# American Institute of Mathematical Sciences

• Previous Article
Nomonotone spectral gradient method for sparse recovery
• IPI Home
• This Issue
• Next Article
Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Stochastic extension of the boundary map
August  2015, 9(3): 791-814. doi: 10.3934/ipi.2015.9.791

## PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem

 1 Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shan'xi, China, China 2 LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, CAS, Beijing 100190, China 3 LSEC, NCMIS, Institute of Systems Science, Academy of Mathematics and Systems Science, CAS, Beijing 100190, China

Received  November 2014 Revised  March 2015 Published  July 2015

This paper concerns the approximation of a Cauchy problem for the elliptic equation. The inverse problem is transformed into a PDE-constrained optimal control problem and these two problems are equivalent under some assumptions. Different from the existing literature which is also based on the optimal control theory, we consider the state equation in the sense of very weak solution defined by the transposition technique. In this way, it does not need to impose any regularity requirement on the given data. Moreover, this method can yield theoretical analysis simply and numerical computation conveniently. To deal with the ill-posedness of the control problem, Tikhonov regularization term is introduced. The regularized problem is well-posed and its solution converges to the non-regularized counterpart as the regularization parameter approaches zero. We establish the finite element approximation to the regularized control problem and the convergence of the discrete problem is also investigated. Then we discuss the first order optimality condition of the control problem further and obtain an efficient numerical scheme for the Cauchy problem via the adjoint state equation. The paper is ended with numerical experiments.
Citation: 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
##### References:
 [1] R. A. Adams, Sobolev Spaces,, Academic Press, (1975).   Google Scholar [2] G. Alessandrini, L. D. Piero and L. Rondi, Stable determination of corrosion by a single electrostatic boundary measurement,, Inverse Problems, 19 (2003), 973.  doi: 10.1088/0266-5611/19/4/312.  Google Scholar [3] G. Alessandrini and E. Sincich, Solving elliptic Cauchy problems and the identification of nonlinear corrosion,, J. Comp. Appl. Math., 198 (2007), 307.  doi: 10.1016/j.cam.2005.06.048.  Google Scholar [4] S. Andrieux and A. A. Ben, Identification of planar cracks by complete overdetermined data: inversion formulae,, Inverse Problems, 12 (1996), 553.  doi: 10.1088/0266-5611/12/5/002.  Google Scholar [5] M. Azaïez, F. B. Belgacem and H. E. Fekih, On Cauchy's problem: II. Completion, regularization and approximation,, Inverse Problems, 22 (2006), 1307.  doi: 10.1088/0266-5611/22/4/012.  Google Scholar [6] F. B. Belgacem and H. E. Fekih, On Cauchy's problem: I. A variational Steklov-Poincaré theory,, Inverse Problems, 21 (2005), 1915.  doi: 10.1088/0266-5611/21/6/008.  Google Scholar [7] M. Berggren, Approximation of very weak solutions to boundary value problems,, SIAM J. Numer. Anal., 42 (2004), 860.  doi: 10.1137/S0036142903382048.  Google Scholar [8] L. Bourgeois and J. Dardé, A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/9/095016.  Google Scholar [9] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods,, $3^{th}$ edition, (2008).  doi: 10.1007/978-0-387-75934-0.  Google Scholar [10] M. Brühl, M. Hanke and M. Pidcock, Crack detection using electrostatic measurements,, M2AN Math. Model. Numer. Anal., 35 (2001), 595.  doi: 10.1051/m2an:2001128.  Google Scholar [11] E. Casas and J. P. Raymond, Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations,, SIAM J. Control Optim., 45 (2006), 1586.  doi: 10.1137/050626600.  Google Scholar [12] A. Chakib and A. Nachaoui, Convergence analysis for finite element approximation to an inverse Cauchy problem,, Inverse Problems, 22 (2006), 1191.  doi: 10.1088/0266-5611/22/4/005.  Google Scholar [13] J. Cheng, Y. Hon, T. Wei and M. Yamamoto, Numerical computation of a Cauchy problem for Laplace's equation,, Z. Angew. Math. Mech., 81 (2001), 665.  doi: 10.1002/1521-4001(200110)81:10<665::AID-ZAMM665>3.0.CO;2-V.  Google Scholar [14] P. G. Ciarlet, The Finite Element Methods for Elliptic Problems,, North-Holland, (1978).  doi: 10.1137/1.9780898719208.  Google Scholar [15] A. Cimetière, F. Delvare, M. Jaoua and F. Pons, Solution of the Cauchy problem using iterated Tikhonov regularization,, Inverse Problems, 17 (2001), 553.  doi: 10.1088/0266-5611/17/3/313.  