January  2018, 23(1): 79-99. doi: 10.3934/dcdsb.2018006

Optimal control of normalized SIMR models with vaccination and treatment

1. 

SYSTEC, DEEC, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

2. 

Universität Münster, Institut für Numerische und Angewandte Mathematik, Einsteinstr. 62,49143 Münster, Germany

3. 

ENSTA-Paris Tech, 828 Boulevard des Maréchaux, 91120 Palaiseau, France

* Corresponding author: MdR de Pinho, mrpinho@fe.up.pt

Received  December 2016 Published  January 2018

We study a model based on the so called SIR model to control the spreading of a disease in a varying population via vaccination and treatment. Since we assume that medical treatment is not immediate we add a new compartment, $M$, to the SIR model. We work with the normalized version of the proposed model. For such model we consider the problem of steering the system to a specified target. We consider both a fixed time optimal control problem with $L^1$ cost and the minimum time problem to drive the system to the target. In contrast to the literature, we apply different techniques of optimal control to our problems of interest.Using the direct method, we first solve the fixed time problem and then proceed to validate the computed solutions using both necessary conditions and second order sufficient conditions. Noteworthy, we perform a sensitivity analysis of the solutions with respect to some parameters in the model. We also use the Hamiltonian Jacobi approach to study how the minimum time function varies with respect to perturbations of the initial conditions. Additionally, we consider a multi-objective approach to study the trade off between the minimum time and the social costs of the control of diseases. Finally, we propose the application of Model Predictive Control to deal with uncertainties of the model.

Citation: Maria do Rosário de Pinho, Helmut Maurer, Hasnaa Zidani. Optimal control of normalized SIMR models with vaccination and treatment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 79-99. doi: 10.3934/dcdsb.2018006
References:
[1]

M. S. AronnaJ. F. BonnansA. V. Dmitruk and P. A. Lotito, Quadratic orderconditions for bang-singular extremals, Numerical Algebra, Control and Optimization, 2 (2012), 511-546.  doi: 10.3934/naco.2012.2.511.  Google Scholar

[2]

M. Assellaou, O. Bokanowski, A. Desilles and H. Zidani, A Hamilton-Jacobi-Bellman Approach for the Optimal Control of an Abort Landing Problem 55th IEEE Conference on Decision and Control, 2016. doi: 10.1109/CDC.2016.7798815.  Google Scholar

[3]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations Systems and Control: Foundations and Applications. Birkhäuser, Boston, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[4]

M. H. A. BiswasL. T. Paiva and M. d. R. de Pinho, A SEIR model for control of infectious diseases with constraints, Mathematical Biosciences and Engineering, 11 (2014), 761-784.  doi: 10.3934/mbe.2014.11.761.  Google Scholar

[5]

O. BokanowskiA. Briani and H. Zidani, Minumum time control problems for non autonomous differential equations, Systems and Control Letters, 58 (2009), 742-746.  doi: 10.1016/j.sysconle.2009.08.003.  Google Scholar

[6]

O. Bokanowski, A. Désilles, H. Zidani and J. Zhao, ROC-HJ software "Reachability, Optimal Control, and Hamilton-Jacobi equations, http://uma.ensta-paristech.fr/soft/ROC-HJ(2016). Google Scholar

[7]

O. BokanowskiN. Forcadel and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption, SIAM Journal on Control and Optimization, 48 (2010), 4292-4316.  doi: 10.1137/090762075.  Google Scholar

[8]

H. Bonnel and Y. C. Kaya, Optimization over the efficient set of multi-objective convex optimal control problems, J. Optim. Theory Appl., 147 (2010), 93-112.  doi: 10.1007/s10957-010-9709-y.  Google Scholar

[9]

C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse Für Optimale Steuerprozesse mit Steuer-und Zustands-Beschränkungen Dissertation, Institut für Numerische Mathematik, Universität Münster, Germany, 1998. Google Scholar

[10]

