# American Institute of Mathematical Sciences

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:

show all references

##### References:
 [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] Sergio Conti, Georg Dolzmann. Optimal laminates in single-slip elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 1-16. doi: 10.3934/dcdss.2020302 [3] Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, 2021, 11 (3) : 555-578. doi: 10.3934/mcrf.2021012 [4] 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 [5] 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, 2011, 29 (2) : 505-522. doi: 10.3934/dcds.2011.29.505 [6] 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 [7] 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 [8] 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 [9] Kazimierz Malanowski, Helmut Maurer. Sensitivity analysis for state constrained optimal control problems. Discrete & Continuous Dynamical Systems, 1998, 4 (2) : 241-272. doi: 10.3934/dcds.1998.4.241 [10] Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control & Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040 [11] Eduardo Casas, Fredi Tröltzsch. Sparse optimal control for the heat equation with mixed control-state constraints. Mathematical Control & Related Fields, 2020, 10 (3) : 471-491. doi: 10.3934/mcrf.2020007 [12] 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 [13] Andrea Bacchiocchi, Germana Giombini. An optimal control problem of monetary policy. Discrete & Continuous Dynamical Systems - B, 2021, 26 (11) : 5769-5786. doi: 10.3934/dcdsb.2021224 [14] Wensheng Yin, Jinde Cao, Yong Ren. Inverse optimal control of regime-switching jump diffusions. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021034 [15] 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 [16] 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 [17] Piernicola Bettiol. State constrained $L^\infty$ optimal control problems interpreted as differential games. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 3989-4017. doi: 10.3934/dcds.2015.35.3989 [18] 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 [19] Luís Tiago Paiva, Fernando A. C. C. Fontes. Adaptive time--mesh refinement in optimal control problems with state constraints. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4553-4572. doi: 10.3934/dcds.2015.35.4553 [20] 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

2020 Impact Factor: 2.425