions (that is, independent of δ ) such that there holds k w δ − w δ (0) k C ,κ ≤ K (cid:18) x,α,β | ℓ | (cid:19) . roof of Proposition 3.1 The proof of points ( i ), ( ii ) and ( iii ) can be found in [3, Theorem 4.1] (see also [4,Theorem II.2] for operators of Bellman type). In fact, the statement of points ( ii ) and ( iii )are equivalent (see [2, Theorem 4], [3, Proposition 2.2] and also [4, Proposition VI.1] forBellman operators) while the first part of point ( i ) is only a sufficient condition for them(see [3, Proposition 7.2]).The proof of point ( iv ) can be easily obtained adapting to HJBI equations thearguments introduced by Arisawa and Lions [4, Theorem II.2] (see also [3, Theorem 4.1]for a similar result). Finally, we refer the reader to [3, proof of Theorem 4.1] for the specialform of the right-hand side. ✷ For i = 1 ,
2, consider the ergodic problemsmin β ∈ B max α ∈ A (cid:8) − tr (cid:0) a i ( x, α, β ) D v i (cid:1) + f i ( x, α, β ) · Dv i + ℓ i ( x, α, β ) (cid:9) = U i in R n (Ei)where the coefficients fulfill assumptions ( A A Theorem 3.1
Let U i be the unique ergodic constant for problem (Ei) ( i = 1 , ). Then,there exist a positive constant ˜ M such that there holds (cid:12)(cid:12) U − U (cid:12)(cid:12) ≤ ˜ M (cid:18) max x,α,β k σ − σ k + max x,α,β | f − f | (cid:19) + ω (max x,α,β k σ − σ k ) + max x,α,β | ℓ − ℓ | . Proof of Theorem 3.1
Let u i be the solution to problem (Pi) ( i = 1 , ii ), the statement follows from a straightforward application of Theorem 2.1provided that the solutions u and u fulfill condition (2.2) with γ = 1. In order to provethis fact, we denote by v i the unique bounded solution to equation (Ei) with v i (0) = 0and we introduce the function w i ( t, x ) := u i ( t, x ) + U i t , which is the unique solution tothe Cauchy problem ( ∂ t w i + min β ∈ B max α ∈ A (cid:8) − tr (cid:0) a i D w i (cid:1) + f i · Dw i + ℓ i (cid:9) = U i in Q ∞ w i (0 , x ) = 0 on R n . (3.3)Let us prove that w i is bounded in Q ∞ arguing as in [4, Theorem II.1]. For k := k v i k ∞ ,the functions v i ( x ) − U i t ± k are respectively a super- and a subsolution to the Cauchyproblem (Pi); hence, the Comparison Principle ensures v i ( x ) − k ≤ w i ( t, x ) ≤ v i ( x ) + k ∀ ( t, x ) ∈ Q ∞ , and, in particular: k w i k ∞ ≤ k . Furthermore, by standard regularity theory for parabolicequations (see [17, 33, 34]) and by Proposition 3.1-( iii ), the function w i fulfill hypothe-sis (2.2) with γ = 1 and, consequently, also u i fulfill hypothesis (2.2) with γ = 1. ✷ Proof of Theorem 3.1: alternative version.