C. Büskens and H. Maurer, SQP-methods for solving optimal control problems with control and state constraints: Adjoint variables, sensitivity analysis and real-time control, Journal of Computational and Applied Mathematics, 120 (2000), 85-108.  doi: 10.1016/S0377-0427(00)00305-8.  Google Scholar

[11]

C. Büskens and H. Maurer, Sensitivity analysis and real-time optimization of parametric nonlinear programming problems, in: Online Optimization of Large Scale Systems (M. Grötschel, S. O. Krumke, J. Rambau, eds.), Springer-Verlag, Berlin, (2001), 3-16.  Google Scholar

[12]

C. Büskens and H. Maurer, Sensitivity analysis and real--time control of parametric optimal control problems using nonlinear programming methods, in: Online Optimization of Large Scale Systems (M. Grötschel, S. O. Krumke, J. Rambau, eds.), Springer-erlag, Berlin, (2001), 57-68.  Google Scholar

[13]

M. d. R. de Pinho and F. N. Nogueira, On application of optimal control to SEIR normalized models: Pros and cons, Mathematical Biosciences and Engineering, 14 (2017), 111-126.  doi: 10.3934/mbe.2017008.  Google Scholar

[14]

G. Eichfelder, Adaptive Scalarization Methods in Multiobjective Optimization Springer, Berlin, Heidelberg, 2008. doi: 10.1007/978-3-540-79159-1.  Google Scholar

[15]

M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations SIAM, Philadelphia, PA, 2014.  Google Scholar

[16]

A. V. Fiacco, Introduction to Sensitivity and Stability Analysis Academic Press, New York, London, 1983.  Google Scholar

[17] R. FourerD. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Brooks-Cole Publishing Company, Duxbury Press, 1993.   Google Scholar
[18]

L. Grüne and J. Pannek, Nonlinear Model Predictive Control. Theory and Algorithms Springer, 2011. doi: 10.1007/978-0-85729-501-9.  Google Scholar

[19]

M. R. Hestenes, Calculus of Variations and Optimal Control Theory John Wiley & Sons, Inc., New York-London-Sydney, 1966.  Google Scholar

[20]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[21]

H. W. Hethcote, The basic epidemiology models: Models, expressions for $ R_0$, parameter estimation, and applications, In:Mathematical Understanding of Infectious Disease Dynamics (S. Ma and Y. Xia, Eds.), 16 (2009), 1-61.  doi: 10.1142/9789812834836_0001.  Google Scholar

[22]

Y. C. Kaya and H. Maurer, A numerical method for nonconvex multi-objective optimal control problems, Computational Optimization and Applications, 57 (2014), 685-702.  doi: 10.1007/s10589-013-9603-2.  Google Scholar

[23]

A. J. Krener, The high order Maximal Principle and its application to singular extremals, SIAM J.on Control and Optimization, 15 (1977), 256-293.  doi: 10.1137/0315019.  Google Scholar

[24]

U. Ledzewicz and H. Schättler, On optimal singular control for a general SIR-model with vaccination and treatment, Discrete and Continuous Dynamical Systems, 2 (2011), 981-990.   Google Scholar

[25]

H. Maurer and M. d. R. de Pinho, Optimal control of epdimiological SEIR models with $ L^1$ objective and control-state constraints, Pacific Journal of Optimization, 12 (2016), 415-436.   Google Scholar

[26]

H. MaurerC. BüskensJ.-H. R. Kim and Y. Kaya, Optimization methods for the verification of second-order sufficient conditions for bang-bang controls, Optimal Control Methods and Applications, 26 (2005), 129-156.  doi: 10.1002/oca.756.  Google Scholar

[27]

R. M. Neilan and S. Lenhart, An introduction to optimal control with an application in disease modeling, DIMACS Series in Discrete Mathematics, 75 (2010), 67-81.   Google Scholar

[28]

