American Institute of Mathematical Sciences

November  2012, 11(6): 2473-2485. doi: 10.3934/cpaa.2012.11.2473

On dual dynamic programming in shape control

 1 University of Lodz, Faculty of Math & Computer Sciences, Banacha 22, 90-238 Lodz, Poland 2 Institut Elie Cartan, UMR 7502 (Nancy Université, CNRS, INRIA), Université Henri Poincaré Nancy I, B.P. 239, 54506 Vandoeuvre-lès-Nancy, Cedex

Received  June 2010 Revised  November 2010 Published  April 2012

We propose a new method for analysis of shape optimization problems. The framework of dual dynamic programming is introduced for a solution of the problems. The shape optimization problem for a linear elliptic boundary value problem is formulated in terms of characteristic functions which define the suport of control. The optimal solution of such a problem can be obtained by solving the sufficient optimality conditions.
Citation: Andrzej Nowakowski, Jan Sokolowski. On dual dynamic programming in shape control. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2473-2485. doi: 10.3934/cpaa.2012.11.2473
References:
 [1] E. Bednarczuk, M. Pierre, E. Rouy and J. Sokolowski, Tangent sets in some functional spaces, Nonlinear Anal., 42 (2000), Ser. A: Theory Methods, 871-886. doi: 10.1016/S0362-546X(99)00134-0.  Google Scholar [2] R. Bellman, "Dynamic Programming," Princeton University Press, Princeton, 1957. doi: 10.1126/science.153.3731.34.  Google Scholar [3] M. C. Delfour and J. P. Zolesio, "Shapes and Geometries - Analysis, Differential Calculus and Optimization," Advances in Design and Control, SIAM, 2001.  Google Scholar [4] W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control," Springer Verlag, New York, NY, 1975.  Google Scholar [5] E. Galewska and A. Nowakowski, A dual dynamic programming for multidimensional elliptic optimal control problems, Numer. Funct. Anal. Optim., 27 (2006), 279-289. doi: 10.1080/01630560600698160.  Google Scholar [6] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer - Verlag, 2001.  Google Scholar [7] P. Hebrard and A. Henrot, Spillover phenomenon in the optimal locations of actuators,, SIAM J. Control Optim., 44 (): 349.  doi: 10.1016/S0167-6911(02)00265-7.  Google Scholar [8] P. Hebrard and A. Henrot, Optimal shape and position of the actuators for the stabilization of a string, Systems and Control Letters, 48 (2003), 199-209.  Google Scholar [9] A. Henrot and M. Pierre, "Variation et optimisation de formes: une analyse g'eométrique," (French) Mathématiques et Applications 48, Springer, 2005.  Google Scholar [10] R. V. Kohn and G. Strang, Optimal design and relaxation of variational problems, Comm. Pure Appl. Math., 39, (1986), 113-137. doi: 10.1002/cpa.3160390107.  Google Scholar [11] F. Murat and L. Tartar, "Calcul des Variations et Homogénéisation, Les Méthodes de Homogénéisation: Théorie et Applications en Physique," (French) Coll. Dir. Etudes et Recherches EDF, 57, Eyrolles, Paris, (1985) 319-369.  Google Scholar [12] A. Münch, Optimal location of the support of the control for the 1-D wave equation: Numerical investigations, Comput. Optim. Appl., 42 (2009), 443-470. doi: 10.1007/s10589-007-9133-x.  Google Scholar [13] A. Münch, Optimal design of the support of the control for the 2-D wave equation: a numerical method, Int. J. Numer. Anal. Model., 5 (2008), 331-351.  Google Scholar [14] A. Nowakowski, The dual dynamic programming, Proceedings of the American Mathematical Society, 116 (1992), 1089-1096. doi: 10.1090/S0002-9939-1992-1102860-3.  Google Scholar [15] A. Nowakowski, Sufficient optimality conditions for Dirichlet boundary control of wave equations, SIAM J. Control Optim., 47 (2008), 92-110. doi: 10.1137/050644008.  Google Scholar [16] F. Periago, Optimal shape and position of the support for the internal exact control of a string, Systems & Control Letters, 58 (2009), 136-140. doi: 10.1016/j.sysconle.2008.08.007.  Google Scholar [17] J. Sokolowski and J. P. Zolesio, "Introduction to Shape Optimization," Springer - Verlag, 1992. doi: 10.1007/978-3-642-58106-9.  Google Scholar [18] J. Sokolowski and A. Zochowski, Topological derivative for elliptic problems, Inverse problems, 15, (1999), 123-134. doi: 10.1088/0266-5611/15/1/016.  Google Scholar

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References:
 [1] E. Bednarczuk, M. Pierre, E. Rouy and J. Sokolowski, Tangent sets in some functional spaces, Nonlinear Anal., 42 (2000), Ser. A: Theory Methods, 871-886. doi: 10.1016/S0362-546X(99)00134-0.  Google Scholar [2] R. Bellman, "Dynamic Programming," Princeton University Press, Princeton, 1957. doi: 10.1126/science.153.3731.34.  Google Scholar [3] M. C. Delfour and J. P. Zolesio, "Shapes and Geometries - Analysis, Differential Calculus and Optimization," Advances in Design and Control, SIAM, 2001.  Google Scholar [4] W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control," Springer Verlag, New York, NY, 1975.  Google Scholar [5] E. Galewska and A. Nowakowski, A dual dynamic programming for multidimensional elliptic optimal control problems, Numer. Funct. Anal. Optim., 27 (2006), 279-289. doi: 10.1080/01630560600698160.  Google Scholar [6] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer - Verlag, 2001.  Google Scholar [7] P. Hebrard and A. Henrot, Spillover phenomenon in the optimal locations of actuators,, SIAM J. Control Optim., 44 (): 349.  doi: 10.1016/S0167-6911(02)00265-7.  Google Scholar [8] P. Hebrard and A. Henrot, Optimal shape and position of the actuators for the stabilization of a string, Systems and Control Letters, 48 (2003), 199-209.  Google Scholar [9] A. Henrot and M. Pierre, "Variation et optimisation de formes: une analyse g'eométrique," (French) Mathématiques et Applications 48, Springer, 2005.  Google Scholar [10] R. V. Kohn and G. Strang, Optimal design and relaxation of variational problems, Comm. Pure Appl. Math., 39, (1986), 113-137. doi: 10.1002/cpa.3160390107.  Google Scholar [11] F. Murat and L. Tartar, "Calcul des Variations et Homogénéisation, Les Méthodes de Homogénéisation: Théorie et Applications en Physique," (French) Coll. Dir. Etudes et Recherches EDF, 57, Eyrolles, Paris, (1985) 319-369.  Google Scholar [12] A. Münch, Optimal location of the support of the control for the 1-D wave equation: Numerical investigations, Comput. Optim. Appl., 42 (2009), 443-470. doi: 10.1007/s10589-007-9133-x.  Google Scholar [13] A. Münch, Optimal design of the support of the control for the 2-D wave equation: a numerical method, Int. J. Numer. Anal. Model., 5 (2008), 331-351.  Google Scholar [14] A. Nowakowski, The dual dynamic programming, Proceedings of the American Mathematical Society, 116 (1992), 1089-1096. doi: 10.1090/S0002-9939-1992-1102860-3.  Google Scholar [15] A. Nowakowski, Sufficient optimality conditions for Dirichlet boundary control of wave equations, SIAM J. Control Optim., 47 (2008), 92-110. doi: 10.1137/050644008.  Google Scholar [16] F. Periago, Optimal shape and position of the support for the internal exact control of a string, Systems & Control Letters, 58 (2009), 136-140. doi: 10.1016/j.sysconle.2008.08.007.  Google Scholar [17] J. Sokolowski and J. P. Zolesio, "Introduction to Shape Optimization," Springer - Verlag, 1992. doi: 10.1007/978-3-642-58106-9.  Google Scholar [18] J. Sokolowski and A. Zochowski, Topological derivative for elliptic problems, Inverse problems, 15, (1999), 123-134. doi: 10.1088/0266-5611/15/1/016.  Google Scholar
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