Abstract

Given a closed subset $S$ of a Banach space $X$, we study the minimal time function to reach a point $x$ of $X$ starting from $S$. We relate this problem to the study of the geodesic distance from $x$ to $S$ associated with a Riemannian metric or a Finsler metric. It appears that both cases can be treated simultaneously and yield solutions to a Hamilton--Jacobi equation. We detect simple links between the general framework we propose and multivalued semigroups. New concepts of infinitesimal generators of such semigroups are considered.

Keywords

  1. dynamical systems
  2. eikonal equation
  3. geodesic distance
  4. Hamilton--Jacobi equation
  5. minimal time function
  6. nonsmooth analysis
  7. semigroups
  8. subdifferential
  9. transfer function
  10. variational analysis

MSC codes

  1. 49L20
  2. 35D05

Get full access to this article

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

References

1.
F. Alvarez, R. Correa, and P. Gajardo, Inner estimation of the eigenvalue set and exponential series solutions of differential inclusions, J. Convex Anal., 12 (2005), pp. 1--11.
2.
A. Amri and A. Seeger, Exponentiating a bundle of linear operators, Set-Valued Anal., 14 (2006), pp. 159--185.
3.
J.-P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, New York, 1984.
4.
J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.
5.
D. Azé and J.-N. Corvellec, Characterizations of error bounds for lower semicontinuous functions on metric spaces, ESAIM Control Optim. Calc. Var., 10 (2004), pp. 409--425.
6.
D. Azé and J.-N. Corvellec, On some variational properties of metric spaces, J. Fixed Point Theory Appl., 5 (2009), pp. 185--200.
7.
D. Azé and J.-P. Penot, On the dependence of fixed point sets of pseudo-contractive multifunctions. Applications to differential inclusions, Nonlinear Dyn. Syst. Theory, 6 (2006), pp. 31--49.
8.
S. Bao, S.-S. Chern, and Z. Shen, An Introduction to Riemann-Finsler Geometry, Grad. Texts in Math. 200, Springer, New York, 2000.
9.
M. Bardi, A boundary value problem for the minimal time function, SIAM J. Control Optim., 27 (1989), pp. 776--785.
10.
M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton--Jacobi--Bellman Equations, Birkhäuser, Basel, 1998.
11.
G. Barles, Solutions de viscosité des équations de Hamilton--Jacobi, Math. et Appl. 17, Springer, Berlin, 1994.
12.
J.M. Borwein and J.R. Giles, The proximal normal formula in Banach space, Trans. Amer. Math. Soc., 302 (1987), pp. 371--381.
13.
J.M. Borwein and Q.J. Zhu, Techniques of Variational Analysis, CMS Books Math./Ouvrages Math. SMC 20, Springer, New York, 2005.
14.
M. Bougeard, J.-P. Penot, and A. Pommellet, Towards minimal assumptions for the infimal convolution regularization, J. Approx. Theory, 64 (1991), pp. 245--272.
15.
F.E. Browder, Nonlinear Functional Analysis, Proc. Sympos. Pure Math. XVIII, Part 1, F. E. Browder, ed., AMS, Providence, RI, 1970.
16.
A. Cabot and A. Seeger, Multivalued exponentiation analysis. I: Maclaurin exponentials, Set-Valued Anal., 14 (2006), pp. 437--379.
17.
A. Cabot and A. Seeger, Multivalued exponentiation analysis. II: Recursive exponentials, Set-Valued Anal., 14 (2006), pp. 381--411.
18.
P. Cannarsa and C. Sinestrari, Convexity properties of the minimum time function, Calc. Var., 3 (1995), pp. 273--298.
19.
P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control, Birkhäuser, Boston, 2004.
20.
O. Cârjă, The minimal time function in infinite dimensions, SIAM J. Control Optim., 31 (1993), pp. 1103--1114.
21.
F. Clarke, A proximal characterization of the reachable set, Systems Control Lett., 27 (1996), pp. 195--197.
22.
F.H. Clarke and P.R. Wolenski, Control of systems to sets and their interiors, J. Optim. Theory Appl., 88 (1996), pp. 3--23.
23.
F.H. Clarke, R.J. Stern, and P.R. Wolenski, Proximal smoothness and the lower-C$^{2}$ property, J. Convex Anal., 2 (1995), pp. 117--144.
24.
F.H. Clarke, Y.S. Ledyaev, R.J. Stern, and P.R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998.
25.
G. Colombo, A. Marigonda, and P.R. Wolenski, Some new regularity properties for the minimal time function, SIAM J. Control Optim., 44 (2006), pp. 2285--2299.
26.
G. Colombo and P.R. Wolenski, Variational analysis for a class of minimal time functions in Hilbert spaces, J. Convex Anal., 11 (2004), pp. 335--361.
27.