N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control Advances in Design and Control, 24. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2012. doi: 10.1137/1.9781611972368.  Google Scholar

[29]

L. S. Pontryagin, V. G. Boltyanski, R. V. Gramkrelidze and E. F. Miscenko, The Mathematical Theory of Optimal Processes (in Russian), Fitzmatgiz, Moscow; English translation: Pergamon Press, New York, 1964.  Google Scholar

[30]

C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, Journal of Computational Physics, 77 (1988), 439-471.  doi: 10.1016/0021-9991(88)90177-5.  Google Scholar

[31]

A. Wächter and L. T. Biegler, On the implementation of a primal-dual interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y.  Google Scholar

show all references

References:
[1]

M. S. AronnaJ. F. BonnansA. V. Dmitruk and P. A. Lotito, Quadratic orderconditions for bang-singular extremals, Numerical Algebra, Control and Optimization, 2 (2012), 511-546.  doi: 10.3934/naco.2012.2.511.  Google Scholar

[2]

M. Assellaou, O. Bokanowski, A. Desilles and H. Zidani, A Hamilton-Jacobi-Bellman Approach for the Optimal Control of an Abort Landing Problem 55th IEEE Conference on Decision and Control, 2016. doi: 10.1109/CDC.2016.7798815.  Google Scholar

[3]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations Systems and Control: Foundations and Applications. Birkhäuser, Boston, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[4]

M. H. A. BiswasL. T. Paiva and M. d. R. de Pinho, A SEIR model for control of infectious diseases with constraints, Mathematical Biosciences and Engineering, 11 (2014), 761-784.  doi: 10.3934/mbe.2014.11.761.  Google Scholar

[5]

O. BokanowskiA. Briani and H. Zidani, Minumum time control problems for non autonomous differential equations, Systems and Control Letters, 58 (2009), 742-746.  doi: 10.1016/j.sysconle.2009.08.003.  Google Scholar

[6]

O. Bokanowski, A. Désilles, H. Zidani and J. Zhao, ROC-HJ software "Reachability, Optimal Control, and Hamilton-Jacobi equations, http://uma.ensta-paristech.fr/soft/ROC-HJ(2016). Google Scholar

[7]

O. BokanowskiN. Forcadel and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption, SIAM Journal on Control and Optimization, 48 (2010), 4292-4316.  doi: 10.1137/090762075.  Google Scholar

[8]

H. Bonnel and Y. C. Kaya, Optimization over the efficient set of multi-objective convex optimal control problems, J. Optim. Theory Appl., 147 (2010), 93-112.  doi: 10.1007/s10957-010-9709-y.  Google Scholar

[9]

C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse Für Optimale Steuerprozesse mit Steuer-und Zustands-Beschränkungen Dissertation, Institut für Numerische Mathematik, Universität Münster, Germany, 1998. Google Scholar

[10]

C. Büskens and H. Maurer, SQP-methods for solving optimal control problems with control and state constraints: Adjoint variables, sensitivity analysis and real-time control, Journal of Computational and Applied Mathematics, 120 (2000), 85-108.  doi: 10.1016/S0377-0427(00)00305-8.  Google Scholar

[11]

C. Büskens and H. Maurer, Sensitivity analysis and real-time optimization of parametric nonlinear programming problems, in: Online Optimization of Large Scale Systems (M. Grötschel, S. O. Krumke, J. Rambau, eds.), Springer-Verlag, Berlin, (2001), 3-16.  Google Scholar

[12]

C. Büskens and H. Maurer, Sensitivity analysis and real--time control of parametric optimal control problems using nonlinear programming methods, in: Online Optimization of Large Scale Systems (M. Grötschel, S. O. Krumke, J. Rambau, eds.), Springer-erlag, Berlin, (2001), 57-68.  Google Scholar

[13]

M. d. R. de Pinho and F. N. Nogueira, On application of optimal control to SEIR normalized models: Pros and cons, Mathematical Biosciences and Engineering, 14 (2017), 111-126.  doi: 10.3934/mbe.2017008.  Google Scholar

