M. Assellaou, O. Bokanowski, and H. Zidani, Probabilistic safety reachability analysis, 2013.

M. Assellaou, O. Bokanowski, and H. Zidani, Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets, Discrete and Continuous Dynamical Systems, vol.35, issue.9, pp.3933-3964, 2015.
DOI : 10.3934/dcds.2015.35.3933
URL : https://hal.archives-ouvertes.fr/hal-00998371

M. Assellaou, O. Bokanowski, A. Desilles, and H. Zidani, Feedback control analysis for state constrained control problem with maximum cost, in preparation, 2015.

M. Assellaou, O. Bokanowski, A. Desilles, and H. Zidani, Optimal feedback control for the abort landing problem in presence of windshear , in preparation, BIBLIOGRAPHIE, 2015.

A. E˜—te, S. Emin, M. €r—ndini, J. Vygeros, and S. ƒ—stry, Computational approaches to reachability analysis of stochastic hybrid systems, Hybrid Systems, p.417, 2007.

A. E˜—te, M. €r—ndini, J. Vygeros, and S. ƒ—stry, Probabilistic reachability and safety for controlled discrete time stochastic hybrid systems

A. Elt—rovi™i, O. Fok—nowski, and H. id—ni, A general hamiltonjacobi framework for non-linear state-constrained control problems, ESAIM : Control, Optimisation and Calculus of Variations, vol.19, issue.4, pp.337357-2013

M. Elthoff and O. ƒturs˜erg, fuss : Safety assessment of autonomous cars using verication techniques, American Control Conference ACC'07, p.41544159, 2007.

M. Elthoff and O. ƒturs˜erg, fuss : Safety assessment for stochastic linear systems using enclosing hulls of probability density functions, 10th European Control Conference, 2009.

S. Emin, A. E˜—te, M. €r—ndini, S. ƒ—stry, and J. Vygeros, Reachability Analysis for Controlled Discrete Time Stochastic Hybrid Systems, Lecture Notes in Computer Science LNCS, issue.3927, p.4963, 2006.
DOI : 10.1007/11730637_7

J. Eu˜in, gellin— : Dierential inclusions. set-valued maps and viability theory, Grundlehren Math. Wiss, issue.264, 1984.

J. P. Eu˜in and H. Pr—nkowsk—, Set-Valued Analysis. Systems and Control : Foundations and Applications, 1990.

J. P. Eu˜in and H. Pr—nkowsk—, The viability kernel algorithm for computing value functions of innite horizon optimal control problems, Journal of mathematical analysis and applications, vol.201, issue.2, p.555576, 1996.

M. F—rdi and I. C. , hol™ett— : Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, 2008.

G. F—rles, Solutions de viscosité des équations de Hamilton-Jacobi, p.164, 1994.

G. F—rles and E. R. T—ko˜sen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations, ESAIM, vol.36, issue.1, p.3354, 2002.

G. F—rles and E. R. T—ko˜sen, Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations, SIAM J. Numer. Anal, vol.43, issue.2, p.540558, 2005.

G. F—rles and E. R. T—ko˜sen, Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations, Mathematics of Computations, issue.240, p.7618611893, 2007.

G. F—rles and P. E. , ƒoug—nidis : Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic analysis, p.271283, 1991.

E. N. F—rron and H. Sshii, The bellman equation for minimizing the maximum cost. Nonlinear Analysis : Theory, Methods & Applications, pp.1067-1090, 1989.

E. N. F—rron and R. Tensen, Semicontinuous viscosity solutions for hamiltonjacobi equations with convex hamiltonians, Communications in Partial Dierential Equations, vol.15, issue.12, p.293309, 1990.

R. E. Fellm—n, On the theory of dynamic programming, Proceedings of the National Academy of Sciences of the United States of America, vol.38, issue.8, p.716, 1952.

R. E. Fellm—n and S. E. Hreyfus, Applied dynamic programming, 1962.

R. E. Fellm—n and R. E. U—l—˜—, Dynamic programming and modern control theory, 1965.

I. H. Fisw—s, E. R. T—ko˜sen, and K. H. , u—rlsen : Dierence-quadrature schemes for nonlinear degenerate parabolic integro-pde, SIAM Journal on Numerical Analysis, vol.48, issue.3, p.11101135, 2010.

I. H. Fisw—s, E. R. T—ko˜sen, and K. H. U—rlsen, Viscosity solutions for a system of integro-pdes and connections to optimal switching and control of jump-diusion processes, Applied Mathematics and Optimization, vol.62, issue.1, p.4780, 2010.

F. FlѪk, Īholes : The pricing of options and corporate liabilities. The journal of political economy, p.637654, 1973.

O. Fok—nowski, A. Fri—ni, and H. id—ni, Minimum time control problems for non-autonomous dierential equations, Systems & Control Letters, issue.10, pp.58-742746, 2009.

O. Fok—nowski, N. Por™—del, and H. id—ni, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption, SIAM Journal on Control and Optimization, vol.48, issue.7, p.42924316, 2010.

O. Fok—nowski, A. €i™—relli, and H. id—ni, Dynamic programming and error estimates for stochastic control problems with maximum cost, 2014.

J. F. Fonn—ns, S. W—roso, and H. id—ni, Error estimates for stochastic differential games : the adverse stopping case, IMA journal of numerical analysis, vol.26, issue.1, p.188212, 2006.

J. F. Fonn—ns, S. W—roso, and H. id—ni, Error estimates for a stochastic impulse control problem, Applied Mathematics and Optimization, vol.55, issue.3, pp.327-357, 2007.

B. Fou™h—rd, R. Ilie, and N. „ouzi, Stochastic target problems with controlled loss, SIAM, vol.48, issue.5, p.31233150, 2008.

A. Fri—ni, F. G—milli, and H. id—ni, Approximation schemes for monotone systems of nonlinear second order partial dierential equations : convergence result and error estimate, Dierential Equations and Applications, p.297317, 2012.

R. Fulirs™h, F. Wontrone, and H. J. €es™h, Abort landing in the presence of windshear as a minimax optimal control problem. I. Necessary conditions

R. Fulirs™h, F. Wontrone, and H. J. €es™h, Abort landing in the presence of windshear as a minimax optimal control problem. II. Multiple shooting and homotopy, J. Optim. Theory Appl, vol.70, issue.2, p.223254, 1991.

L. A. G—ff—relli and P. E. , ƒoug—nidis : A rate of convergence for monotone nite dierence approximations to fully nonlinear, uniformly elliptic pdes, Communications on Pure and Applied Mathematics, vol.61, issue.1, p.117, 2008.

F. G—milli, p—l™one : An approximation scheme for the optimal control of diusion processes. RAIRO Modél, Math. Anal. Numér, vol.29, issue.1, pp.97-122, 1995.

F. G—milli and E. R. T—ko˜sen, A nite element like scheme for integropartial dierential hamilton-jacobi-bellman equations, SIAM Journal on Numerical Analysis, vol.47, issue.4, p.24072431, 2009.

P. G—rd—li—guet, M. uin™—mpoix, and P. ƒ—inte€ierre, Optimal times for constrained nonlinear control problems without local controllability, Applied Mathematics and Optimization, vol.36, issue.1, p.2142, 1997.

P. G—rd—li—guet and M. uin™—mpoix, ƒ—intE€ierre : Numerical schemes for discontinuous value functions of optimal control. Set-Valued Analysis, p.111126, 2000.

F. H. Gl—rke, Optimization and nonsmooth analysis, Siam, vol.5, 1990.

R. Gour—nt, E. Ss——™son, and M. ‚ees, On the solution of nonlinear hyperbolic dierential equations by nite dierences, Communications on Pure and Applied Mathematics, vol.5, issue.3, p.243255, 1952.

M. G. Gr—nd—ll, H. Sshii, and P. L. , vions : User's guide to viscosity solutions of second order partial dierential equations, Bulletin of the American Mathematical Society, vol.27, issue.1, p.167, 1992.

