# Hamilton–Jacobi equations and two-person zero-sum differential games with unbounded controls

ESAIM: Control, Optimisation and Calculus of Variations (2013)

- Volume: 19, Issue: 2, page 404-437
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topQiu, Hong, and Yong, Jiongmin. "Hamilton–Jacobi equations and two-person zero-sum differential games with unbounded controls." ESAIM: Control, Optimisation and Calculus of Variations 19.2 (2013): 404-437. <http://eudml.org/doc/272807>.

@article{Qiu2013,

abstract = {A two-person zero-sum differential game with unbounded controls is considered. Under proper coercivity conditions, the upper and lower value functions are characterized as the unique viscosity solutions to the corresponding upper and lower Hamilton–Jacobi–Isaacs equations, respectively. Consequently, when the Isaacs’ condition is satisfied, the upper and lower value functions coincide, leading to the existence of the value function of the differential game. Due to the unboundedness of the controls, the corresponding upper and lower Hamiltonians grow super linearly in the gradient of the upper and lower value functions, respectively. A uniqueness theorem of viscosity solution to Hamilton–Jacobi equations involving such kind of Hamiltonian is proved, without relying on the convexity/concavity of the Hamiltonian. Also, it is shown that the assumed coercivity conditions guaranteeing the finiteness of the upper and lower value functions are sharp in some sense.},

author = {Qiu, Hong, Yong, Jiongmin},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {two-person zero-sum differential games; unbounded control; Hamilton–Jacobi equation; viscosity solution; Hamilton-Jacobi equation},

language = {eng},

number = {2},

pages = {404-437},

publisher = {EDP-Sciences},

title = {Hamilton–Jacobi equations and two-person zero-sum differential games with unbounded controls},

url = {http://eudml.org/doc/272807},

volume = {19},

year = {2013},

}

TY - JOUR

AU - Qiu, Hong

AU - Yong, Jiongmin

TI - Hamilton–Jacobi equations and two-person zero-sum differential games with unbounded controls

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2013

PB - EDP-Sciences

VL - 19

IS - 2

SP - 404

EP - 437

AB - A two-person zero-sum differential game with unbounded controls is considered. Under proper coercivity conditions, the upper and lower value functions are characterized as the unique viscosity solutions to the corresponding upper and lower Hamilton–Jacobi–Isaacs equations, respectively. Consequently, when the Isaacs’ condition is satisfied, the upper and lower value functions coincide, leading to the existence of the value function of the differential game. Due to the unboundedness of the controls, the corresponding upper and lower Hamiltonians grow super linearly in the gradient of the upper and lower value functions, respectively. A uniqueness theorem of viscosity solution to Hamilton–Jacobi equations involving such kind of Hamiltonian is proved, without relying on the convexity/concavity of the Hamiltonian. Also, it is shown that the assumed coercivity conditions guaranteeing the finiteness of the upper and lower value functions are sharp in some sense.

LA - eng

KW - two-person zero-sum differential games; unbounded control; Hamilton–Jacobi equation; viscosity solution; Hamilton-Jacobi equation

UR - http://eudml.org/doc/272807

ER -

## References

top- [1] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997). Zbl0890.49011MR1484411
- [2] M. Bardi and F. Da Lio, On the Bellman equation for some unbounded control problems. NoDEA4 (1997) 491–510. Zbl0894.49017MR1485734
- [3] G. Barles, Existence results for first order Hamilton-Jacobi equations. Ann. Inst. Henri Poincaré1 (1984) 325–340. Zbl0574.70019MR779871
- [4] S. Biton, Nonlinear monotone semigroups and viscosity solutions. Ann. Inst. Henri Poincaré18 (2001) 383–402. Zbl1002.35060MR1831661
- [5] M.G. Crandall and P.L. Lions, Viscosity solutions of Hamilton-Jacobi equations. Trans. AMS277 (1983) 1–42. Zbl0599.35024MR690039
- [6] M.G. Crandall and P.L. Lions, On existence and uniqueness of solutions of Hamilton-Jacobi equations. Nonlinear Anal.10 (1986) 353–370. Zbl0603.35016MR836671
- [7] M.G. Crandall and P.L. Lions, Remarks on the existence and uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations. Ill. J. Math.31 (1987) 665–688. Zbl0678.35009MR909790
- [8] F. Da Lio, On the Bellman equation for infinite horizon problems with unblounded cost functional. Appl. Math. Optim.41 (2000) 171–197. Zbl0952.49023MR1731417
- [9] F. Da Lio and O. Ley, Uniqueness results for second-order Bellman-Isaacs equations under quadratic growth assumptions and applications. SIAM J. Control Optim.45 (2006) 74–106. Zbl1116.49017MR2225298
- [10] F. Da Lio and O. Ley, Convex Hamilton-Jacobi equations under superlinear growth conditions on data. Appl. Math. Optim.63 (2011) 309–339. Zbl1223.35115MR2784834
- [11] R.J. Elliott and N.J. Kalton, The existence of value in differential games. Amer. Math. Soc., Providence, RI. Memoirs of AMS 126 (1972). Zbl0262.90076MR359845
- [12] L.C. Evans and P.E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations. Indiana Univ. Math. J.5 (1984) 773–797. Zbl1169.91317MR756158
- [13] W.H. Fleming and P.E. Souganidis, On the existence of value functions of two-players, zero-sum stochastic differential games. Indiana Univ. Math. J.38 (1989) 293–314. Zbl0686.90049MR997385
- [14] A. Friedman and P.E. Souganidis, Blow-up solutions of Hamilton-Jacobi equations. Commun. Partial Differ. Equ.11 (1986) 397–443. Zbl0593.35063MR829323
- [15] M. Garavello and P. Soravia, Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost. NoDEA11 (2004) 271–298. Zbl1053.49026MR2090274
- [16] H. Ishii, Uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations. Indiana Univ. Math. J.33 (1984) 721–748. Zbl0551.49016MR756156
- [17] H. Ishii, Representation of solutions of Hamilton-Jacobi equations. Nonlinear Anal.12 (1988) 121–146. Zbl0687.35025MR926208
- [18] P.L. Lions, Generalized Solutions of Hamilton-Jacobi equations. Pitman, London (1982). Zbl0497.35001MR667669
- [19] P.L. Lions and P.E. Souganidis, Differential games, optimal conrol and directional derivatives of viscosity solutions of Bellman’s and Isaacs’ equations. SIAM J. Control Optim.23 (1985) 566–583. Zbl0569.49019MR791888
- [20] W. McEneaney, A uniqueness result for the Isaacs equation corresponding to nonlinear H∞ control. Math. Control Signals Syst.11 (1998) 303–334. Zbl0919.93027MR1662969
- [21] F. Rampazzo, Differential games with unbounded versus bounded controls. SIAM J. Control Optim.36 (1998) 814–839. Zbl0909.90286MR1613865
- [22] P. Soravia, Equivalence between nonlinear ℋ∞ control problems and existence of viscosity solutions of Hamilton-Jacobi-Isaacs equations. Appl. Math. Optim. 39 (1999) 17–32. Zbl0920.93016MR1654550
- [23] P.E. Souganidis, Existence of viscosity solution of Hamilton-Jacobi equations. J. Differ. Equ.56 (1985) 345–390. Zbl0506.35020MR780496
- [24] J. Yong, Zero-sum differential games involving impusle controls. Appl. Math. Optim.29 (1994) 243–261. Zbl0808.90142MR1264011
- [25] Y. You, Syntheses of differential games and pseudo-Riccati equations. Abstr. Appl. Anal.7 (2002) 61–83. Zbl1066.91013MR1891031

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.