We study optimal control problems for partial differential equations (focusing on the multidimensional differential equation) with control functions in the Dirichlet boundary conditions under pointwise control (and we admit state - by assuming weak hypotheses) constraints.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1080,
author = {Andrzej Nowakowski},
title = {Sufficient optimality conditions for multivariable control problems},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
volume = {27},
year = {2007},
pages = {135-150},
zbl = {1191.49023},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1080}
}
Andrzej Nowakowski. Sufficient optimality conditions for multivariable control problems. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 27 (2007) pp. 135-150. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1080/
[000] [1] N. Arada and J.P. Raymond, Optimality conditions for state-constrained Dirichlet boundary control problems, J. Optim. Theory Appl. 102 (1999), 51-68. | Zbl 0951.49029
[001] [2] N. Arada and J.P. Raymond, Dirichlet boundary control of semilinear parabolic equations, Part 2: Problems with pointwise state constraints, Appl. Math. Optim. 45 (2002), 145-167. | Zbl 1037.49023
[002] [3] V. Barbu, Analysis and control of nonlinear infinite dimensional systems, Academic Press, Boston 1993. | Zbl 0776.49005
[003] [4] V. Barbu, The dynamic programming equation for the time-optimal control problem in infinite dimensions, SIAM J. Control Optim. 29 (1991), 445-456. | Zbl 0737.49022
[004] [5] V. Barbu and G. Da Prato, Hamilton-Jacobi equations in Hilbert spaces, Pitman Advanced Publishing Program, Boston 1983. | Zbl 0508.34001
[005] [6] P. Cannarsa and O. Carja, On the Bellman equation for the minimum time problem in infinite dimensions, SIAM J. Control Optim. 43 (2004), 532-548. | Zbl 1095.49023
[006] [7] E. Casas, Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations, SIAM J. Control Optim. 35 (1997), 1297-1327. | Zbl 0893.49017
[007] [8] H.O. Fattorini, Infinite-Dimensional Optimization and Control Theory, Cambridge University Press, Cambridge, 1999. | Zbl 0931.49001
[008] [9] H.O. Fattorini and T. Murphy, Optimal control for nonlinear parabolic boundary control systems: the Dirichlet boundary conditions, Diff. Integ. Equ. 7 (1994), 1367-1388. | Zbl 0814.93080
[009] [10] A.V. Fursikov, Optimal control of distributed systems. Theory and applications, American Mathematical Society, Providence, RI 2000. | Zbl 1027.93500
[010] [11] F. Gozzi and M.E. Tessitore, Optimality conditions for Dirichlet boundary control problems of parabolic type, J. Math. Systems Estim. Control (1998), 8. | Zbl 0895.35009
[011] [12] E. Galewska and A. Nowakowski, Multidimensional Dual Dynamic Programming, Journal of Optimization Theory and Applications 124 (2005), 175-186. | Zbl 1137.49309
[012] [13] E. Galewska and A. Nowakowski, A dual dynamic programming for multidimensional elliptic optimal control problems, Numerical Functional Analysis and Optimization-accepted in July 2005. | Zbl 1137.49309
[013] [14] I. Lasiecka, J.L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl. 65 (1986), 149-192. | Zbl 0631.35051
[014] [15] X. Li and J. Yong, Optimal control theory for infinite dimensional systems, Birkhauser, Boston 1994. | Zbl 0817.49001
[015] [16] B.S. Mordukhovich and K. Zhang, Minimax control of parabolic systems with Dirichlet boundary conditions and state constraints, Appl. Math. Optim. 36 (1997), 323-360. | Zbl 0890.49008
[016] [17] B.S. Mordukhovich and K. Zhang, Dirichlet boundary control of parabolic systems with pointwise state constraints in International Conference on Control and Estimations of Distributed Parameter Systems, in Control and Estimation of Distributed Parameter Systems (Vorau, 1996), Intentational Series of Numerrical Mathematics, Vol. 126, Birkhäuser. Basel (1998), 223-236. | Zbl 0905.49004
[017] [18] B.S. Mordukhovich and J.P. Raymond, Dirichlet boundary control of hyperbolic equations in the presence of state constraints, Appl. Math. Optim. 49 (2004), 145-157 | Zbl 1056.49025
[018] [19] B.S. Mordukhovich and J.P. Raymond, Neumann boundary control of hyperbolic equations with pointwise state constraints, SIAM J. Control Optim. 43 (2004), 1354-1372. | Zbl 1101.49018
[019] [20] P. Neittaanmaki and D. Tiba, Optimal control of nonlinear parabolic systems, Marcel Dekker, New York 1994. | Zbl 0812.49001
[020] [21] A. Nowakowski, The dual dynamic programming, Proc. Amer. Math. Soc. 116 (1992), 1089-1096. | Zbl 0769.49024
[021] [22] J.P. Raymond, Nonlinear boundary control of semilinear parabolic problems with pointwise state constraints, Discrete Contin. Dynam. Systems 3 (1997), 341-370. | Zbl 0953.49026
[022] [23] L.W. White, Control of a hyperbolic problem with pointwise stress constraints, J. Optim. Theory Appl. 41 (1983), 359-369. | Zbl 0501.49003
[023] [24] L.W. White, Distributed control of a hyperbolic problem with control and stress constraints, J. Math. Anal. Appl. 106 (1985), 41-53. | Zbl 0599.49004