G. Colombo and P.R. Wolenski, The subgradient formula for the minimal time function in the case of constant dynamics in Hilbert spaces, J. Global Optim., 28 (2004), pp. 269--282.
28.
M.G. Crandall and P.-L. Lions, Hamilton--Jacobi equations in infinite dimensions, Part I, Uniqueness of viscosity solutions, J. Funct. Anal., 62 (1985), pp. 379--396.
29.
M.G. Crandall and P.-L. Lions, Hamilton--Jacobi equations in infinite dimensions, Part II, Existence of viscosity solutions, J. Funct. Anal., 65 (1986) pp. 368--405.
30.
M.G. Crandall and P.-L. Lions, Hamilton--Jacobi equations in infinite dimensions, Part III, J. Funct. Anal., 68 (1986), pp. 214--247.
31.
M.G. Crandall and P.-L. Lions, Hamilton--Jacobi equations in infinite dimensions, Part IV, Unbounded Linear Terms, J. Funct. Anal., 90 (1990), pp. 237--283.
32.
M.G. Crandall and P.-L. Lions, Hamilton--Jacobi equations in infinite dimensions, Part V, B-continuous solutions, J. Funct. Anal., 97 (1991), pp. 417--465.
33.
K. Deimling, Multivalued Differential Equations, De Gruyter, Berlin, 1992.
34.
T. Donchev and A.L. Dontchev, Extensions of Clarke's proximal characterization for reachable mappings of differential inclusions, J. Math. Anal. Appl., 348 (2008), pp. 454--460.
35.
T. Donchev, E. Farkhi, and P. Wolenski, Characterizations of reachable sets for a class of differential inclusions, Funct. Differ. Equ., 10 (2003), pp. 473--483.
36.
T. Donchev, V. Ríos, and P. Wolenski, Strong invariance and one-sided Lipschitz multifunctions, Nonlinear Anal., 60 (2005), pp. 849--852.
37.
K.-J. Engel and R. Nagel, A Short Course on Operator Semigroups, Universitext, Springer, New York, 2006.
38.
S. Fitzpatrick, Metric projection and the differentiability of the distance function, Bull. Austral. Math. Soc., 22 (1980), pp. 773--802.
39.
H. Frankowska, Local controllability and infinitesimal generators of semigroups of set-valued maps, SIAM J. Control Optim., 25 (1987), pp. 412--432.
40.
H. Frankowska, Contingent cones to reachable sets of control systems, SIAM J. Control Optim., 27 (1989), pp. 170--198.
41.
H. Frankowska, How set-valued maps pop up in control theory, in Systems and Control in the Twenty-First Century, Progr. Systems Control Theory, Birkhäuser, Boston, 1997, pp. 155--169.
42.
M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhäuser, Boston, 1998.
43.
Kh. G. Guseinov, S.A. Duzce, and O. Özer, The construction of differential inclusions with prescribed attainable sets, J. Dynam. Control Systems, 14 (2008) pp. 441--452.
44.
O. Hajek, Geometric theory of time-optimal control, SIAM J. Control Optim., 9 (1971), pp. 339--350.
45.
Y. He and K.F. Ng, Subdifferentials of a minimal time function in Banach spaces, J. Math. Anal. Appl., 321 (2006), pp. 896--910.
46.
H. Hermes and J.P. Lasalle, Functional Analysis and Time Optimal Control, Academic Press, New York, 1969.
47.
J.-B. Hiriart-Urruty and J. Malick, A fresh variational-analysis look at the positive semidefinite matrices world, J. Optim. Theory Appl., 153 (2012), pp. 551--577.
48.
A.D. Ioffe, Euler-Lagrange and Hamiltonian formalisms in dynamic optimization, Trans. Amer. Math. Soc., 349 (1997), pp. 2871--2900.
49.
A. Ioffe, Towards metric theory of metric regularity, in Approximation, Optimization and Mathematical Economics, M. Lassonde, ed., Physica-Verlag, Heidelberg, 2001, pp. 165--176.
50.
A. Ioffe, Existence and relaxation theorems for unbounded differential inclusions, J. Convex Anal., 13 (2006), pp. 353--362.
51.
C. Li and R. Ni, Derivatives of generalized distance functions and existence of generalized nearest points, J. Approx. Theory, 115 (2002), pp. 44--55.
52.
Y.Y. Li and L. Nirenberg, The Distance Function to the Boundary, Finsler Geometry and the Singular Set of Viscosity Solutions of Some Hamilton-Jacobi Equations, preprint.
53.
J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation des Systèmes Distribués, tome 1: Contrôlabilité Exacte, Masson, Paris, 1988.
54.
P.-L. Lions, Generalized Solutions of Hamilton--Jacobi Equations, Res. Notes Math. 69, Pitman, London, 1982.
55.
C. Lobry, Sur l'ensemble des points atteignables par les solutions d'une équation différentielle multivoque, Publ. Math. Bordeaux, 1 (1973), pp. 43--60.
56.
P. Loreti and A. Siconolfi, Semigroup approach for the approximation of a control problem with unbounded dynamics, J. Optim. Theory Appl., 79 (1993), pp. 599--610.
