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ϑ (T, x) = Φ (x) in R d
where H(t, x, p, Q) := sup a∈U −b(t, x, a) · p − Tr([σσ T ](t, x, a)Q). In **the** case when
**the** **the** drift b and **the** diffusion σ are bounded and when **the** value function ϑ is itself
bounded, error estimates **of** monotone schemes have been obtained **first** by Krylov [ 27 ] for a case where σ is a constant function. These results were developed further in [ 28 , 29 , 6 , 7 , 8 ] by introducing new tools that allow to consider **the** case where σ can depend on time, space and also on **the** control variable. Several other extensions **of** **the** theory have been analysed in **the** literature, let us mention some **of** these extensions for stopping-game problems [ 13 ], for impulsive control systems [ 14 ], for integro-partial differential HJB **equations** [ 18 , 9 , 10 ], and for a general class **of** coupled HJB systems [ 16 ]. Note also that **the** case **of** fully uniformly elliptic operators have been also studied in [ 17 ] using a different approach than **the** one introduced by Krylov. Here, we extend **the** theory **of** error estimates to an unbounded Lipschitz setting. **The** proof is still based on “Krylov regularization” and on some refined consistency estimates. To **the** best **of** our knowledge, this is **the** **first** result in **the** case where b, σ and **the** solution to **the** HJB equation itself are unbounded with respect to **the** space variable (with linear growth).

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We show that if ˜ V 6 V is not equal to V , then **the** set **of** differentiability points x 0 such that H(x 0 , ∇ ˜ V ) > 0 is **of** positive measure (Proposition 8.11). To find such points, we randomly generate a number **of** points and select those at which **the** value **of** Hamiltonian is positive. This also takes a linear computation time (with respect to **the** current number **of** basis functions) for each sampled point. **The** advantages **of** this method are as follows. **First** adding a basis function only requires a number **of** arithmetical operations that is linear to **the** number **of** basis functions. Thus, such an ele- mentary step can be done quickly. Secondly, **the** time discretization step is automatically adapted by **the** method, and each added basis function is guaranteed to be useful in some region **of** **the** state space, at least at **the** step when we add it. **Numerical** experiments (Section 8.4) show that to reach a back- substitution **of** **the** same **order**, **the** new algorithm takes much less time than **the** SDP based pruning algorithm (103s vs >10h for **the** instance used in [MDG08]). We might make a connection between this randomized method for infinite horizon optimal control problems and **the** so-called "point-base value iteration" [CLZ97, ZZ01] for Partially Observable Markov Decision Process (POMDP). Al- though developed in very different settings, **the** two methods share **the** idea that improving quickly **the** approximate value function at randomly generated witness points leads to a better performance than improving uniformly but slowly **the** approximated value function.

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Example 5.6. We consider again **the** problem **of** Example 5.5 and use **the** filtering scheme in which Π A
k was chosen as **the** cubic spline interpolation already presented in Example
5.5, cf. Figure 5.2 (right). **The** **numerical** parameters were chosen as space and time step k = h = 0.06, resulting in 51 nodes, and **the** minimum in (4.2) was computed over a discrete set **of** controls U discretizing **the** interval [0, 0.4] with 51 equidistant values. Figure 5.3 shows **the** evolution **of** **the** value iteration plotting **the** difference kw j+1 − w j k depending on j for different filtering parameter ε. One clearly observes that **the** iteration converges to increasingly smaller sets for shrinking filtering parameter, i.e., for increasing weight on **the** **first** **order** monotone scheme. Obviously, **the** filtering significantly improves **the** convergence behavior **of** **the** value iteration.

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For Lemma 4.2, we refer **the** reader to [1, Theorem 9.2, p.90]. References
[1] Yves Achdou, Guy Barles, Hitoshi Ishii, and Grigory L. Litvinov. **Hamilton**-**Jacobi** equa- tions: approximations, **numerical** analysis and applications, volume 2074 **of** Lecture Notes in Mathematics. Springer, Heidelberg; Fondazione C.I.M.E., Florence, 2013. Lecture Notes from **the** CIME Summer School held in Cetraro, August 29–September 3, 2011, Edited by Paola Loreti and Nicoletta Anna Tchou, Fondazione CIME/CIME Foundation Subseries. [2] M. Bardi and I. Capuzzo Dolcetta. Optimal control and viscosity solutions **of** **Hamilton**-

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guaranted when f is bounded even for simple ordinary (y ′ (t) = −1) or partial
(u t − u xx + |u x | = −1) **equations**. It is then be hopeless for fully nonlinear PDE as
we can note from (1.6) where u(x, t) could go to −∞ for some f < 0.
Taking this into account, we find that **the** study **of** **the** long time behavior **of** **the** solution u **of** (1.1)-(1.2)-(1.3) **first** lead us to **the** study **of** a stationary ergodic problem. More precisely, like in **the** work **of** lasry & Lions [37], we are interested in findind an appropriate constant c such that **the** function u(·, t)+ct remains bounded and c is **the** unique constant for which **the** stationnary equation with state constraint boundary condition

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Abstract. We introduce a new class **of** “filtered” schemes for some **first** **order** non-linear **Hamilton**-**Jacobi**- Bellman **equations**. **The** work follows recent ideas **of** Froese and Oberman (SIAM J. Numer. Anal., Vol 51, pp.423- 444, 2013). **The** proposed schemes are not monotone but still satisfy some -monotone property. Convergence results and precise error estimates are given, **of** **the** **order** **of** √ ∆x where ∆x is **the** mesh size. **The** framework allows to construct finite difference discretizations that are easy to implement, high–**order** in **the** domains where **the** solution is smooth, and provably convergent, together with error estimates. **Numerical** tests on several examples are given to validate **the** approach, also showing how **the** filtered technique can be applied to stabilize an otherwise unstable high–**order** scheme.

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where A may be degenerate. In both cases, we provide regularizing effects for bounded from below solutions. **The** main improvement in **the** assumptions is easy to describe in **the** coercive case since we just require that A, f are continuous in x (no uniform continuity assumptions) and, in particular, f may have some growth at infinity. In **the** non-coercive case, analogous results hold except that we have to impose far more restrictive assumptions on **the** t-dependence **of** **the** equation.

tions **of** controls that drive **the** solution from a prescribed initial state to zero at a large enough controllability time. In [Cindea & M¨ unch, A mixed formulation for **the** direct ap- proximation **of** **the** control **of** minimal L 2 -norm for linear type wave **equations**], we have
introduced a space-time variational approach ensuring strong convergent approximations with respect to **the** discretization parameter. **The** method, which relies on generalized ob- servability inequality, requires H 2 -finite element **approximation** both in time and space.

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If H is convex in p and A is independent **of** p, our proof implies that, for any p ∈ R d , **the**
limit lim δ→9 δv δ,p (0, ·) exists in probability; see Proposition 3.10. This result and its **the**
proof are very much in **the** flavor **of** [14].
Finally we note that our arguments also yield **the** existence **of** a corrector in some directions and, thus, homogenization, for nonconvex Hamiltonians and p dependent A. More precisely, for any direction p, there exists a constant c such that p belongs to **the** convex hull **of** directions p ′ for which a corrector exists with associated homogenized constant equal to c;

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Both [1] and [14] contain **the** **first** comparison and uniqueness results: in [1], suitably modified geodesic distances are used in **the** doubling variables method for proving comparison theorems under rather strong continuity as- sumptions. In [14], Imbert, Monneau and Zidani used a completely different argument based on **the** explicit solution **of** a related optimal control problem, which could be obtained because it was assumed that **the** Hamil- tonians associated with each edge did not depend on **the** state variable.

1.1. General framework. In this section we give **the** general setting **of** this article. 1.1.1. Assumptions on **the** space. In all **the** paper, (X, d) will be a complete and separable metric space in which closed balls are compact. This latter assumption could be removed at **the** expense **of** additional standard technicalities. We will sometimes assume that (X, d) is a geodesic space, meaning that for every two points x, y ∈ X there is at least one curve (γ t ) t∈[0,1] with γ 0 = x, γ 1 = y and such that d(γ s , γ t ) = |t − s|d(x, y) for all s, t ∈ [0, 1]. Such

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Very few results applicable to highway traffic are available for control **of** **first** **order** hyperbolic conservation laws. Differential flatness [47] has been successfully applied to Burgers equation (and therefore to **the** Lighthill-Whitham-Richards equation) in [72] **order** to avoid **the** formation **of** such shockwaves. This analysis does not so far extend to **the** presence **of** shocks. Lyapunov based techniques have also been applied to **the** Burgers equation [59]. Adjoint based methods have been successfully applied to networks **of** Lighthill- Whitham-Richards **equations** in [54]; these results seem so far **the** most promising, but they do not have guarantees to provide an optimal control policy. Questions **of** interest in controlling **first**-**order** partial differential **equations**, and in particular, Lighthill-Whitham-Richards **equations**, are still open and difficult to solve due to **the** presence **of** shocks occurring in **the** solutions **of** these partial differential **equations**.

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Now, we turn to **the** proof **of** **the** interior H¨older estimates on v with respect to **the** x-variable uniformly in t ∈ (0, +∞) which is based on an idea introduced by Ishii and Lions [12]. This idea has been already used for instance in Barles [1] and Barles and Souganidis [7] to show gradient estimates **of** viscosity solutions to quasilinear elliptic ans parabolic PDE with Lipschitz initial conditions, by Barles and Da Lio [5] to prove local H¨older estimates up to **the** boundary **of** bounded solutions to fully non linear elliptic PDE with Neumann boundary conditions and by Da Lio [10] to obtain C 0,ν -

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