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.
Keywords: two-person zero-sum differential games, unbounded control, Hamilton-Jacobi equation, viscosity solution
@article{COCV_2013__19_2_404_0, author = {Qiu, Hong and Yong, Jiongmin}, title = {Hamilton-Jacobi equations and two-person zero-sum differential games with unbounded controls}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {404--437}, publisher = {EDP-Sciences}, volume = {19}, number = {2}, year = {2013}, doi = {10.1051/cocv/2012015}, mrnumber = {3049717}, zbl = {1263.49024}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2012015/} }
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 SP - 404 EP - 437 VL - 19 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2012015/ DO - 10.1051/cocv/2012015 LA - en ID - COCV_2013__19_2_404_0 ER -
%0 Journal Article %A Qiu, Hong %A Yong, Jiongmin %T Hamilton-Jacobi equations and two-person zero-sum differential games with unbounded controls %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 404-437 %V 19 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2012015/ %R 10.1051/cocv/2012015 %G en %F COCV_2013__19_2_404_0
Qiu, Hong; Yong, Jiongmin. Hamilton-Jacobi equations and two-person zero-sum differential games with unbounded controls. ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 2, pp. 404-437. doi : 10.1051/cocv/2012015. http://archive.numdam.org/articles/10.1051/cocv/2012015/
[1] Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997). | MR | Zbl
and ,[2] On the Bellman equation for some unbounded control problems. NoDEA 4 (1997) 491-510. | MR | Zbl
and ,[3] Existence results for first order Hamilton-Jacobi equations. Ann. Inst. Henri Poincaré 1 (1984) 325-340. | Numdam | MR | Zbl
,[4] Nonlinear monotone semigroups and viscosity solutions. Ann. Inst. Henri Poincaré 18 (2001) 383-402. | Numdam | MR | Zbl
,[5] Viscosity solutions of Hamilton-Jacobi equations. Trans. AMS 277 (1983) 1-42. | MR | Zbl
and ,[6] On existence and uniqueness of solutions of Hamilton-Jacobi equations. Nonlinear Anal. 10 (1986) 353-370. | MR | Zbl
and ,[7] Remarks on the existence and uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations. Ill. J. Math. 31 (1987) 665-688. | MR | Zbl
and ,[8] On the Bellman equation for infinite horizon problems with unblounded cost functional. Appl. Math. Optim. 41 (2000) 171-197. | MR | Zbl
,[9] Uniqueness results for second-order Bellman-Isaacs equations under quadratic growth assumptions and applications. SIAM J. Control Optim. 45 (2006) 74-106. | MR | Zbl
and ,[10] Convex Hamilton-Jacobi equations under superlinear growth conditions on data. Appl. Math. Optim. 63 (2011) 309-339. | MR | Zbl
and ,[11] The existence of value in differential games. Amer. Math. Soc., Providence, RI. Memoirs of AMS 126 (1972). | MR | Zbl
and ,[12] Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations. Indiana Univ. Math. J. 5 (1984) 773-797. | MR | Zbl
and ,[13] On the existence of value functions of two-players, zero-sum stochastic differential games. Indiana Univ. Math. J. 38 (1989) 293-314. | MR | Zbl
and ,[14] Blow-up solutions of Hamilton-Jacobi equations. Commun. Partial Differ. Equ. 11 (1986) 397-443. | MR | Zbl
and ,[15] Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost. NoDEA 11 (2004) 271-298. | MR | Zbl
and ,[16] Uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations. Indiana Univ. Math. J. 33 (1984) 721-748. | MR | Zbl
,[17] Representation of solutions of Hamilton-Jacobi equations. Nonlinear Anal. 12 (1988) 121-146. | MR | Zbl
,[18] Generalized Solutions of Hamilton-Jacobi equations. Pitman, London (1982). | MR | Zbl
,[19] Differential games, optimal conrol and directional derivatives of viscosity solutions of Bellman's and Isaacs' equations. SIAM J. Control Optim. 23 (1985) 566-583. | MR | Zbl
and ,[20] A uniqueness result for the Isaacs equation corresponding to nonlinear H∞ control. Math. Control Signals Syst. 11 (1998) 303-334. | MR | Zbl
,[21] Differential games with unbounded versus bounded controls. SIAM J. Control Optim. 36 (1998) 814-839. | MR | Zbl
,[22] Equivalence between nonlinear ℋ∞ control problems and existence of viscosity solutions of Hamilton-Jacobi-Isaacs equations. Appl. Math. Optim. 39 (1999) 17-32. | MR | Zbl
,[23] Existence of viscosity solution of Hamilton-Jacobi equations. J. Differ. Equ. 56 (1985) 345-390. | MR | Zbl
,[24] Zero-sum differential games involving impusle controls. Appl. Math. Optim. 29 (1994) 243-261. | MR | Zbl
,[25] Syntheses of differential games and pseudo-Riccati equations. Abstr. Appl. Anal. 7 (2002) 61-83. | MR | Zbl
,Cited by Sources: