\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

(Almost) Everything you always wanted to know about deterministic control problems in stratified domains

Abstract Related Papers Cited by
  • We revisit the pioneering work of Bressan & Hong on deterministic control problems in stratified domains, i.e. control problems for which the dynamic and the cost may have discontinuities on submanifolds of $\mathbb{R}^N$. By using slightly different methods, involving more partial differential equations arguments, we $(i)$ slightly improve the assumptions on the dynamic and the cost; $(ii)$ obtain a comparison result for general semi-continuous sub and supersolutions (without any continuity assumptions on the value function nor on the sub/supersolutions); $(iii)$ provide a general framework in which a stability result holds.
    Mathematics Subject Classification: Primary: 49L20, 49L25; Secondary: 35F21.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    Y. Achdou, F. Camilli, A. Cutri and N. Tchou, Hamilton-Jacobi equations constrained on networks, NoDea Nonlinear Differential Equations Appl., 20 (2013), 413-445.doi: 10.1007/s00030-012-0158-1.

    [2]

    Adimurthi, S. Mishra and G. D. Veerappa Gowda, Explicit Hopf-Lax type formulas for Hamilton-Jacobi equations and conservation laws with discontinuous coefficients, J. Differential Equations, 241 (2007), 1-31.doi: 10.1016/j.jde.2007.05.039.

    [3]

    J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2. Birkhäuser Boston, Inc., Boston, MA, 1990.

    [4]

    M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi- Bellman Equations, Systems & Control: Foundations & Applications, Birkhauser Boston Inc., Boston, MA, 1997.doi: 10.1007/978-0-8176-4755-1.

    [5]

    G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi, Springer-Verlag, Paris, 1994.

    [6]

    G. Barles, A. Briani and E. Chasseigne, A Bellman approach for two-domains optimal control problems in $\mathbbR^N$, ESAIM COCV, 19 (2013), 710-739.doi: 10.1051/cocv/2012030.

    [7]

    G. Barles, A. Briani and E. Chasseigne, A Bellman approach for regional optimal control problems in $\mathbbR^N$, SIAM J. Control Optim., 52 (2014), 1712-1744.doi: 10.1137/130922288.

    [8]

    G. Barles, A. Briani, E. Chasseigne and N. Tchou, Homogenization Results for a Deterministic Multi-domains Periodic Control Problem, preprint, arXiv:1405.0661.

    [9]

    G. Barles and E. R. Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations, M2AN, 36 (2002), 33-54.doi: 10.1051/m2an:2002002.

    [10]

    G. Barles and B. Perthame, Exit time problems in optimal control and vanishing viscosity method, SIAM J. in Control and Optimisation, 26 (1988), 1133-1148.doi: 10.1137/0326063.

    [11]

    R. Barnard and P. Wolenski, Flow invariance on stratified domains, Set-Valued and Variational Analysis, 21 (2013), 377-403.doi: 10.1007/s11228-013-0230-y.

    [12]

    A. Bressan and Y. Hong, Optimal control problems on stratified domains, Netw. Heterog. Media., 2 (2007), 313-331 (electronic) and Errata corrige: Optimal control problems on stratified domains. Netw. Heterog. Media., 8 (2013), p625.doi: 10.3934/nhm.2007.2.313.

    [13]

    F. Camilli and D. Schieborn, Viscosity solutions of Eikonal equations on topological networks, Calc. Var. Partial Differential Equations, 46 (2013), 671-686.doi: 10.1007/s00526-012-0498-z.

    [14]

    F. Camilli, C. Marchi and D. Schieborn, Eikonal equations on ramified spaces, Interfaces Free Bound, 15 (2013), 121-140.doi: 10.4171/IFB/297.

    [15]

    F Camilli and A. Siconolfi, Time-dependent measurable Hamilton-Jacobi equations, Comm. in Par. Diff. Eq., 30 (2005), 813-847.doi: 10.1081/PDE-200059292.

    [16]

    G. Coclite and N. Risebro, Viscosity solutions of Hamilton-Jacobi equations with discontinuous coefficients, J. Hyperbolic Differ. Equ., 4 (2007), 771-795.doi: 10.1142/S0219891607001355.

    [17]

    C. De Zan and P. Soravia, Cauchy problems for noncoercive Hamilton-Jacobi-Isaacs equations with discontinuous coefficients, Interfaces Free Bound, 12 (2010), 347-368.doi: 10.4171/IFB/238.

    [18]

    K. Deckelnick and C. Elliott, Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities, Interfaces Free Bound, 6 (2004), 329-349.doi: 10.4171/IFB/103.

    [19]

    P. Dupuis, A numerical method for a calculus of variations problem with discontinuous integrand, Applied stochastic analysis (New Brunswick, NJ, 1991), 90-107, Lecture Notes in Control and Inform. Sci., 177, Springer, Berlin, 1992.doi: 10.1007/BFb0007050.

    [20]

    W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Applications of Mathematics, Springer-Verlag, New York, 1993.

    [21]

    M. Garavello and P. Soravia, Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost, NoDEA Nonlinear Differential Equations Appl. 11 (2004), 271-298.doi: 10.1007/s00030-004-1058-9.

    [22]

    M. Garavello and P. Soravia, Representation formulas for solutions of the HJI equations with discontinuous coefficients and existence of value in differential games, J. Optim. Theory Appl., 130 (2006), 209-229.doi: 10.1007/s10957-006-9099-3.

    [23]

    Y. Giga, P. Gòrka and P. Rybka, A comparison principle for Hamilton-Jacobi equations with discontinuous Hamiltonians, Proc. Amer. Math. Soc., 139 (2011), 1777-1785.doi: 10.1090/S0002-9939-2010-10630-5.

    [24]

    C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM: COCV, 19 (2013), 129-166.doi: 10.1051/cocv/2012002.

    [25]

    C. Imbert and R. Monneau, Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case, preprint, arXiv:1410.3056.

    [26]

    C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, preprint , arXiv:1306.2428.

    [27]

    H. Ishii, Hamilton-Jacobi Equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Fac. Sci. Eng. Chuo Univ., 28 (1985), 33-77.

    [28]

    Z. Rao and H. Zidani, Hamilton-Jacobi-Bellman Equations on Multi-Domains, Control and Optimization with PDE Constraints, International Series of Numerical Mathematics, 164, Birkhäuser Basel, 2013.doi: 10.1007/978-3-0348-0631-2_6.

    [29]

    Z. Rao, A. Siconolfi and H. Zidani, Transmission conditions on interfaces for Hamilton-Jacobi-Bellman equations, J. Differential Equations, 257 (2014), 3978-4014.doi: 10.1016/j.jde.2014.07.015.

    [30]

    P. Soravia, Degenerate eikonal equations with discontinuous refraction index, ESAIM COCV, 12 (2006), 216-230.doi: 10.1051/cocv:2005033.

    [31]

    H. Whitney, Tangents to an analytic variety, Annals of Mathematics, 81 (1965), 496-549.doi: 10.2307/1970400.

    [32]

    H. Whitney, Local properties of analytic varieties, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pp. 205-244, Princeton Univ. Press, Princeton, N. J., 1965.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(192) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return