[14]

G. Eichfelder, Adaptive Scalarization Methods in Multiobjective Optimization Springer, Berlin, Heidelberg, 2008. doi: 10.1007/978-3-540-79159-1.  Google Scholar

[15]

M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations SIAM, Philadelphia, PA, 2014.  Google Scholar

[16]

A. V. Fiacco, Introduction to Sensitivity and Stability Analysis Academic Press, New York, London, 1983.  Google Scholar

[17] R. FourerD. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Brooks-Cole Publishing Company, Duxbury Press, 1993.   Google Scholar
[18]

L. Grüne and J. Pannek, Nonlinear Model Predictive Control. Theory and Algorithms Springer, 2011. doi: 10.1007/978-0-85729-501-9.  Google Scholar

[19]

M. R. Hestenes, Calculus of Variations and Optimal Control Theory John Wiley & Sons, Inc., New York-London-Sydney, 1966.  Google Scholar

[20]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[21]

H. W. Hethcote, The basic epidemiology models: Models, expressions for $ R_0$, parameter estimation, and applications, In:Mathematical Understanding of Infectious Disease Dynamics (S. Ma and Y. Xia, Eds.), 16 (2009), 1-61.  doi: 10.1142/9789812834836_0001.  Google Scholar

[22]

Y. C. Kaya and H. Maurer, A numerical method for nonconvex multi-objective optimal control problems, Computational Optimization and Applications, 57 (2014), 685-702.  doi: 10.1007/s10589-013-9603-2.  Google Scholar

[23]

A. J. Krener, The high order Maximal Principle and its application to singular extremals, SIAM J.on Control and Optimization, 15 (1977), 256-293.  doi: 10.1137/0315019.  Google Scholar

[24]

U. Ledzewicz and H. Schättler, On optimal singular control for a general SIR-model with vaccination and treatment, Discrete and Continuous Dynamical Systems, 2 (2011), 981-990.   Google Scholar

[25]

H. Maurer and M. d. R. de Pinho, Optimal control of epdimiological SEIR models with $ L^1$ objective and control-state constraints, Pacific Journal of Optimization, 12 (2016), 415-436.   Google Scholar

[26]

H. MaurerC. BüskensJ.-H. R. Kim and Y. Kaya, Optimization methods for the verification of second-order sufficient conditions for bang-bang controls, Optimal Control Methods and Applications, 26 (2005), 129-156.  doi: 10.1002/oca.756.  Google Scholar

[27]

R. M. Neilan and S. Lenhart, An introduction to optimal control with an application in disease modeling, DIMACS Series in Discrete Mathematics, 75 (2010), 67-81.   Google Scholar

[28]

N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control Advances in Design and Control, 24. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2012. doi: 10.1137/1.9781611972368.  Google Scholar

[29]

L. S. Pontryagin, V. G. Boltyanski, R. V. Gramkrelidze and E. F. Miscenko, The Mathematical Theory of Optimal Processes (in Russian), Fitzmatgiz, Moscow; English translation: Pergamon Press, New York, 1964.  Google Scholar

[30]

C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, Journal of Computational Physics, 77 (1988), 439-471.  doi: 10.1016/0021-9991(88)90177-5.  Google Scholar

[31]

A. Wächter and L. T. Biegler, On the implementation of a primal-dual interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y.  Google Scholar

