June  2019, 24(6): 2577-2612. doi: 10.3934/dcdsb.2018266

Optimal control problems for the Gompertz model under the Norton-Simon hypothesis in chemotherapy

Dep. Matemáticas, Estadística y Computación, Universidad de Cantabria, Avda. de los Castros, s/n, 39005 Santander, Spain

Received  November 2017 Revised  April 2018 Published  October 2018

We study a collection of problems associated with the optimization of cancer chemotherapy treatments, under the assumptions of Gomperztian-type tumor growth and that the drug killing effect is proportional to the rate of growth for the untreated tumor (Norton-Simon hypothesis). Classical pharmacokinetics and different pharmacodynamics (Skipper and Emax) are considered, together with a toxicity limit or the penalization of the accumulated drug effect. Existence and uniqueness of the optimal control is proved in some cases, while in others the total amount of drug is the unique relevant aspect to take into account and the existence of an infinite number of optimal controls is shown. In all cases, explicit expressions for the solutions are derived in terms of the problem data. Finally, numerical results of illustrative examples and some conclusions are presented.

Citation: Luis A. Fernández, Cecilia Pola. Optimal control problems for the Gompertz model under the Norton-Simon hypothesis in chemotherapy. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2577-2612. doi: 10.3934/dcdsb.2018266
References:
[1]

S. Benzekry, C. Lamont, A. Beheshti, A. Tracz, J. M. L. Ebos, L. Hlatky and P. Hahnfeldt, Classical Mathematical Models for Description and Prediction of Experimental Tumor Growth, PLOS Comput. Biol., 10 (2014), e1003800. doi: 10.1371/journal.pcbi.1003800.  Google Scholar

[2]

S. BenzekryE. PasquierD. BarbolosiB. LacarelleF. BarlésiN. André and J. Ciccolini, Metronomic reloaded: Theoretical models bringing chemotherapy into the era of precision medicine, Semin. Cancer Biol., 35 (2015), 53-61.  doi: 10.1016/j.semcancer.2015.09.002.  Google Scholar

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011. doi: 10.1007/978-0-387-70914-7.  Google Scholar

[4]

L. Cesari, Optimization—Theory and Applications, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8165-5.  Google Scholar

[5]

J. Clairambault, Modelling physiological and pharmacological control on cell proliferation to optimise cancer treatments, Math. Model. Nat. Phenom., 4 (2009), 12-67.  doi: 10.1051/mmnp/20094302.  Google Scholar

[6]

C. L. DarbyW. W. Hager and A. V. Rao, An hp-adaptive pseudospectral method for solving optimal control problems, Optimal Control Appl. Methods, 32 (2011), 476-502.  doi: 10.1002/oca.957.  Google Scholar

[7]

C. FaivreD. BarbolosiE. Pasquier and N. André, A mathematical model for the administration of temozolomide: comparative analysis of conventional and metronomic chemotherapy regimens, Cancer Chemother. Pharmacol., 71 (2013), 1013-1019.  doi: 10.1007/s00280-013-2095-z.  Google Scholar

[8]

L. A. Fernández and C. Pola, Catalog of the optimal controls in cancer chemotherapy for the Gompertz model depending on PK/PD and the integral constraint, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1563-1588.  doi: 10.3934/dcdsb.2014.19.1563.  Google Scholar

[9]

K. R. Fister and J. C. Panetta, Optimal control applied to competing chemotherapeutic cell-kill strategies, SIAM J. Appl. Math., 63 (2003), 1954-1971.  doi: 10.1137/S0036139902413489.  Google Scholar

[10]

R. Langreth, The mathematics of cancer, Forbes Magazine, 2010, https://www.forbes.com/forbes/2010/0315/opinions-health-cancer-larry-norton-ideas-opinions.html#5469c17c5519 Google Scholar

[11]

U. LedzewiczH. Maurer and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy, Math. Biosci. Eng., 8 (2011), 307-323.  doi: 10.3934/mbe.2011.8.307.  Google Scholar

[12]

R. Martin and K. L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy, World Scientific, Singapore, 1994. doi: 10.1142/2048.  Google Scholar

[13]

J. M. Murray, Some optimal control problems in cancer chemotherapy with a toxicity limit, Math. Biosci., 100 (1990), 49-67.  doi: 10.1016/0025-5564(90)90047-3.  Google Scholar

[14]

L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treat Rep., 61 (1977), 1307-1317.   Google Scholar

[15]

M. C. Perry, D. C. Doll and C. E. Freter, Perry's The Chemotherapy Source Book, Wolters Kluwer Health, Philadelphia, 2012. Google Scholar

[16]