Google Scholar [16] F. P. Colli and E. Magenes, On the inverse potential problem of electrocardiology,, Calcolo, 16 (1979), 459.  doi: 10.1007/BF02576643.  Google Scholar [17] K. Deckelnick, A. Günther and M. Hinze, Finite element approximation of Dirichlet boundary control for elliptic PDEs on two- and three-dimensional curved domains,, SIAM J. Control Optim., 48 (2009), 2798.  doi: 10.1137/080735369.  Google Scholar [18] H. W. Engl and A. Leitão, A Mann iterative regularization method for elliptic Cauchy problems,, Numer. Funct. Anal. Optim., 22 (2001), 861.  doi: 10.1081/NFA-100108313.  Google Scholar [19] D. A. French and J. T. King, Approximation of an elliptic control problem by the finite element method,, Numer. Funct. Anal. Optim., 12 (1991), 299.  doi: 10.1080/01630569108816430.  Google Scholar [20] A. Friedman and M. S. Vogelius, Determining cracks by boundary measurements,, J. Ind. Univ. Math., 38 (1989), 527.  doi: 10.1512/iumj.1989.38.38025.  Google Scholar [21] A. V. Fursikov, The Cauchy problem for a second-order elliptic equation in a conditionally well-posed formulation,, Tr. Mosk. Mat. Obs., 52 (1991), 139.   Google Scholar [22] H. Han and H. J. Reinhardt, Some stability estimates for Cauchy problems for elliptic equations,, J. Inverse Ill-Posed Problems, 5 (1997), 437.  doi: 10.1515/jiip.1997.5.5.437.  Google Scholar [23] W. Han, W. Cong and G. Wang, Mathematical theory and numerical analysis of bioluminescence tomography,, Inverse Problems, 22 (2006), 1659.  doi: 10.1088/0266-5611/22/5/008.  Google Scholar [24] W. Han, J. Huang, K. Kazmi and Y. Chen, A numerical method for a Cauchy problem for elliptic partial differential equations,, Inverse Problems, 23 (2007), 2401.  doi: 10.1088/0266-5611/23/6/008.  Google Scholar [25] M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case,, Comput. Optim. Appl., 30 (2005), 45.  doi: 10.1007/s10589-005-4559-5.  Google Scholar [26] M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints,, Springer, (2009).   Google Scholar [27] G. Inglese, An inverse problem in corrosion detection,, Inverse Problems, 13 (1997), 977.  doi: 10.1088/0266-5611/13/4/006.  Google Scholar [28] V. Isakov, Inverse Problems for Partial Differential Equations,, $2^{nd}$ edition, (2006).   Google Scholar [29] M. Jourhmane and A. Nachaoui, Convergence of an alternating method to solve the Cauchy problem for Poisson's equation,, Appl. Anal., 81 (2002), 1065.  doi: 10.1080/0003681021000029819.  Google Scholar [30] S. I. Kabanikhin and A. L. Karchevsky, Optimizational method for solving the Cauchy problem for an elliptic equation,, J. Inverse Ill-Posed problems, 3 (1995), 21.  doi: 10.1515/jiip.1995.3.1.21.  Google Scholar [31] M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace's equation,, SIAM J. Appl. Math., 51 (1991), 1653.  doi: 10.1137/0151085.  Google Scholar [32] R. V. Kohn and M. S. Vogelius, Determining conductivity by boundary measurements: II. Interior results,, Comm. Pure Appl. Math., 38 (1985), 643.  doi: 10.1002/cpa.3160380513.  Google Scholar [33] V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations,, U.S.S.R. Comput. Math. and Math. Phys., 31 (1991), 45.   Google Scholar [34] R. Lattès and J.-L. Lions, Méthode de Quasi-réversibilité et Applications, (French) [The Quasi-Reversibility Methods and Applications],, Travaux et Recherches Mathématiques, (1967).   Google Scholar [35] R. Li and W. Liu, The C++ software library AFEPack., Available from: , ().   Google Scholar [36] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer-Verlag, (1971).   Google Scholar [37] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications,, Vol. II, (1972).   Google Scholar [38] M. Maher, K. Moez and T. Anis, A Cauchy problem for an inverse problem in image inpainting,, in Industrial Engineering and Systems Management (IESM), (2013).   Google Scholar [39] S. Pereverzev and E. Schock, Morozov's discrepancy principle for Tikhonov regularization of severely ill-posed problems in finite-dimensional subspaces,, Numer. Funct. Anal. Optim., 21 (2000), 901.  doi: 10.1080/01630560008816993.  Google Scholar [40] N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations,, Akademie Verlag, (1995).   Google Scholar [41] A. N. Tikhonov and V. A. Arsenin, Solutions of Ill-Posed Problems,, Winston and Sons, (1977).   Google Scholar [42] W. Weikl, H. Andra and E. Schnack, An alternating iterative algorithm for the reconstruction of internal cracks in a three-dimensional solid body,, Inverse Problems, 17 (2001), 1957.  doi: 10.1088/0266-5611/17/6/325.  Google Scholar [43] K. Yosida, Functional Analysis,, $5^{th}$ edition, (1978).   Google Scholar