57.
J.E. Martínez-Legaz, Level sets and the minimal time function of linear control processes, Numer. Funct. Anal. Optim., 9 (1987), pp. 105--129.
58.
F. Mignanego and G. Pieri, On a generalized Bellman equation for the optimal-time problem, Systems Control Lett., 3 (1983), pp. 235--241.
59.
Y. Nesterov and A.S. Nemirovski, Interior-Point Methods Polynomial Algorithms in Convex Programming, SIAM Stud. Appl. Math. 13, SIAM, Philadelphia, 1994.
60.
H.V. Ngai and J.-P. Penot, Approximately convex sets, J. Nonlinear Convex Anal., 8 (2007), pp. 337--371.
61.
H.V. Ngai and J.-P. Penot, Paraconvex functions and paraconvex sets, Studia Math., 184 (2008), pp. 1--29.
62.
L. Nirenberg, The distance function to the boundary and the singular set of viscosity solutions of Hamilton-Jacobi equations, in Variational Analysis and Applications, Nonconvex Optim. Appl. 79, Springer, New York, 2005, pp. 765--772.
63.
C. Nour, The bilateral minimal time function, J. Convex Anal., 13 (2006), pp. 61--80.
64.
R.S. Palais, Lusternik-Schnirelman theory on Banach manifolds, Topology, 5 (1966) pp. 115--132.
65.
J.-P. Penot, A characterization of tangential regularity, Nonlinear Anal., 5 (1981), pp. 625--643.
66.
J.-P. Penot, Proximal mappings, J. Approx. Theory, 94 (1998), pp. 203--221.
67.
J.-P. Penot, Differentiability properties of optimal value functions, Canad. J. Math., 56 (2004), pp. 825--842.
68.
J.-P. Penot, Calculus Without Derivatives, Grad. Texts in Math., Springer, New York, 2013.
69.
J.-P. Penot and R. Ratsimahalo, Subdifferentials of distance functions, approximations and enlargements, Acta Math. Sin. (Engl. Ser.), English Series, 23 (2007), pp. 507--520.
70.
J.-P. Penot and M. Volle, Duality methods for the study of Hamilton-Jacobi equations, ESAIM: Proc., 17 (2007), pp. 96--142.
71.
J. Renegar, A Mathematical View of Interior-Point Methods in Convex Optimization, MPS SIAM Ser. Optim., SIAM, Philadelphia, 2001.
72.
E. Roxin, Stability in general control systems, J. Differential Equations, 1 (1965), pp. 115--150.
73.
E. Roxin, On the generalized dynamical systems defined by contingent equations, J. Differential Equations, 1 (1965), pp. 188--205.
74.
A. Seeger, Smoothing a nondifferentiable convex function: The technique of the rolling ball, Rev. Mat. Appl., 17 (1996), pp. 45--60.
75.
A. Siconolfi, Metric character of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 355 (2003), pp. 1987--2009, p. 4265.
76.
P. Soravia, Generalized motion of a front propagating along its normal direction: A differential games approach, Nonlinear Anal., 22 (1994), pp. 1247--1262.
77.
M. Spivak, A Comprehensive Introduction to Differential Geometry, 2nd ed., Publish or Perish, Wilmington, DE, 1979.
78.
V. Staicu, Minimal time function and viscosity solutions, J. Optim. Theory Appl., 60 (1989), pp. 81--91.
79.
A. Tolstonogov, Differential Inclusions in a Banach Space, Kluwer, Dordrecht, Netherlands, 2000.
80.
V. Veliov, Lipschitz continuity of the value function in optimal control, J. Optim. Theory Appl., 94 (1997), pp. 335--363.
81.
P.R. Wolenski, A uniqueness theorem for differential inclusions, J. Differential Equations, 84 (1990), pp. 165--182.
82.
P. R. Wolenski, The exponential formula for the reachable set of a Lipschitz differential inclusion, SIAM J. Control Optim., 28 (1990), pp. 1148--1161.
83.
P.R. Wolenski, The semigroup property of value functions in Lagrange problems, Trans. Amer. Math. Soc., 335 (1993), pp. 131--154.
84.
P.R. Wolenski and Y. Zhuang, Proximal analysis and the minimal time function, SIAM J. Control Optim., 36 (1998), pp. 1048--1072.

Information & Authors

Information

Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 2839 - 2868
ISSN (online): 1095-7138

History

Submitted: 1 June 2012
Accepted: 8 March 2013
Published online: 3 July 2013

Keywords

  1. dynamical systems
  2. eikonal equation
  3. geodesic distance
  4. Hamilton--Jacobi equation
  5. minimal time function
  6. nonsmooth analysis
  7. semigroups
  8. subdifferential
  9. transfer function
  10. variational analysis

MSC codes

  1. 49L20
  2. 35D05

Authors

Affiliations

Metrics & Citations

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.

Cited By

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

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.