We shall follow the arguments forthe Comparison Principle (see [16] and also [27] for continuous dependence estimates).For every positive η , define ψ ( x, y ) := w δ ( x ) − w δ ( y ) − η | x − y | (3.4)10here w iδ ( i = 1 ,
2) is the unique bounded (and periodic) solution to δw iδ + min β ∈ B max α ∈ A (cid:8) − tr (cid:0) a i ( x, α, β ) D w iδ (cid:1) + f i ( x, α, β ) · Dw iδ + ℓ i ( x, α, β ) (cid:9) = 0 in R n . (3.5)(Here, taking advantage of the periodicity of w iδ , the penalization term is simpler than theone in the proof of Theorem 2.1.) Owing to these properties of w iδ , we deduce that thereexists a point ( x , y ) ∈ R n × R n where the function ψ attains its global maximum.Let us now claim that, for C := 2 K (1 + max x,α,β,i | ℓ i | ) (where K is the constant intro-duced in Proposition 3.1-( iv )), there holds η | x − y | ≤ C. (3.6)Actually, we observe that the inequality ψ ( x , x ) + ψ ( y , y ) ≤ ψ ( x , y ) gives η | x − y | ≤ [ w δ ( x ) − w δ ( y )] + [ w δ ( x ) − w δ ( y )] ≤ K (1 + max x,α,β,i | ℓ i | ) | x − y | where the latter inequality is due to Proposition 3.1-( iv ); whence, estimate (3.6) easilyfollows.By [16, Theorem 3.2], for every ν >
0, there exist matrices
X, Y ∈ S n such that( η ( x − y ) , X ) ∈ J , + w δ ( x ) , ( η ( x − y ) , Y ) ∈ J , − w δ ( y ) (3.7) (cid:18) X − Y (cid:19) ≤ η (1 + 2 νη ) (cid:18) I − I − I I (cid:19) . Moreover, by the same arguments as those in the proof of Theorem 2.1, for every ( α, β ) ∈ A × B , from last inequality we deducetr ( a ( x , α, β ) X ) − tr ( a ( y , α, β ) Y ) ≤ η (1 + 2 νη ) k σ ( x , α, β ) − σ ( y , α, β ) k . Since w δ (respectively, w δ ) is a subsolution (resp., a supersolution) to equation (3.5) with i = 1 (resp., i = 2), by relations (3.7), we infer δw δ ( x ) + min β ∈ B max α ∈ A {− tr ( a ( x , α, β ) X ) + ηf ( x , α, β ) · ( x − y ) + ℓ ( x , α, β ) } ≤ δw δ ( y ) + min β ∈ B max α ∈ A {− tr ( a ( y , α, β ) Y ) + ηf ( y , α, β ) · ( x − y ) + ℓ ( y , α, β ) } ≥ . Taking into account the last three inequalities, by the same calculations as before, weobtain δ (cid:0) w δ ( x ) − w δ ( y ) (cid:1) ≤ η (1 + 2 νη ) max α,β k σ ( x , α, β ) − σ ( y , α, β ) k + η | x − y | max α,β | f ( x , α, β ) − f ( y , α, β ) | + max α,β | ℓ ( x , α, β ) − ℓ ( y , α, β ) | . Letting ν → + , by the regularity of the coefficients (see assumption ( A δ (cid:0) w δ ( x ) − w δ ( y ) (cid:1) ≤ η (cid:18) C σ | x − y | + max x,α,β k σ − σ k (cid:19) + ηC f | x − y | + η | x − y | max x,α,β | f − f | + ω ( | x − y | ) + max x,α,β | ℓ − ℓ |≤ C (2 C σ + C f ) η − + 2 η max x,α,β k σ − σ k + C max x,α,β | f − f | + ω ( Cη − ) + max x,α,β | ℓ − ℓ | . (3.8)11e separately consider the cases σ = σ and σ = σ . If σ = σ , as η → + ∞ , lastinequality reads δ (cid:0) w δ ( x ) − w δ ( y ) (cid:1) ≤ C max x,α,β | f − f | + max x,α,β | ℓ − ℓ | ;finally, as δ → + , we conclude U − U ≤ C max x,α,β | f − f | + max x,α,β | ℓ − ℓ | . If σ = σ , we choose η = C/ max x,α,β k σ − σ k ; even though, in general, this is not theoptimal choice for minimizing the right-hand side of (3.8), the final estimate will behavewith respect to C in the desired manner for the purposes of section 3.1. For ˜ C := 2 C σ +2 + C f , we have: δ (cid:0) w δ ( x ) − w δ ( y ) (cid:1) ≤ C (cid:18) ˜ C max x,α,β k σ − σ k + max x,α,β | f − f | (cid:19) + ω (max x,α,β k σ − σ k )+ max x,α,β | ℓ − ℓ | ;finally, as δ → + , we conclude U − U ≤ C (cid:18) ˜ C max x,α,β k σ − σ k + max x,α,β | f − f | (cid:19) + ω (max x,α,β k σ − σ k ) + max x,α,β | ℓ − ℓ | . Hence, one side of the inequality of our statement is proved. Reversing the role of w δ and w δ , one can easily obtain the other side; therefore, we shall omit its proof. ✷ Remark 3.1
