American Institute of Mathematical Sciences

Advanced
doi: 10.3934/jgm.2020032

Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model

Bernard Bonnard 1, and  Jérémy Rouot 2,,

1. 

McTAO team, INRIA Sophia Antipolis, 2004 Route des Lucioles, 06902 Valbonne, France
2. 

L@bISEN, Yncréa Ouest, 20 rue Cuirassé Bretagne, 29200 Brest, France

* Corresponding author: Jérémy Rouot

Received  May 2020 Revised  September 2020 Published  November 2020

Fund Project: This research paper benefited from the support of the FMJH Program PGMO and from the support of EDF, Thales, Orange and the authors are partially supported by the Labex AMIES

A recent force-fatigue parameterized mathematical model, based on the seminal contributions of V. Hill to describe muscular activity, allows to predict the muscular force response to external electrical stimulation (FES) and it opens the road to optimize the FES-input to maximize the force response to a pulse train, to track a reference force while minimizing the fatigue for a sequence of pulse trains or to follow a reference joint angle trajectory to produce motion in the non-isometric case. In this article, we introduce the geometric frame to analyze the dynamics and we present Pontryagin types necessary optimality conditions adapted to digital controls, used in the experiments, vs permanent control and which fits in the optimal sampled-data control frame. This leads to Hamiltonian differential variational inequalities, which can be numerically implemented vs direct optimization schemes.

Keywords: Biomechanics force-fatigue models, geometric optimal control, sampled-data control problem, Pontryagin type necessary conditions, Hamiltonian differential variational inequality.
Mathematics Subject Classification: 49K15, 93B07, 92B05.
Citation: 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, doi: 10.3934/jgm.2020032
References:
[1]

A. A. Agrachëv and R. V. Gamkrelidze, Symplectic geometry for optimal control, in Nonlinear Controllability and Optimal Control, Monogr. Textbooks Pure Appl. Math., 133, Dekker, New York, 1990,263-277.  Google Scholar

[2]

T. Bakir, B. Bonnard, L. Bourdin and J. Rouot, Direct and indirect methods to optimize the muscular force response to a pulse train of electrical stimulation, in progress. Google Scholar

[3]

T. Bakir, B. Bonnard, L. Bourdin and J. Rouot, Pontryagin-type conditions for optimal muscular force response to functional electrical stimulations, J. Optim. Theory Appl., 184 (2020), 581-602.  doi: 10.1007/s10957-019-01599-4.  Google Scholar

[4]

T. Bakir, B. Bonnard and S. Othman, Predictive control based on nonlinear observer for muscular force and fatigue model, 2018 Annual American Control Conference (ACC), Milwaukee, WI, 2018, 2157-2162. doi: 10.23919/ACC.2018.8430962.  Google Scholar

[5]

T. Bakir, B. Bonnard and J. Rouot, A case study of optimal input-output system with sampled-data control: Ding et al. force and fatigue muscular control model, Netw. Heterog. Media, 14 (2019), 79-100.  doi: 10.3934/nhm.2019005.  Google Scholar

[6] G. A. Bliss, Lectures on the Calculus of Variations, University of Chicago Press, Chicago, Ill., 1946.   Google Scholar
[7]

B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory, Mathematics & Applications, 40, Springer-Verlag, Berlin, 2003.  Google Scholar

[8]

B. Bonnard and I. Kupka, Generic properties of singular trajectories, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 167-186.  doi: 10.1016/S0294-1449(97)80143-6.  Google Scholar

[9]

U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, Mathematics & Applications, 43, Springer-Verlag, Berlin, 2004.  Google Scholar

[10]

N. Bourbaki, Lie groups and Lie algebras. Chapters 4-6, in Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-540-89394-3.  Google Scholar

[11]

L. Bourdin and G. Dhar, Continuity/constancy of the Hamiltonian function in a Pontryagin maximum principle for optimal sampled-data control problems with free sampling times, Math. Control Signals Systems, 31 (2019), 503-544.  doi: 10.1007/s00498-019-00247-6.  Google Scholar

[12]

L. Bourdin and E. Trélat, Optimal sampled-data control, and generalizations on time scales, Math. Control Relat. Fields, 6 (2016), 53-94.  doi: 10.3934/mcrf.2016.6.53.  Google Scholar

[13]