Figure 1.  SIMR compartmental model with treatment and vaccination.
Figure 2.  Case 1: state variables for $(P_1)$ with parameters as in Table 1.
Figure 4.  Case 1 with parameters as in Table 1. (left) comparison between the computed values $u_*(t)$ and the values of the singular control $u_{sing}(x_*(t),p_*(t))$ (19), (right) control $v_*(t)$ and switching function $\phi_2$ showing that the control law (14) for $v_*$ is precisely satisfied.
Figure 5.  Optimal controls for $(P_1)$. Left: Control $u_*(t)$ for Case 1 in red and, for Case 2, control $u_*(t)$ in blue and its switching function $\phi_1(t)$ in dashed blue. Right: Control $v_*(t)$ for Case 1 in red and, for Case 2, control $v_*(t)$ in blue and its switching function $\phi_2(t)$ in dashed blue,
Figure 3.  Case 1: controls for $(P_1)$ with parameters as in Table 1.
Figure 6.  Value of the minimum time function for different values of $(s_0,i_0)$. The vertical axis corresponds to the values of the minimum time.
Figure 7.  State variables for optimal trajectories corresponding to different values of the initial position $(s_0,i_0,m_0)$.
Figure 8.  Case 1 :$A=10, B=1, C=3$. Minimize the scalarized objective $J^{(w)}$ (23). Top row: (left) Pareto front $(J_1,T)$ for $w \in [0,1]$, (right) objective $J_1$ as function of $w \in [0,1]$. Bottom row: (left) Distance of Pareto curve to the origin, (right) objective $J_2=T$ as function of $w \in [0,1]$.
Figure 9.  Case 2 : $A=10, B=3, C=1$. Minimize the scalarized objective $J^{(w)}$ (23) for $w \in [0,1]$. Top row: (left) Pareto front $(J_1,J_2)$ for $w \in [0,1]$, (right) objective $J_1(u,v)$ as function of $w \in [0,1]$. Bottom row: (left) Distance of Pareto curve to the origin, (right) objective $J_2=T$ as function of $w \in [0,1]$.
Figure 10.  Perturbed incidence rate $c_p$.
Figure 11.  Six computed values of the state variables for the Fixed Horizon MPC described above assuming that the "real" system differs from the model of $(P_1)$ solely with respect to the incidence rate $c$. From left to right on the first row, we show the $s$ and $i$ state variables; on the second row we show the $m$ state variable. All the three graphs have different scales.
Table 1.  Parameters for the normalized SIMR model (7)-(10)
Parameter Description Value
$ b$ Natural birth rate 0.01
$ c$ Incidence coefficient 1.1
$ a_1$ Infection induced death rate 0.08
$ a_2$ Treatment induced death rate 0.005
$ g_1$ Recovery rate of those infected 0.02
$ g_2$ Recovery rate of those under treatment 0.5
$ \eta$ Efficiency of vaccine 0.8
$ u_{max}$ Maximum rate of vaccination 0.8
$ v_{max}$ Maximum rate of treatment 0.8
$ s_0$ Initial percentage of susceptible population 0.95
$ i_0$ Initial percentage of infected population 0.05
$ m_0$ Initial percentage of infected population under treatment 0
Parameter Description Value
$ b$ Natural birth rate 0.01
$ c$ Incidence coefficient 1.1
$ a_1$ Infection induced death rate 0.08
$ a_2$ Treatment induced death rate 0.005
$ g_1$ Recovery rate of those infected 0.02
$ g_2$ Recovery rate of those under treatment 0.5
$ \eta$ Efficiency of vaccine 0.8
$ u_{max}$ Maximum rate of vaccination 0.8
$ v_{max}$ Maximum rate of treatment 0.8
$ s_0$ Initial percentage of susceptible population 0.95
$ i_0$ Initial percentage of infected population 0.05
$ m_0$ Initial percentage of infected population under treatment 0
Table2 
Numerical results for $(P_1)$ with $ A=10, B=1, C=3 : {\text{Cost}} \ J_1= 22.02571 ,$