M. G. Gr—nd—ll and P. L. , vions : Condition d'unicité pour les solutions généralisées des équations de hamilton-jacobi du premier ordre

M. G. Gr—nd—ll and P. L. , vions : Viscosity solutions of hamilton-jacobi equations. Transactions of the, p.142, 1983.

M. G. Gr—nd—ll and P. L. , vions : Two approximations of solutions of hamilton-jacobi equations, Mathematics of Computation, vol.43, issue.167, p.119, 1984.

F. H—-vio and O. Vey, Uniqueness results for convex Hamilton-Jacobi equations under p > 1 growth conditions on data, Applied Math. and Optimization, vol.63, issue.3, p.309339, 2011.

K. He˜r—˜—nt and E. R. T—ko˜sen, Semi-Lagrangian schemes for linear and fully non-linear diusion equations, Mathematics of Computations, vol.82, issue.283, p.14331462, 2013.

F. Hel—rue and S. Wenozzi, Density estimates for a random noise propagating through a chain of dierential equations, J. Funct. Anal, vol.259, issue.6, pp.1577-1630, 2010.

I. C. Hol™ett— and H. Sshii, Approximate solutions of the bellman equation of deterministic control theory, Applied Mathematics and Optimization, vol.11, issue.1, p.161181, 1984.

I. C. Hol™ett— and P. L. , vions : Hamilton-jacobi equations with state constraints. Transactions of the, pp.643-683, 1990.

M. P—l™one, A numerical approach to the innite horizon problem of deterministic control theory, Applied Mathematics and Optimization, vol.15, issue.1, p.113, 1987.

M. P—l™one, The minimum time problem and its applications to front propagation. Motion by mean curvature and related topics (Trento, 1992), de Gruyter, p.7088, 1994.

M. P—l™one, Numerical solution of dynamic programming equations. Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman equations, 1997.

M. P—l™one, Recent results in the approximation of nonlinear optimal control problems, Large-Scale Scientic Computing, p.1532, 2014.

M. P—l™one and R. Perretti, Discrete time high-order schemes for viscosity solutions of hamilton-jacobi-bellman equations, Numerische Mathematik, vol.67, issue.3, p.315344, 1994.

M. P—l™one and R. Perretti, Semi-Lagrangian approximation schemes for linear and Hamilton-Jacobi equations, SIAM, 2013.

M. P—l™one and T. Qiorgi, An approximation scheme for evolutive hamiltonjacobi equations In Stochastic analysis, control, optimization and applications, Bibliography, vol.167, p.289303, 1999.

M. P—l™one, T. Qiorgi, and P. Voreti, Level sets of viscosity solutions : some applications to fronts and rendez-vous problems, SIAM Journal on Applied Mathematics, vol.54, issue.5, p.13351354, 1994.

W. H. Pleming and M. H. ƒoner, Controlled Markov processes and viscosity solutions, volume 25 de Stochastic Modelling and Applied Probability, 2006.

H. Pöllmer and P. Veukert, Quantile hedging, SIAM, vol.3, issue.3, p.251273, 1999.

H. Pr—nkowsk— and S. €l—sk—™z, Semicontinuous solutions of hamilton jacobi bellman equations with degenerate state constraints, Journal of Mathematical Analysis and Applications, vol.251, issue.2, pp.818-838, 2000.

H. Pr—nkowsk— and R. B. , †inter : Existence of neighboring feasible trajectories : Applications to dynamic programming for state-constrained optimal control problems, Journal of Optimization Theory and Applications, vol.104, issue.1, p.2040, 2000.

D. Qore—™ and O. ƒere—, Mayer and optimal stopping stochastic control problems with discontinuous cost, J. Math. Anal. Appl, vol.380, issue.1, p.327342, 2011.

A. R—rten, B. Ingquist, S. Ysher, and S. R. Gh—kr—v—rthy, Uniformly high order accurate essentially non-oscillatory schemes, iii, Journal of computational physics, vol.71, issue.2, p.231303, 1987.

U. G. R—ussm—nn and J. P. Vepeltier, On the existence of optimal controls, SIAM Journal on Control and Optimization, vol.28, issue.4, p.851902, 1990.

H. Sshii, Uniqueness of unbounded viscosity solution of hamilton-jacobi equations, Indiana University Mathematics Journal, vol.33, issue.5, p.721748, 1984.

