We prove that the viscosity solution to a Hamilton--Jacobi equation with a smooth convex Hamiltonian of the form $H(x,p)$ is differentiable with respect to the initial condition. Moreover, the directional Gâteaux derivatives can be explicitly computed almost everywhere in $\mathbb{R}^N$ by means of the optimality system of the associated optimal control problem. We also prove that, in the one-dimensional case in space and in the quadratic case in any space dimension, these directional Gâteaux derivatives actually correspond to the unique duality solution to the linear transport equation with discontinuous coefficient, resulting from the linearization of the Hamilton--Jacobi equation. The motivation behind these differentiability results arises from the following optimal inverse-design problem: given a time horizon $T>0$ and a target function $u_T$, construct an initial condition such that the corresponding viscosity solution at time $T$ minimizes the $L^2$-distance to $u_T$. Our differentiability results allow us to derive a necessary first-order optimality condition for this optimization problem and the implementation of gradient-based methods to numerically approximate the optimal inverse design.


  1. Hamilton--Jacobi equation
  2. inverse design problems
  3. Gâteaux derivatives

MSC codes

  1. 35F21
  2. 58C20
  3. 35R30,49K20
  4. 35Q49

Get full access to this article

View all available purchase options and get full access to this article.


O. Alvarez, E. N. Barron, and H. Ishii, Hopf-Lax formulas for semicontinuous data, Indiana Univ. Math. J., 48 (1999), pp. 993--1035.
L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math., 158 (2004), pp. 227--260.
L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Courier Corporation, North Chelmsford, MA, 2000.
F. Ancona, P. Cannarsa, and K. T. Nguyen, Compactness estimates for Hamilton--Jacobi equations depending on space, Bull. Inst. Math. Acad. Sin. (N.S.), 11 (2016), pp. 63--113.
F. Ancona, P. Cannarsa, and K. T. Nguyen, Quantitative compactness estimates for Hamilton--Jacobi equations, Arch. Ration. Mech. Anal., 219 (2016), pp. 793--828.
F. Ancona and M. T. Chiri, Attainable profiles for conservation laws with flux function spatially discontinuous at a single point, ESAIM Control Optim. Calc. Var., 26 (2020), 124, https://doi.org/10.1051/cocv/2020044.
M. Bardi and L. C. Evans, On Hopf's formulas for solutions of Hamilton-Jacobi equations, Nonlinear Anal., 8 (1984), pp. 1373--1381.
E. Barron, P. Cannarsa, R. Jensen, and C. Sinestrari, Regularity of Hamilton--Jacobi equations when forward is backward, Indiana Univ. Math. J., 48 (1999), pp. 385--409.
R. Bellman, Dynamic programming and Lagrange multipliers, Proc. Natl. Acad. Sci. USA, 42 (1956), pp. 767--769.
P. Bernhard and A. Rapaport, On a theorem of Danskin with an application to a theorem of Von Neumann-Sion, Nonlinear Anal., 24 (1995), pp. 1163--1181.
D. Bertsekas, Reinforcement and Optimal Control, Athena Scientific, Nashua, NH, 2019.
S. Bianchini, On the shift differentiability of the flow generated by a hyperbolic system of conservation laws, Discrete Contin. Dyn. Syst., 6 (2000), pp. 329--350.
S. Bianchini and D. Tonon, SBV regularity for Hamilton--Jacobi equations with Hamiltonian depending on $(t,x)$, SIAM J. Math. Anal., 44 (2012), pp. 2179--2203.
F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Anal., 32 (1998), pp. 891--933.
F. Bouchut and F. James, Differentiability with respect to initial data for a scalar conservation law, in Hyperbolic Problems: Theory, Numerics, Applications, Springer, New York, 1999, pp. 113--118.
F. Bouchut, F. James, and S. Mancini, Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (5), 4 (2005), pp. 1--25.
A. Bressan and G. Guerra, Shift-differentiabilitiy of the flow generated by a conservation law, Discrete Contin. Dyn. Syst., 3 (1997), pp. 35--58.
