April  2016, 9(2): 599-611. doi: 10.3934/dcdss.2016014

On estimation of internal state by an optimal control approach for elastoplastic material

1. 

GeM, UMR CNRS 6183,1 rue de la Noe, F-44321 Nantes, EdF-CEA-ENSTA UMR CNRS 8193, 1 avenue General Leclerc, F- 92141 Clamart, France

Received  October 2014 Revised  November 2015 Published  March 2016

After a general formulation of the evolution of an elastoplastic body using duality based on the constitutive behaviour, some classes of inverse problems (estimation of the internal state, determination of an unknown history, ...) for such materials are investigated. A general formulation based on optimal control is proposed, the control variables are related to the internal state. In each class of inverse problem, the solution is obtained by introducing a adjoin state and a suitable cost function.
Citation: Claude Stolz. On estimation of internal state by an optimal control approach for elastoplastic material. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 599-611. doi: 10.3934/dcdss.2016014
References:
[1]

P. Ballard and A. Constantinescu, On the inversion of subsurface residual stresses from surface stress measurements,, J. Mech. Phys. Solids, 42 (1994), 1767. doi: 10.1016/0022-5096(94)90071-X.

[2]

H. D. Bui, Introduction Aux Problèmes Inverses en Mécanique des Matériaux,, Eyrolles, (1993).

[3]

B. Halphen, Stress accommodation in elastic perfectly plastic and viscoplastic structures,, Mech. Res. Comm., 2 (1975), 273. doi: 10.1016/0093-6413(75)90057-9.

[4]

J. L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations Aux Dérivées Partielles,, Avant propos de P. Lelong Dunod, (1968).

[5]

Q. S. Nguyen, Bifurcation et stabilité des systèmes irréversibles obéissant au principe de dissipation maximale,, J. Mécanique Théorique et appliquée, 3 (1984), 41.

[6]

M. Peigney and C. Stolz, An optimal control approach to the analysis of inelastic structures under cyclic loading,, J. Mech. Phys. Solids, 51 (2003), 575. doi: 10.1016/S0022-5096(02)00104-7.

[7]

M. Peigney and C. Stolz, Approche par contrôle optimal des structures élastoviscoplastique sous chargement cyclique,, C. R. Mécanique, 339 (2001), 643.

[8]

C. Stolz, Optimal control approach in non linear mechanics,, C. R. Mécanique, 336 (2008), 238.

[9]

C. Stolz, Some applications of optimal control to inverse problems in elastoplasticity,, J. of Mechanics of Materials and Structures, 20 (2015), 411. doi: 10.2140/jomms.2015.10.411.

show all references

References:
[1]

P. Ballard and A. Constantinescu, On the inversion of subsurface residual stresses from surface stress measurements,, J. Mech. Phys. Solids, 42 (1994), 1767. doi: 10.1016/0022-5096(94)90071-X.

[2]

H. D. Bui, Introduction Aux Problèmes Inverses en Mécanique des Matériaux,, Eyrolles, (1993).

[3]

B. Halphen, Stress accommodation in elastic perfectly plastic and viscoplastic structures,, Mech. Res. Comm., 2 (1975), 273. doi: 10.1016/0093-6413(75)90057-9.

[4]

J. L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations Aux Dérivées Partielles,, Avant propos de P. Lelong Dunod, (1968).

[5]

Q. S. Nguyen, Bifurcation et stabilité des systèmes irréversibles obéissant au principe de dissipation maximale,, J. Mécanique Théorique et appliquée, 3 (1984), 41.

[6]

M. Peigney and C. Stolz, An optimal control approach to the analysis of inelastic structures under cyclic loading,, J. Mech. Phys. Solids, 51 (2003), 575. doi: 10.1016/S0022-5096(02)00104-7.

[7]

M. Peigney and C. Stolz, Approche par contrôle optimal des structures élastoviscoplastique sous chargement cyclique,, C. R. Mécanique, 339 (2001), 643.

[8]

C. Stolz, Optimal control approach in non linear mechanics,, C. R. Mécanique, 336 (2008), 238.

[9]

C. Stolz, Some applications of optimal control to inverse problems in elastoplasticity,, J. of Mechanics of Materials and Structures, 20 (2015), 411. doi: 10.2140/jomms.2015.10.411.

[1]

Lili Chang, Wei Gong, Guiquan Sun, Ningning Yan. PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem. Inverse Problems & Imaging, 2015, 9 (3) : 791-814. doi: 10.3934/ipi.2015.9.791

[2]

Cristiana J. Silva, Helmut Maurer, Delfim F. M. Torres. Optimal control of a Tuberculosis model with state and control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 321-337. doi: 10.3934/mbe.2017021

[3]

Maria do Rosário de Pinho, Ilya Shvartsman. Lipschitz continuity of optimal control and Lagrange multipliers in a problem with mixed and pure state constraints. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 505-522. doi: 10.3934/dcds.2011.29.505

[4]

Ciro D'Apice, Peter I. Kogut, Rosanna Manzo. On relaxation of state constrained optimal control problem for a PDE-ODE model of supply chains. Networks & Heterogeneous Media, 2014, 9 (3) : 501-518. doi: 10.3934/nhm.2014.9.501

[5]

Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial & Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967

[6]

Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control & Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185

[7]

Kazimierz Malanowski, Helmut Maurer. Sensitivity analysis for state constrained optimal control problems. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 241-272. doi: 10.3934/dcds.1998.4.241

[8]

Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. An optimal control problem in HIV treatment. Conference Publications, 2013, 2013 (special) : 311-322. doi: 10.3934/proc.2013.2013.311

[9]

Changzhi Wu, Kok Lay Teo, Volker Rehbock. Optimal control of piecewise affine systems with piecewise affine state feedback. Journal of Industrial & Management Optimization, 2009, 5 (4) : 737-747. doi: 10.3934/jimo.2009.5.737

[10]

Piernicola Bettiol. State constrained $L^\infty$ optimal control problems interpreted as differential games. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3989-4017. doi: 10.3934/dcds.2015.35.3989

[11]

Luís Tiago Paiva, Fernando A. C. C. Fontes. Adaptive time--mesh refinement in optimal control problems with state constraints. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4553-4572. doi: 10.3934/dcds.2015.35.4553

[12]

Christian Clason, Barbara Kaltenbacher. Avoiding degeneracy in the Westervelt equation by state constrained optimal control. Evolution Equations & Control Theory, 2013, 2 (2) : 281-300. doi: 10.3934/eect.2013.2.281

[13]

Theodore Tachim-Medjo. Optimal control of a two-phase flow model with state constraints. Mathematical Control & Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006

[14]

Vincenzo Basco, Piermarco Cannarsa, Hélène Frankowska. Necessary conditions for infinite horizon optimal control problems with state constraints. Mathematical Control & Related Fields, 2018, 8 (3&4) : 535-555. doi: 10.3934/mcrf.2018022

[15]

Ana P. Lemos-Paião, Cristiana J. Silva, Delfim F. M. Torres. A sufficient optimality condition for delayed state-linear optimal control problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2293-2313. doi: 10.3934/dcdsb.2019096

[16]

V.N. Malozemov, A.V. Omelchenko. On a discrete optimal control problem with an explicit solution. Journal of Industrial & Management Optimization, 2006, 2 (1) : 55-62. doi: 10.3934/jimo.2006.2.55

[17]

Urszula Ledzewicz, Heinz Schättler. Drug resistance in cancer chemotherapy as an optimal control problem. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 129-150. doi: 10.3934/dcdsb.2006.6.129

[18]

Akram Kheirabadi, Asadollah Mahmoudzadeh Vaziri, Sohrab Effati. Solving optimal control problem using Hermite wavelet. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 101-112. doi: 10.3934/naco.2019008

[19]

Lijuan Wang, Qishu Yan. Optimal control problem for exact synchronization of parabolic system. Mathematical Control & Related Fields, 2019, 9 (3) : 411-424. doi: 10.3934/mcrf.2019019

[20]

Yujing Wang, Changjun Yu, Kok Lay Teo. A new computational strategy for optimal control problem with a cost on changing control. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 339-364. doi: 10.3934/naco.2016016

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]