H. Sshii, A boundary value problem of the dirichlet type for hamilton-jacobi equations, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, p.105135, 1989.

H. Sshii, On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic pde's. Communications on pure and applied mathematics, p.1545, 1989.

H. Sshii, Fully nonlinear oblique derivative problems for nonlinear second-order elliptic pdes. Duke Math, J, vol.62, issue.3, p.633661, 1991.

H. Sshii and S. Uoike, A new formulation of state constraint problems for rst-order pdes, SIAM Journal on Control and Optimization, vol.34, issue.2, p.554571, 1996.

H. Sshii and P. L. , vions : Viscosity solutions of fully nonlinear second-order elliptic partial dierential equations, Journal of Dierential equations, vol.83, issue.1, p.2678, 1990.

R. Tensen, The maximum principle for viscosity solutions of fully nonlinear second order partial dierential equations. Archive for Rational Mechanics and Analysis, p.127, 1988.

N. V. Urylov, On the rate of convergence of nite dierence approximation for Bellman's equation, St. Petersburg Math. J, vol.9, p.639650, 1997.

N. V. Urylov, On the rate of convergence of nite-dierence approximations for Bellman's equations with variable coecients. Probability Theory and Related Fields, p.116, 2000.

N. V. Urylov, Mean value theorems for stochastic integrals, Ann. Probab, vol.29, issue.1, p.385410, 2001.

H. J. Uushner, Existence results for optimal stochastic controls, Journal of Optimization Theory and Applications, vol.15, issue.4, p.347359, 1975.

H. J. Uushner and P. G. , hupuis : Numerical methods for stochastic control problems in continuous time, volume 24 de Applications of mathematics, 2001.

P. D. Vyi‡ix, Existence theory for a stochastic bolza problem, IMA Journal of Mathematical Control and Information, vol.4, issue.4, p.301320, 1987.

A. Wiele, T. ‡—ng, and W. Welvin, Quasi-steady ight to quasi-steady ight transition for abort landing in a windshear : Trajectory optimization and guidance, p.165207, 1988.

A. Wiele, T. ‡—ng, and C. Y. „zeng, welvin : Optimal abort landing trajectories in the presence of windshear, p.165202, 1987.

I. Wit™hell, A. F—yen, and C. , „omlin : A time dependant hamilton jacobi formulation of reachable sets for continous dynamic games, Transactions on Automatic Control, 2002.

M. Wott—, On nonlinear optimal control problems with state constraints, SIAM journal on control and optimization, vol.33, issue.5, p.14111424, 1995.

M. Wott— and F. ‚—mp—zzo, Multivalued dynamics on a closed domain with absorbing boundary. applications to optimal control problems with integral constraints. Nonlinear Analysis : Theory, Methods & Applications, vol.41, issue.5, pp.631-647, 2000.

R. Wunos and H. id—ni, Consistency of a simple multidimensional scheme for hjb equations, C. R. Acad. Sci. Paris, 2005.

S. Ysher and J. A. , ƒethi—n : Fronts propagating with curvature-dependent speed : algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys, vol.79, issue.1, p.1249, 1988.

M. uin™—mpoix and O. S. ƒere—, A viability approach for optimal control with inmum cost, Annals. Stiint. Univ. Al. I. Cuza Iasi, vol.48, pp.113-132, 2002.

J. D. ‚owl—nd and R. B. , †inter : Construction of optimal feedback controls, pp.357-367, 1991.

O. S. ƒi‚ie, Discontinuous dierential games and control systems with supremum cost, Journal of mathematical analysis and applications, vol.270, issue.2, p.519542, 2002.

H. ƒoner, Optimal control with state-space constraint i, SIAM Journal on Control and Optimization, vol.24, issue.3, p.552561, 1986.

H. ƒoner, Optimal control with state-space constraint. ii, SIAM Journal on Control and Optimization, vol.24, issue.6, p.11101122, 1986.

N. „ouzi, Optimal stochastic control, stochastic target problems, and backward SDE, 2012.

J. ‰yxq and X. Y. ry…, Stochastic Controls : Hamiltonian Systems and HJB Equations. Stochastic Modelling and Applied Probability, 1999.