P. Brunovský, A classification of linear controllable systems, Kybernetika (Prague), 6 (1970), 173-188.   Google Scholar

[14]

J.-B. Caillau, O. Cots and J. Gergaud, Differential continuation for regular optimal control problems, Optim. Methods Softw., 27 (2012), 177-196.  doi: 10.1080/10556788.2011.593625.  Google Scholar

[15]

J. Ding, A. S. Wexler and S. A. Binder-Macleod, A predictive fatigue model. Ⅰ. Predicting the effect of stimulation frequency and pattern on fatigue, IEEE Transactions on Neural Systems and Rehabilitation Engineering, 10 (2002), 48-58.  doi: 10.1109/TNSRE.2002.1021586.  Google Scholar

[16]

J. Ding, A. S. Wexler and S. A. Binder-Macleod, A predictive fatigue model. Ⅱ. Predicting the effect of resting times on fatigue, IEEE Transactions on Neural Systems and Rehabilitation Engineering, 10 (2002), 59-67.  doi: 10.1109/TNSRE.2002.1021587.  Google Scholar

[17]

J. Ding, A. S. Wexler and S. A. Binder-Macleod, Development of a mathematical model that predicts optimal muscle activation patterns by using brief trains, J. Appl. Physiol., 88 (2000), 917-925.  doi: 10.1152/jappl.2000.88.3.917.  Google Scholar

[18]

B. D. Doll, N. A. Kirsch and N. Sharma, Optimization of a stimulation train based on a predictive model of muscle force and fatigue, IFAC-PapersOnLine, 48 (2015), 338-342.  doi: 10.1016/j.ifacol.2015.10.162.  Google Scholar

[19]

R. V. Gamkrelidze, Discovery of the maximum principle, J. Dynam. Control Systems, 5 (1999), 437-451.  doi: 10.1023/A:1021783020548.  Google Scholar

[20]

I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1963.  Google Scholar

[21]

R. Gesztelyi, J. Zsuga, A. Kemeny-Beke, B. Varga, B. Juhasz and A. Tosaki, The Hill equation and the origin of quantitative pharmacology, Arch. Hist. Exact Sci., 66 (2012), 427-438.  doi: 10.1007/s00407-012-0098-5.  Google Scholar

[22]

S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Graduate Studies in Mathematics, 34, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/034.  Google Scholar

[23]

R. Hermann and A. J. Krener, Nonlinear controllability and observability, IEEE Trans. Automatic Control, AC-22 (1977), 728-740.  doi: 10.1109/tac.1977.1101601.  Google Scholar

[24]

A. Isidori, Non-Linear Control Systems, Communications and Control Engineering Series, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-1-84628-615-5.  Google Scholar

[25]

E. B. Lee and L. Markus, Foundations of Optimal Control Theory, Robert E. Kreiger Publishing Co., Inc., Melbourne, FL, 1986.  Google Scholar

[26]

M. S. Marion, Predicting Fatigue During Electrically Stimulated Non-Isometric Contractions, Ph.D thesis, University of California Davis, 2010.  Google Scholar

[27]

M. S. Marion, A. S. Wexler and M. L. Hull, Predicting non-isometric fatigue induced by electrical stimulation pulse trains as a function of pulse duration, J. Neuroengineering Rehab., 10 (2013). doi: 10.1186/1743-0003-10-13.  Google Scholar

[28]

V. Renault, M. Thieullen and E. Trélat, Minimal time spiking in various ChR2-controlled neuron models, J. Math. Biol., 76 (2018), 567-608.  doi: 10.1007/s00285-017-1101-1.  Google Scholar

[29]

H. Schättler and U. Ledzewicz, Geometric Optimal Control. Theory, Methods and Examples, Interdisciplinary Applied Mathematics, 38, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2.  Google Scholar

[30]

H. J. Sussmann and V. Jurdjevic, Controllability of nonlinear systems, J. Differential Equations, 12 (1972), 95-116.  doi: 10.1016/0022-0396(72)90007-1.  Google Scholar

[31]

R. Vinter, Optimal Control, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2000.  Google Scholar

show all references

References:
[1]

A. A. Agrachëv and R. V. Gamkrelidze, Symplectic geometry for optimal control, in Nonlinear Controllability and Optimal Control, Monogr. Textbooks Pure Appl. Math., 133, Dekker, New York, 1990,263-277.  Google Scholar