By the calculations of the proof above, a good choice is ˜ M = 2 K (1 +max x,α,β,i | ℓ i | )(2 C σ + 2 + C f ) , where K is the constant introduced in Proposition 3.1-( iv )while C σ and C f are the Lipschitz constants of σ and f respectively (see assumption ( A )). We consider the following singular perturbation problems ( ∂ t u ε + H (cid:16) x, y, D x u ε , D y u ε ε , D xx u ε , D yy u ε ε , D xy u ε √ ε (cid:17) = 0 in (0 , T ) × R n × R m u ε (0 , x, y ) = h ( x ) on R n × R m (3.9)where u ε = u ε ( t, x, y ) is a real function, ε ∈ (0 ,
1) and H ( x, y, p, q, X, Y, Z ) := min β ∈ B max α ∈ A {− tr( M X ) − tr( N Y ) − EZ ) + F · q + G · p + L } with φ = φ ( x, y, α, β ) for every φ = M, N, E, F, G, L . The aim of this section is to studythe asymptotic behavior of u ε as ε → + . For the wide literature on this matter, we referthe reader to the monographs by Bensoussan [10], Dontchev and Zolezzi [19], Kokotovi´c,Khalil and O’Reilly [29], Alvarez and Bardi [3] and references therein. Let us only recallthat these problems arise in zero-sum two-persons stochastic differential games (1.3)-(1.4)12here the state variable “splits” in the slow one x and in the fast one y . For the controlsystem dx s = G ( x s , y s , α s , β s ) + √ x s , y s , α s , β s ) dW s , x = xdy s = ε − f ( x s , y s , α s , β s ) + √ ε − Σ( x s , y s , α s , β s ) dW s , y = y and the cost functional P ( t, x, y, α, β ) := E ( x,y ) (cid:20)Z t L ( x s , y s , α s , β s ) ds + h ( x t , y t ) (cid:21) , the lower value function u ε is a viscosity solution to problem (3.9) with M = T ΞΞ, N = T ΣΣ and E = T ΣΞ.Throughout this section, we shall assume:( S A and B are two compact metric spaces.( S M = T ΞΞ, N = T ΣΣ, E = T ΣΞ. The functions Ξ, Σ, F , G , L and h are boundedcontinuous functions in R n × R m × A × B with values respectively in M n,p , M m,p , R m , R n , R and R , namely, for some C >
0, there holds: k φ k ∞ ≤ C for φ = M, N, E, F, G, L .All these functions are Z m -periodic in y .( S
3) The functions Ξ, Σ, F and G (respectively, L and h ) are Lipschitz (resp., uniformly)continuous in ( x, y ) uniformly in ( α, β ) that is: there exists a positive constant C φ and a modulus of continuity ω ψ such that | φ ( x , y , α, β ) − φ ( x , y , α, β ) | ≤ C φ ( | x − x | + | y − y | ) | ψ ( x , y , α, β ) − ψ ( x , y , α, β ) | ≤ ω ψ ( | x − x | + | y − y | )for every ( x i , y i ) ∈ R n × R m ( i = 1 ,
2) and ( α, β ) ∈ A × B , with φ = Ξ , Σ , F, G and ψ = L, h .( S
4) There exists ν > x, y, α, β ) ∈ R n × R m × A × B , there holds M ( x, y, α, β ) ≥ νI, N ( x, y, α, β ) ≥ νI. Let us recall from [2, 3] the definition of the effective
Hamiltonian H : for ev-ery ( x, p, X ) ∈ R n × R n × S n fixed, the value − H ( x, p, X ) is the ergodic constant for H ( x, y, p, q, X, Y,
0) with respect to the variable y . In other words, for δ >
0, the problem δw δ + H (cid:0) x, y, p, D y w, X, D yy w, (cid:1) = 0 in R m , w δ = w δ ( y ) periodic (3.10)admits exactly one continuous solution and moreover, as δ → + , δw δ converges uniformlyin y to the value − H ( x, p, X ). We refer the reader to Proposition 3.1 for several prop-erties of problem (3.10). In particular, let us observe (see also [3, Theorem 4.1]) thatProposition 3.1-( iv ) can be stated as follows: there exist κ ∈ (0 ,
1] and
K > k w δ − w δ (0) k C ,κ ≤ K (cid:0) | p | + k X k (cid:1) . (3.11)13 roposition 3.2 The solution u ε to problem (3.9) converges locally uniformly in [0 , T ) × R n × R m to the unique solution u = u ( t, x ) to the effective problem (cid:26) u t + H ( x, D x u, D xx u ) = 0 in (0 , T ) × R n u (0 , x ) = h ( x ) on R n . (3.12) Proof of Proposition 3.2
We shall argue using several results established by Alvarezand Bardi [2, 3]. Invoking [3, Theorem 2.9] (see also [2, Corollary 2]), it suffices to provethat the Comparison Principle applies to the effective problem (3.12). To this end, byvirtue of the results by Ishii and Lions [24], we need the following two properties: ( i ) H isuniformly elliptic, ( ii ) for some constant K and for some modulus of continuity ¯ ω , thereholds (cid:12)(cid:12) H ( x , p , X ) − H ( x , p , X ) (cid:12)(cid:12) ≤ C k X − X k + C | p − p | + ¯ ω ( | x − x | )+ K | x − x | (1 + | p | ∨ | p | + k X k ∨ k X k ) (3.13)for every ( x i , p i , X i ) ∈ R n × R n × S n ( i = 1 , i = 1 , − H ( x i , p i , X i ) is the ergodic constant for the problemmin β ∈ B max α ∈ A (cid:8) − tr( N ( x i , y, α, β ) D yy w i ) + D y w i · F ( x i , y, α, β ) − tr( M ( x i , y, α, β ) X i )+ p i · G ( x i , y, α, β ) + L ( x i , y, α, β ) } = − H ( x i , p i , X i ) . Applying Theorem 3.1 with the variable x replaced by y and σ i ( · , α, β ) = Σ( x i , · , α, β ) , f i ( · , α, β ) = F ( x i , · , α, β ) ℓ i ( · , α, β ) = − tr( M ( x i , · , α, β ) X i ) + p i · G ( x i , · , α, β ) + L ( x i , · , α, β ) ω ( r ) = [ C M ( k X k ∨ k X k ) + C G ( | p | ∨ | p | )] r + ω L ( r )for some constant ˜ M , we infer (cid:12)(cid:12) H ( x , p , X ) − H ( x , p , X ) (cid:12)(cid:12) ≤ ˜ M (cid:18) max y,α,β k Σ( x , y, α, β ) − Σ( x , y, α, β ) k + max y,α,β | F ( x , y, α, β ) − F ( x , y, α, β ) | (cid:19) + [ C M ( k X k ∨ k X k ) + C G ( | p | ∨ | p | )] max y,α,β k Σ( x , y, α, β ) − Σ( x , y, α, β ) k + ω L (cid:18) max y,α,β k Σ( x , y, α, β ) − Σ( x , y, α, β ) k (cid:19) + max y,α,β | tr [ M ( x , y, α, β ) X − M ( x , y, α, β ) X ] | + max y,α,β | p · G ( x , y, α, β ) − p · G ( x , y, α, β ) | + max y,α,β | L ( x , y, α, β ) − L ( x , y, α, β ) | . Taking into account the regularity of the coefficients (see assumption ( S (cid:12)(cid:12) H ( x , p , X ) − H ( x , p , X ) (cid:12)(cid:12) ≤ C k X − X k + C | p − p | + ω L ( C Σ | x − x | ) + ω L ( | x − x | ) + ˜ M ( C Σ + C F ) | x − x | + | x − x | [ C M ( k X k ∨ k X k ) + C G ( | p | ∨ | p | )] C Σ + C M | x − x | ( k X k ∧ k X k )+ C G | x − x | ( | p | ∧ | p | ) . x,α,β,i | ℓ i | ≤ C (1 + | p | ∨ | p | + k X k ∨ k X k ), by Remark 3.1, wecan choose ˜ M := 2 K ( C + 1)(2 C + 2 + C F ) (1 + | p | ∨ | p | + k X k ∨ k X k ) . Hence, the previous inequality becomes (cid:12)(cid:12) H ( x , p , X ) − H ( x , p , X ) (cid:12)(cid:12) ≤ C k X − X k + C | p − p | + ω L ( C Σ | x − x | )+ ω L ( | x − x | )+ K | x − x | (1 + | p | ∨ | p | + k X k ∨ k X k )for some constant K independent of ( x i , p i , X i ). Finally, choosing ¯ ω ( r ) := ω L ( C Σ r )+ ω L ( r ),our claim (3.13) is completely proved. ✷ References [1] V.I. Arnold and A. Avez.
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