A. Bressan and M. Lewicka, Shift Differentials of Maps in BV Spaces, Chapman & Hall/CRC Res. Notes Math., Chapman & Hall/CRC, Boca Raton, FL, 1999, pp. 47--62.
A. Bressan and W. Shen, Optimality conditions for solutions to hyperbolic balance, in Control Methods in PDE-Dynamical Systems Contemp. Math., 426, AMS, 2007, pp. 129--152.
P. Cannarsa and H. Frankowska, From pointwise to local regularity for solutions of Hamilton--Jacobi equations, Calc. Var. Partial Differential Equations, 49 (2014), pp. 1061--1074.
P. Cannarsa, A. Mennucci, and C. Sinestrari, Regularity results for solutions of a class of Hamilton-Jacobi equations, Arch. Ration. Mech. Anal., 140 (1997), pp. 197--223.
P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progr. Nonlinear Differential Equations Appl. 58, Springer, New York, 2004.
R. M. Colombo and A. Groli, On the optimization of the initial boundary value problem for a conservation law, J. Math. Anal. Appl., 291 (2004), pp. 82--99.
R. M. Colombo, M. Herty, and M. Mercier, Control of the continuity equation with a non local flow, ESAIM Control Optim. Calc. Var., 17 (2011), pp. 353--379.
R. M. Colombo and V. Perrollaz, Initial data identification in conservation laws and Hamilton--Jacobi equations, J. Math. Pures Appl. (9), 138 (2020), pp. 1--27.
M. G. Crandall, H. Ishii, and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), pp. 1--67.
M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), pp. 1--42.
J. M. Danskin, The theory of max-min, with applications, SIAM J. Appl. Math., 14 (1966), pp. 641--664.
R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), pp. 511--547.
C. Esteve and E. Zuazua, The inverse problem for Hamilton--Jacobi equations and semiconcave envelopes, SIAM J. Math. Anal., 52 (2020), pp. 5627--5657.
L. C. Evans, Partial differential equations, Grad. Stud. Math., 19 (1998).
W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Stoch. Model. Appl. Probab. 25, Springer, New York, 2006.
T. Liard and E. Zuazua, Analysis and Numerics Solvability of Backward-Forward Conservation Laws, preprint, hal-02389808, 2020; also available online from https://cmc.deusto.eus/wp-content/uploads/2020/12/Inverse_design_Burgers.pdf.
T. Liard and E. Zuazua, Initial data identification for the one-dimensional Burgers equation, IEEE Trans. Automat. Control, 67 (2022), pp. 3098--3104.
P.-L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Res. Notes Math. 69, Pitman, London, 1982.
P.-L. Lions and P. E. Souganidis, New regularity results for Hamilton--Jacobi equations and long time behavior of pathwise (stochastic) viscosity solutions, Res. Math. Sci., 7 (2020), pp. 1--18.
A. Misztela and S. Plaskacz, An initial condition reconstruction in Hamilton--Jacobi equations, Nonlinear Anal., 200 (2020), 112082.
S. Ulbrich, A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms, SIAM J. Control Optim., 41 (2002), pp. 740--797.
S. Ulbrich, Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws, Systems Control Lett., 48 (2003), pp. 313--328.

Information & Authors


Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 5388 - 5423
ISSN (online): 1095-7154


Submitted: 5 January 2022
Accepted: 12 July 2022
Published online: 13 September 2022


  1. Hamilton--Jacobi equation
  2. inverse design problems
  3. Gâteaux derivatives

MSC codes

  1. 35F21
  2. 58C20
  3. 35R30,49K20
  4. 35Q49



Funding Information

Transregio 154 Project
COST Action grant : CA18232
Alexander von Humboldt-Stiftung https://doi.org/10.13039/100005156
Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : C08
European Research Council https://doi.org/10.13039/501100000781 : 694126-DyCon
Horizon 2020 Framework Programme https://doi.org/10.13039/100010661 : 765579-ConFlex
Ministerio de Economía, Industria y Competitividad, Gobierno de España https://doi.org/10.13039/501100010198 : PID2020-112617GB-C22

Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.







Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.