[2]

T. Bakir, B. Bonnard, L. Bourdin and J. Rouot, Direct and indirect methods to optimize the muscular force response to a pulse train of electrical stimulation, in progress. Google Scholar

[3]

T. Bakir, B. Bonnard, L. Bourdin and J. Rouot, Pontryagin-type conditions for optimal muscular force response to functional electrical stimulations, J. Optim. Theory Appl., 184 (2020), 581-602.  doi: 10.1007/s10957-019-01599-4.  Google Scholar

[4]

T. Bakir, B. Bonnard and S. Othman, Predictive control based on nonlinear observer for muscular force and fatigue model, 2018 Annual American Control Conference (ACC), Milwaukee, WI, 2018, 2157-2162. doi: 10.23919/ACC.2018.8430962.  Google Scholar

[5]

T. Bakir, B. Bonnard and J. Rouot, A case study of optimal input-output system with sampled-data control: Ding et al. force and fatigue muscular control model, Netw. Heterog. Media, 14 (2019), 79-100.  doi: 10.3934/nhm.2019005.  Google Scholar

[6] G. A. Bliss, Lectures on the Calculus of Variations, University of Chicago Press, Chicago, Ill., 1946.   Google Scholar
[7]

B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory, Mathematics & Applications, 40, Springer-Verlag, Berlin, 2003.  Google Scholar

[8]

B. Bonnard and I. Kupka, Generic properties of singular trajectories, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 167-186.  doi: 10.1016/S0294-1449(97)80143-6.  Google Scholar

[9]

U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, Mathematics & Applications, 43, Springer-Verlag, Berlin, 2004.  Google Scholar

[10]

N. Bourbaki, Lie groups and Lie algebras. Chapters 4-6, in Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-540-89394-3.  Google Scholar

[11]

L. Bourdin and G. Dhar, Continuity/constancy of the Hamiltonian function in a Pontryagin maximum principle for optimal sampled-data control problems with free sampling times, Math. Control Signals Systems, 31 (2019), 503-544.  doi: 10.1007/s00498-019-00247-6.  Google Scholar

[12]

L. Bourdin and E. Trélat, Optimal sampled-data control, and generalizations on time scales, Math. Control Relat. Fields, 6 (2016), 53-94.  doi: 10.3934/mcrf.2016.6.53.  Google Scholar

[13]

P. Brunovský, A classification of linear controllable systems, Kybernetika (Prague), 6 (1970), 173-188.   Google Scholar

[14]

J.-B. Caillau, O. Cots and J. Gergaud, Differential continuation for regular optimal control problems, Optim. Methods Softw., 27 (2012), 177-196.  doi: 10.1080/10556788.2011.593625.  Google Scholar

[15]

J. Ding, A. S. Wexler and S. A. Binder-Macleod, A predictive fatigue model. Ⅰ. Predicting the effect of stimulation frequency and pattern on fatigue, IEEE Transactions on Neural Systems and Rehabilitation Engineering, 10 (2002), 48-58.  doi: 10.1109/TNSRE.2002.1021586.  Google Scholar

[16]

J. Ding, A. S. Wexler and S. A. Binder-Macleod, A predictive fatigue model. Ⅱ. Predicting the effect of resting times on fatigue, IEEE Transactions on Neural Systems and Rehabilitation Engineering, 10 (2002), 59-67.  doi: 10.1109/TNSRE.2002.1021587.  Google Scholar

[17]

J. Ding, A. S. Wexler and S. A. Binder-Macleod, Development of a mathematical model that predicts optimal muscle activation patterns by using brief trains, J. Appl. Physiol., 88 (2000), 917-925.  doi: 10.1152/jappl.2000.88.3.917.  Google Scholar

[18]

B. D. Doll, N. A. Kirsch and N. Sharma, Optimization of a stimulation train based on a predictive model of muscle force and fatigue, IFAC-PapersOnLine, 48 (2015), 338-342.  doi: 10.1016/j.ifacol.2015.10.162.  Google Scholar

[19]

R. V. Gamkrelidze, Discovery of the maximum principle, J. Dynam. Control Systems, 5 (1999), 437-451.  doi: 10.1023/A:1021783020548.  Google Scholar

[20]

I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1963.  Google Scholar

[21]