show all references

##### References:
 [1] R. A. Adams, Sobolev Spaces,, Academic Press, (1975).   Google Scholar [2] G. Alessandrini, L. D. Piero and L. Rondi, Stable determination of corrosion by a single electrostatic boundary measurement,, Inverse Problems, 19 (2003), 973.  doi: 10.1088/0266-5611/19/4/312.  Google Scholar [3] G. Alessandrini and E. Sincich, Solving elliptic Cauchy problems and the identification of nonlinear corrosion,, J. Comp. Appl. Math., 198 (2007), 307.  doi: 10.1016/j.cam.2005.06.048.  Google Scholar [4] S. Andrieux and A. A. Ben, Identification of planar cracks by complete overdetermined data: inversion formulae,, Inverse Problems, 12 (1996), 553.  doi: 10.1088/0266-5611/12/5/002.  Google Scholar [5] M. Azaïez, F. B. Belgacem and H. E. Fekih, On Cauchy's problem: II. Completion, regularization and approximation,, Inverse Problems, 22 (2006), 1307.  doi: 10.1088/0266-5611/22/4/012.  Google Scholar [6] F. B. Belgacem and H. E. Fekih, On Cauchy's problem: I. A variational Steklov-Poincaré theory,, Inverse Problems, 21 (2005), 1915.  doi: 10.1088/0266-5611/21/6/008.  Google Scholar [7] M. Berggren, Approximation of very weak solutions to boundary value problems,, SIAM J. Numer. Anal., 42 (2004), 860.  doi: 10.1137/S0036142903382048.  Google Scholar [8] L. Bourgeois and J. Dardé, A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/9/095016.  Google Scholar [9] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods,, $3^{th}$ edition, (2008).  doi: 10.1007/978-0-387-75934-0.  Google Scholar [10] M. Brühl, M. Hanke and M. Pidcock, Crack detection using electrostatic measurements,, M2AN Math. Model. Numer. Anal., 35 (2001), 595.  doi: 10.1051/m2an:2001128.  Google Scholar [11] E. Casas and J. P. Raymond, Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations,, SIAM J. Control Optim., 45 (2006), 1586.  doi: 10.1137/050626600.  Google Scholar [12] A. Chakib and A. Nachaoui, Convergence analysis for finite element approximation to an inverse Cauchy problem,, Inverse Problems, 22 (2006), 1191.  doi: 10.1088/0266-5611/22/4/005.  Google Scholar [13] J. Cheng, Y. Hon, T. Wei and M. Yamamoto, Numerical computation of a Cauchy problem for Laplace's equation,, Z. Angew. Math. Mech., 81 (2001), 665.  doi: 10.1002/1521-4001(200110)81:10<665::AID-ZAMM665>3.0.CO;2-V.  Google Scholar [14] P. G. Ciarlet, The Finite Element Methods for Elliptic Problems,, North-Holland, (1978).  doi: 10.1137/1.9780898719208.  Google Scholar [15] A. Cimetière, F. Delvare, M. Jaoua and F. Pons, Solution of the Cauchy problem using iterated Tikhonov regularization,, Inverse Problems, 17 (2001), 553.  doi: 10.1088/0266-5611/17/3/313.  Google Scholar [16] F. P. Colli and E. Magenes, On the inverse potential problem of electrocardiology,, Calcolo, 16 (1979), 459.  doi: 10.1007/BF02576643.  Google Scholar [17] K. Deckelnick, A. Günther and M. Hinze, Finite element approximation of Dirichlet boundary control for elliptic PDEs on two- and three-dimensional curved domains,, SIAM J. Control Optim., 48 (2009), 2798.  doi: 10.1137/080735369.  Google Scholar [18] H. W. Engl and A. Leitão, A Mann iterative regularization method for elliptic Cauchy problems,, Numer. Funct. Anal. Optim., 22 (2001), 861.  doi: 10.1081/NFA-100108313.  Google Scholar [19] D. A. French and J. T. King, Approximation of an elliptic control problem by the finite element method,, Numer. Funct. Anal. Optim., 12 (1991), 299.  doi: 10.1080/01630569108816430.  Google Scholar [20] A. Friedman and M. S. Vogelius, Determining cracks by boundary measurements,, J. Ind. Univ. Math., 38 (1989), 527.  doi: 10.1512/iumj.1989.38.38025.  Google Scholar [21] A. V. Fursikov, The Cauchy problem for a second-order elliptic equation in a conditionally well-posed formulation,, Tr. Mosk. Mat. Obs., 52 (1991), 139.   Google Scholar [22] H. Han and H. J. Reinhardt, Some stability estimates for Cauchy problems for elliptic equations,, J. Inverse Ill-Posed Problems, 5 (1997), 437.  doi: 10.1515/jiip.1997.5.5.437.  Google Scholar [23] W. Han, W. Cong and G. Wang, Mathematical theory and numerical analysis of bioluminescence tomography,, Inverse Problems, 22 (2006), 1659.  doi: 10.1088/0266-5611/22/5/008.  Google Scholar [24] W. Han, J. Huang, K. Kazmi and Y. Chen, A numerical method for a Cauchy problem for elliptic partial differential equations,, Inverse Problems, 23 (2007), 2401.  doi: 10.1088/0266-5611/23/6/008.  Google Scholar [25] M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case,, Comput. Optim. Appl., 30 (2005), 45.  doi: 10.1007/s10589-005-4559-5.  Google Scholar [26] M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints,, Springer, (2009).   Google Scholar [27] G. Inglese, An inverse problem in corrosion detection,, Inverse Problems, 13 (1997), 977.  doi: 10.1088/0266-5611/13/4/006.  Google Scholar [28] V. Isakov, Inverse Problems for Partial Differential Equations,, $2^{nd}$ edition, (2006).   Google Scholar [29] M. Jourhmane and A. Nachaoui, Convergence of an alternating method to solve the Cauchy problem for Poisson's equation,, Appl. Anal., 81 (2002), 1065.  doi: 10.1080/0003681021000029819.  Google Scholar [30] S. I. Kabanikhin and A. L. Karchevsky, Optimizational method for solving the Cauchy problem for an elliptic equation,, J. Inverse Ill-Posed problems, 3 (1995), 21.  doi: 10.1515/jiip.1995.3.1.21.  Google Scholar [31] M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace's equation,, SIAM J. Appl. Math., 51 (1991), 1653.  doi: 10.1137/0151085.  Google Scholar [32] R. V. Kohn and M. S. Vogelius, Determining conductivity by boundary measurements: II. Interior results,, Comm. Pure Appl. Math., 38 (1985), 643.  doi: 10.1002/cpa.3160380513.  Google Scholar [33] V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations,, U.S.S.R. Comput. Math. and Math. Phys., 31 (1991), 45.   Google Scholar [34] R. Lattès and J.-L. Lions, Méthode de Quasi-réversibilité et Applications, (French) [The Quasi-Reversibility Methods and Applications],, Travaux et Recherches Mathématiques, (1967).   Google Scholar [35] R. Li and W. Liu, The C++ software library AFEPack., Available from: , ().   Google Scholar [36] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer-Verlag, (1971).   Google Scholar [37] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications,, Vol. II, (1972).   Google Scholar [38] M. Maher, K. Moez and T. Anis, A Cauchy problem for an inverse problem in image inpainting,, in Industrial Engineering and Systems Management (IESM), (2013).   Google Scholar [39] S. Pereverzev and E. Schock, Morozov's discrepancy principle for Tikhonov regularization of severely ill-posed problems in finite-dimensional subspaces,, Numer. Funct. Anal. Optim., 21 (2000), 901.  doi: 10.1080/01630560008816993.  Google Scholar [40] N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations,, Akademie Verlag, (1995).   Google Scholar [41] A. N. Tikhonov and V. A. Arsenin, Solutions of Ill-Posed Problems,, Winston and Sons, (1977).   Google Scholar [42] W. Weikl, H. Andra and E. Schnack, An alternating iterative algorithm for the reconstruction of internal cracks in a three-dimensional solid body,, Inverse Problems, 17 (2001), 1957.  doi: 10.1088/0266-5611/17/6/325.  Google Scholar [43] K. Yosida, Functional Analysis,, $5^{th}$ edition, (1978).   Google Scholar
 [1] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 [2] Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351 [3] Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120 [4] Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019 [5] Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099 [6] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [7] Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108 [8] Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 [9] Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046 [10] Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107 [11] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [12] Hassan Mohammad. A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2021, 17 (1) : 101-116. doi: 10.3934/jimo.2019101 [13] Shuai Huang, Zhi-Ping Fan, Xiaohuan Wang. Optimal financing and operational decisions of capital-constrained manufacturer under green credit and subsidy. Journal of Industrial & Management Optimization, 2021, 17 (1) : 261-277. doi: 10.3934/jimo.2019110 [14] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340 [15] Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347 [16] Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213 [17] Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110 [18] Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119 [19] Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 [20] Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032

2019 Impact Factor: 1.373