A dual dynamic programming for minimax optimal control problems governed by parabolic equation

References

• Barbu , V . 1993 . Analysis and Control of Nonlinear Infinite Dimensional Systems , Boston : Academic Press .
   Google Scholar
• Barbu , V . 1991 . The dynamic programming equation for the time-optimal control problem in infinite dimensions . SIAM J. Control Optim. , 29 : 445 – 456 .
   Web of Science ®Google Scholar
• Barbu , V and Da Prato , G . 1983 . Hamilton-Jacobi Equations in Hilbert Spaces , Boston : Pitman Advanced Publishing Program .
   Google Scholar
• Berkovitz , LD . 1986 . Differential games of generalized pursuit and evasion . SIAM J. Control Optim. , 24 : 361 – 373 .
   Google Scholar
• Cannarsa , P and Carja , O . 2004 . On the Bellman equation for the minimum time problem in infinite dimensions . SIAM J. Control Optim. , 43 : 532 – 548 .
   Google Scholar
• Casas , E . 1997 . Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations . SIAM J. Control Optim. , 35 : 1297 – 1327 .
   Web of Science ®Google Scholar
• Denkowski , Z , Migorski , S and Papageorgiou , NS . 2003 . An Introduction to Nonlinear Analysis: Applications , New York : Kluwer .
   Google Scholar
• Fattorini , HO . 1999 . Infinite-dimensional Optimization and Control Theory , Cambridge : Cambridge University Press .
   Google Scholar
• Fattorini , HO and Murphy , T . 1994 . Optimal control for nonlinear parabolic boundary control systems: The Dirichlet boundary conditions . Diff. Integ. Equ. , 7 : 1367 – 1388 .
   Google Scholar
• Fursikov , AV . 2000 . Optimal Control of Distributed Systems. Theory and Applications , Providence, RI : American Mathematical Society .
   Google Scholar
• Galewska , E and Nowakowski , A . 2005 . Multidimensional dual dynamic programming . J. Optim. Theory Appl. , 124 : 175 – 186 .
   Web of Science ®Google Scholar
• Galewska , E and Nowakowski , A . 2006 . A dual dynamic programming for multidimensional elliptic optimal control problems . Numer. Funct. Anal. Optim. , 27 : 279 – 289 .
   Web of Science ®Google Scholar
• Gozzi , F and Tessitore , ME . 1998 . Optimality conditions for Dirichlet boundary control problems of parabolic type . J. Math. Systems Estim. Control , 8 : 143 – 146 .
   Google Scholar
• Krasovskii , NN and Subbotin , AI . 1987 . Game Theoretical Control Problems , New York : Springer-Verlag .
   Google Scholar
• Li , X and Yong , J . 1994 . Optimal Control Theory for Infinite Dimensional Systems , Boston : Birkhauser .
   Google Scholar
• Mordukhovich , BS and Zhang , K . 1997 . Minimax control of parabolic systems with Dirichlet boundary conditions and state constraints . Appl. Math. Optim. , 36 : 323 – 360 .
   Web of Science ®Google Scholar
• Neittaanmaki , P and Tiba , D . 1994 . Optimal Control of Nonlinear Parabolic Systems , New York : Marcel Dekker .
   Google Scholar
• Nowakowski , A . 1992 . The dual dynamic programming . Proc. Amer. Math. Soc. , 116 : 1089 – 1096 .
   Web of Science ®Google Scholar
• Nowakowski , A . 2008 . Sufficient optimality conditions for Dirichlet boundary conditions of wave equation . SIAM J. Control Optim. , 47 : 92 – 110 .
   Google Scholar
• A. Nowakowski, A Neumann boundary control for multidimensional parabolic ‘minmax’ control problems, Annals of Dynamic Games, Gaitsgory et al. – Part I, Chapter 4, Birkhauser, 2008, pp. 71–83.
   Google Scholar
• Subbotin , AI . 1995 . Generalized Solutions of First-order PDEs. The Dynamical Optimization Perspective. Translated from the Russian, Systems & Control: Foundations & Applications , Boston, MA : Birkhäuser Boston, Inc. .
   Google Scholar
• Subbotin , AI . 1999 . “ Constructive Theory of Positional Differential Games and Generalized Solutions to Hamilton-Jacobi Equations ” . In Stochastic and Differential Games, Ann. Internat. Soc. Dynam. Games , Vol. 4 , 3 – 67 . Boston, MA : Birkhäuser Boston .
   Google Scholar