R. Gesztelyi, J. Zsuga, A. Kemeny-Beke, B. Varga, B. Juhasz and A. Tosaki, The Hill equation and the origin of quantitative pharmacology, Arch. Hist. Exact Sci., 66 (2012), 427-438.  doi: 10.1007/s00407-012-0098-5.  Google Scholar

[22]

S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Graduate Studies in Mathematics, 34, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/034.  Google Scholar

[23]

R. Hermann and A. J. Krener, Nonlinear controllability and observability, IEEE Trans. Automatic Control, AC-22 (1977), 728-740.  doi: 10.1109/tac.1977.1101601.  Google Scholar

[24]

A. Isidori, Non-Linear Control Systems, Communications and Control Engineering Series, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-1-84628-615-5.  Google Scholar

[25]

E. B. Lee and L. Markus, Foundations of Optimal Control Theory, Robert E. Kreiger Publishing Co., Inc., Melbourne, FL, 1986.  Google Scholar

[26]

M. S. Marion, Predicting Fatigue During Electrically Stimulated Non-Isometric Contractions, Ph.D thesis, University of California Davis, 2010.  Google Scholar

[27]

M. S. Marion, A. S. Wexler and M. L. Hull, Predicting non-isometric fatigue induced by electrical stimulation pulse trains as a function of pulse duration, J. Neuroengineering Rehab., 10 (2013). doi: 10.1186/1743-0003-10-13.  Google Scholar

[28]

V. Renault, M. Thieullen and E. Trélat, Minimal time spiking in various ChR2-controlled neuron models, J. Math. Biol., 76 (2018), 567-608.  doi: 10.1007/s00285-017-1101-1.  Google Scholar

[29]

H. Schättler and U. Ledzewicz, Geometric Optimal Control. Theory, Methods and Examples, Interdisciplinary Applied Mathematics, 38, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2.  Google Scholar

[30]

H. J. Sussmann and V. Jurdjevic, Controllability of nonlinear systems, J. Differential Equations, 12 (1972), 95-116.  doi: 10.1016/0022-0396(72)90007-1.  Google Scholar

[31]

R. Vinter, Optimal Control, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2000.  Google Scholar

Figure 1.  Extremal of the Ding et al. model defined on $ [0,T] $, $ T = 100ms $, with four sampling times and maximizing the final force
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Sensitivity analysis. Time evolution of the Jacobi fields component $ \delta F(\cdot) $ according to Definition 2.7 and associated to the trajectory of the force-fatigue model
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Sensitivity analysis. Time evolution of the Jacobi fields component $ \delta F(\cdot) $ where their initializations are $ 10\% $ of the physical values $ A_{\text{rest}},K_{m,\text{rest}},\tau_{1,\text{rest}} $ associated to the trajectory of the force-fatigue model
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Time evolution on $ [0,T],\, T = 0.4s $ of the state of the Ding et al. model computed with the indirect approach maximizing $ \varphi(x(T)) = F(T) $ with $ n = 6 $, $ I_m = 20 $ms. The optimal value is the same as the optimal value computed with the direct method (see Fig.5). The model parameters are given in Table 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Time evolution on $ [0,T],\, T = 0.4s $ of the normalized state of the Ding et al. model computed with the direct approach maximizing $ \varphi(x(T)) = F(T) $ with $ n = 6 $, $ I_m = 20 $ms. The model parameters are given in Table 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Time evolution on $ [0,T],\, T = 0.9s $ of the normalized state of the Marion et al. model computed with the direct approach maximizing $ \varphi(x(T)) = F(T)/F_{ext}+A_{90}(T)/A_{90,0} $ with $ n = 10 $, $ I_m = 20 $ms. The model parameters are given in Table 2
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Time evolution on $ [0,T],\, T = 0.9s $ of the normalized state of the Marion et al. model with the direct method when considering $ t_i = T - (n-i+1)I_m,\ i = 1,\dots,n $, $ n = 10 $, $ I_m = 20ms $. The model parameters are given in Table 2
Figure Options

