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), 871.  doi: 10.1016/S0362-546X(99)00134-0.  Google Scholar

[2]

R. Bellman, "Dynamic Programming,", Princeton University Press, (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, (2001).   Google Scholar

[4]

W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control,", Springer Verlag, (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.  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.   Google Scholar

[9]

A. Henrot and M. Pierre, "Variation et optimisation de formes: une analyse g'eométrique,", (French) Math\'ematiques et Applications 48, (2005).   Google Scholar

[10]

R. V. Kohn and G. Strang, Optimal design and relaxation of variational problems,, Comm. Pure Appl. Math., 39 (1986), 113.  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, (1985), 319.   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.  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.   Google Scholar

[14]

A. Nowakowski, The dual dynamic programming,, Proceedings of the American Mathematical Society, 116 (1992), 1089.  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.  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.  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.  doi: 10.1088/0266-5611/15/1/016.  Google Scholar

show all references

References:
[1]

E. Bednarczuk, M. Pierre, E. Rouy and J. Sokolowski, Tangent sets in some functional spaces,, Nonlinear Anal., 42 (2000), 871.  doi: 10.1016/S0362-546X(99)00134-0.  Google Scholar

[2]

R. Bellman, "Dynamic Programming,", Princeton University Press, (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, (2001).   Google Scholar

[4]

W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control,", Springer Verlag, (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.  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.   Google Scholar

[9]

A. Henrot and M. Pierre, "Variation et optimisation de formes: une analyse g'eométrique,", (French) Math\'ematiques et Applications 48, (2005).   Google Scholar

[10]

R. V. Kohn and G. Strang, Optimal design and relaxation of variational problems,, Comm. Pure Appl. Math., 39 (1986), 113.  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, (1985), 319.   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.  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.   Google Scholar

[14]

A. Nowakowski, The dual dynamic programming,, Proceedings of the American Mathematical Society, 116 (1992), 1089.  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.  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.  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.  doi: 10.1088/0266-5611/15/1/016.  Google Scholar

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