$ s(T)= 1.201593 e-01 ,$ $i( T)= 5e-04 ,$ $m(T)= 8.932 084 e-06$
$p_s(0)= -6.3486 ,$ $p_i(0) = -7.6544e+01,$ $p_m(0) = -6.2821e-02,$
$p_s(T)= -3.2237e-03,$ $p_i(T) = -5.8613e+03,$ $p_m(T) = -1.4648e-05 .$
Numerical results for $(P_1)$ with $ A=10, B=1, C=3 : {\text{Cost}} \ J_1= 22.02571 ,$
$ s(T)= 1.201593 e-01 ,$ $i( T)= 5e-04 ,$ $m(T)= 8.932 084 e-06$
$p_s(0)= -6.3486 ,$ $p_i(0) = -7.6544e+01,$ $p_m(0) = -6.2821e-02,$
$p_s(T)= -3.2237e-03,$ $p_i(T) = -5.8613e+03,$ $p_m(T) = -1.4648e-05 .$
Table3 
Numerical results fors $(P_1)$ with $ A=10, B=3, C=1 : {\text{Cost}} J_1= 15.56816 ,$
$s(T)= 3.337 73e-01 ,$ $i( T)= 5e-04 ,$ $m(T)= 1.370 10 e-05 $
$p_s(0)= -5.614 47 ,$ $p_i(0) = -3.432 49e+01,$ $p_m(0) = -5.360 07 e-02,$
$p_s(T)= -5.4088e-04 ,$ $p_i(T) = -1.9668 e+03 ,$ $p_m(T) = -2.4584e-06.$
Numerical results fors $(P_1)$ with $ A=10, B=3, C=1 : {\text{Cost}} J_1= 15.56816 ,$
$s(T)= 3.337 73e-01 ,$ $i( T)= 5e-04 ,$ $m(T)= 1.370 10 e-05 $
$p_s(0)= -5.614 47 ,$ $p_i(0) = -3.432 49e+01,$ $p_m(0) = -5.360 07 e-02,$
$p_s(T)= -5.4088e-04 ,$ $p_i(T) = -1.9668 e+03 ,$ $p_m(T) = -2.4584e-06.$
Table4 
parameter $dt_1/dq$ $dt_2/dq$ $dt_3/dq$ $du_c/dq$
$ q = B$ -0.5892, 0.7584, 0.7584, -0.1868,
$ q = C$ 0.1520, -0.6556, -0.6556, 0.07068,
$q = a_1 $ -1.960, -23.52, -23.52, 0.4620.
parameter $dt_1/dq$ $dt_2/dq$ $dt_3/dq$ $du_c/dq$
$ q = B$ -0.5892, 0.7584, 0.7584, -0.1868,
$ q = C$ 0.1520, -0.6556, -0.6556, 0.07068,
$q = a_1 $ -1.960, -23.52, -23.52, 0.4620.
Table5 
parameter $dt_1/dq$ $dt_2/dq$
$ q = B$ -0.49896, 1.45249
$ q = C$ 1.4525, -5.4816
$q = a_1 $ 0.27758, -24.112
parameter $dt_1/dq$ $dt_2/dq$
$ q = B$ -0.49896, 1.45249
$ q = C$ 1.4525, -5.4816
$q = a_1 $ 0.27758, -24.112
[1]

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

[2]

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

[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]

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

[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]

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

[7]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[8]

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

[9]

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

[10]

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

[11]

Tien-Yu Lin, Bhaba R. Sarker, Chien-Jui Lin. An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts. Journal of Industrial & Management Optimization, 2021, 17 (1) : 467-484. doi: 10.3934/jimo.2020043

[12]

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

[13]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379

[14]

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

[15]

José Madrid, João P. G. Ramos. On optimal autocorrelation inequalities on the real line. Communications on Pure & Applied Analysis, 2021, 20 (1) : 369-388. doi: 10.3934/cpaa.2020271

[16]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[17]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267

[18]

Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020

[19]

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

[20]

Haili Yuan, Yijun Hu. Optimal investment for an insurer under liquid reserves. Journal of Industrial & Management Optimization, 2021, 17 (1) : 339-355. doi: 10.3934/jimo.2019114

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (220)
  • HTML views (121)
  • Cited by (2)

[Back to Top]