Download full-size image

Download as PowerPoint slide

Table 2.  Numerical values for parameters of the Marion et al. model
Variable Value Unit Variable Value Unit
$ F_{load} $ 45.4 N $ F_M $ 247.5 N
$ \tau_c $ 20×-3 s $ \tau_{fat} $ 99.4 s
$ A_{90,0} $ 2100 N s−1 $ K_{m,0} $ 3.52×-1
$ \tau_{1,0} $ 3.61×-2 s $ \tau_2 $ 5.21×-2 s
$ a $ 4.49×-4 deg−2 $ b $ 3.44×-2 deg−1
$ v_1 $ 3.71×-1 N deg−2 $ v_2 $ 2.29×-2 deg−1
$ \ell = L/I $ 9.85 kg m−1 $ \alpha_A $ -4.02×-1 s−2
$ \alpha_{K_m} $ 1.36×-5 s−1 N−1 $ \alpha_{\tau_1} $ 2.93×-5 N−1
$ \beta_{\tau_1} $ 8.54×-7 s deg−1 N−1 $ R_0 $ 1.143
$ \beta_{90} $ 0 deg−1 s−1 $ \beta_{K_m} $ 0 deg−1 N−1
Table Options

Download as excel

Variable Value Unit Variable Value Unit
$ F_{load} $ 45.4 N $ F_M $ 247.5 N
$ \tau_c $ 20×-3 s $ \tau_{fat} $ 99.4 s
$ A_{90,0} $ 2100 N s−1 $ K_{m,0} $ 3.52×-1
$ \tau_{1,0} $ 3.61×-2 s $ \tau_2 $ 5.21×-2 s
$ a $ 4.49×-4 deg−2 $ b $ 3.44×-2 deg−1
$ v_1 $ 3.71×-1 N deg−2 $ v_2 $ 2.29×-2 deg−1
$ \ell = L/I $ 9.85 kg m−1 $ \alpha_A $ -4.02×-1 s−2
$ \alpha_{K_m} $ 1.36×-5 s−1 N−1 $ \alpha_{\tau_1} $ 2.93×-5 N−1
$ \beta_{\tau_1} $ 8.54×-7 s deg−1 N−1 $ R_0 $ 1.143
$ \beta_{90} $ 0 deg−1 s−1 $ \beta_{K_m} $ 0 deg−1 N−1
Table 1.  Numerical values for parameters of the Ding et al. model
Variable Value Unit Variable Value Unit
$ \tau_c $ 20×-3 s $ \tau_{fat} $ 127 s
$ A_{rest} $ 3.009 kN s−1 $ K_{m,rest} $ 1.03×-1
$ \tau_{1,rest} $ 5.1×-2 s $ \tau_2 $ 1.24×-1 s
$ \alpha_A $ -4×-1 s−2 $ \alpha_{K_m} $ 1.9×-2 s−1 kN−1
$ \alpha_{\tau_1} $ 2.1×-2 kN−1 $ R_0 $ 2
Table Options

Download as excel

Variable Value Unit Variable Value Unit
$ \tau_c $ 20×-3 s $ \tau_{fat} $ 127 s
$ A_{rest} $ 3.009 kN s−1 $ K_{m,rest} $ 1.03×-1
$ \tau_{1,rest} $ 5.1×-2 s $ \tau_2 $ 1.24×-1 s
$ \alpha_A $ -4×-1 s−2 $ \alpha_{K_m} $ 1.9×-2 s−1 kN−1
$ \alpha_{\tau_1} $ 2.1×-2 kN−1 $ R_0 $ 2
[1]

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
[2]

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
[3]

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
[4]

Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051
[5]

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
[6]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033
[7]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052
[8]

Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1711-1722. doi: 10.3934/dcdsb.2020179
[9]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045
[10]

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
[11]

Ali Wehbe, Rayan Nasser, Nahla Noun. Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020050
[12]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133
[13]

Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021022
[14]

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
[15]

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
[16]

Lars Grüne, Roberto Guglielmi. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields, 2021, 11 (1) : 169-188. doi: 10.3934/mcrf.2020032
[17]

Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026
[18]

Arthur Fleig, Lars Grüne. Strict dissipativity analysis for classes of optimal control problems involving probability density functions. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020053
[19]

Xiuli Xu, Xueke Pu. Optimal convergence rates of the magnetohydrodynamic model for quantum plasmas with potential force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 987-1010. doi: 10.3934/dcdsb.2020150
[20]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

2019 Impact Factor: 0.649

Tools

Metrics

  • PDF downloads (19)
  • HTML views (76)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences