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76 Bokanowski et al.1. General frame work: motivationWe deal with optimal control problems with state constraints in finite horizon:P x,s⎧⎨⎩min ϕ(y x,s (T )),ẏ x,s (t) = f(y x,s (t), u(t)) for t ∈ [s, T ] and y x,s (s) = x, x ∈ IR nu(t) ∈ U a.e .where ϕ : IR n → IR is bounded lower semi continuous, f : IR n × IR m → IR n is boundedlipshitz, U is a compact of IR m and y x,s is the trajectory starting at x at time s and evolvingunder the control u.Let us define the value function: V (x, s) = val P x,s .If we set W (x, s) = V (x, T − s), then W is a solution in a viscosity sense [1]-[2] of theHamilton-Jacobi-EquationW t (x, t) − minu∈U [f(x, u).W x(x, t)] = 0, W (x, 0) = ϕ(x), x ∈ IR n . (1)We are interested in solving the above HJB equation and then reconstructing the optimaltrajectories for P x,s . In our case, the exact value function is discontinuous. Thisoccurs in the case of “Rendez-Vous problem” where the aim is to reach a target C at agiven time T. The associated final cost ϕ satisfies{0 if ∃u such that yx,s (T ) ∈ C,ϕ(x) =1 otherwise.Note that finite difference and semi-lagrangian schemes give good approximations of thevalue function when it is continuous. In fact the numerical diffusion induced by interpolationin these schemes stands acceptable in this case. But as we deal with discontinuousvalue functions, such schemes become no more suitable.In our approach, we use an antidissipative scheme for HJB equations with discontinuoussolutions. This scheme is called UltraBee scheme. It was studied by Désprès andLagoutière [4] for transport equations and generalized by Bokanowski and Zidani [3] forHJB equations. An interesting feature of the UltraBee scheme is a uniform convergenceproperty with respect to time (satisfied by noother known scheme) and an exact advectionproperty for a particular class of step functions.Our idea is to use the UltraBee scheme on an adaptative grid. This choice is motivatedby many practical and numerical reasons: the UltraBee scheme approximates naturallydiscontinuities with a very good precision. As we apply it on an adaptative grid, we gainan additional precision and optimize the number of cells on the grid. Management of thegrid cells is facilitated by the use of linear quadtrees. We apply our method to an optimalTAMTAM –Tunis– 2005