A. V. Rao, D. A. Benson, C. Darby, M. A. Patterson, C. Francolin, I. Sanders and G. T. Huntington, Algorithm 902: GPOPS, a MATLAB software for solving multiple-phase optimal control problems using the Gauss pseudospectral method, ACM Trans. Math. Software, 41 (2014), Art. 1, 37 pp. doi: 10.1145/1731022.1731032.  Google Scholar

[17]

R. Simon and L. Norton, The Norton-Simon hypothesis: Designing more effective and less toxic chemotherapeutic regimens, Nat. Clin. Pract. Oncol., 3 (2006), 406-407.  doi: 10.1038/ncponc0560.  Google Scholar

[18]

G. W. Swan, Role of optimal control theory in cancer chemotherapy, Math. Biosci., 101 (1990), 237-284.  doi: 10.1016/0025-5564(90)90021-P.  Google Scholar

[19]

A. SwierniakM. Kimmel and J. Smieja, Mathematical modeling as a tool for planning anticancer therapy, Eur. J. Pharmacol., 625 (2009), 108-121.  doi: 10.1016/j.ejphar.2009.08.041.  Google Scholar

[20]

T. A. TrainaU. DuganB. HigginsK. KolinskyM. TheodoulouC. A. Hudis and L. Norton, Optimizing chemotherapy dose and schedule by Norton-Simon mathematical modeling, Breas Dis., 31 (2010), 7-18.  doi: 10.3233/BD-2009-0290.  Google Scholar

show all references

References:
[1]

S. Benzekry, C. Lamont, A. Beheshti, A. Tracz, J. M. L. Ebos, L. Hlatky and P. Hahnfeldt, Classical Mathematical Models for Description and Prediction of Experimental Tumor Growth, PLOS Comput. Biol., 10 (2014), e1003800. doi: 10.1371/journal.pcbi.1003800.  Google Scholar

[2]

S. BenzekryE. PasquierD. BarbolosiB. LacarelleF. BarlésiN. André and J. Ciccolini, Metronomic reloaded: Theoretical models bringing chemotherapy into the era of precision medicine, Semin. Cancer Biol., 35 (2015), 53-61.  doi: 10.1016/j.semcancer.2015.09.002.  Google Scholar

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011. doi: 10.1007/978-0-387-70914-7.  Google Scholar

[4]

L. Cesari, Optimization—Theory and Applications, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8165-5.  Google Scholar

[5]

J. Clairambault, Modelling physiological and pharmacological control on cell proliferation to optimise cancer treatments, Math. Model. Nat. Phenom., 4 (2009), 12-67.  doi: 10.1051/mmnp/20094302.  Google Scholar

[6]

C. L. DarbyW. W. Hager and A. V. Rao, An hp-adaptive pseudospectral method for solving optimal control problems, Optimal Control Appl. Methods, 32 (2011), 476-502.  doi: 10.1002/oca.957.  Google Scholar

[7]

C. FaivreD. BarbolosiE. Pasquier and N. André, A mathematical model for the administration of temozolomide: comparative analysis of conventional and metronomic chemotherapy regimens, Cancer Chemother. Pharmacol., 71 (2013), 1013-1019.  doi: 10.1007/s00280-013-2095-z.  Google Scholar

[8]

L. A. Fernández and C. Pola, Catalog of the optimal controls in cancer chemotherapy for the Gompertz model depending on PK/PD and the integral constraint, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1563-1588.  doi: 10.3934/dcdsb.2014.19.1563.  Google Scholar

[9]

K. R. Fister and J. C. Panetta, Optimal control applied to competing chemotherapeutic cell-kill strategies, SIAM J. Appl. Math., 63 (2003), 1954-1971.  doi: 10.1137/S0036139902413489.  Google Scholar

[10]

R. Langreth, The mathematics of cancer, Forbes Magazine, 2010, https://www.forbes.com/forbes/2010/0315/opinions-health-cancer-larry-norton-ideas-opinions.html#5469c17c5519 Google Scholar

[11]

U. LedzewiczH. Maurer and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy, Math. Biosci. Eng., 8 (2011), 307-323.  doi: 10.3934/mbe.2011.8.307.  Google Scholar

[12]

R. Martin and K. L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy, World Scientific, Singapore, 1994. doi: 10.1142/2048.  Google Scholar

[13]

J. M. Murray, Some optimal control problems in cancer chemotherapy with a toxicity limit, Math. Biosci., 100 (1990), 49-67.  doi: 10.1016/0025-5564(90)90047-3.  Google Scholar

[14]

L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treat Rep., 61 (1977), 1307-1317.   Google Scholar

[15]

M. C. Perry, D. C. Doll and C. E. Freter, Perry's The Chemotherapy Source Book, Wolters Kluwer Health, Philadelphia, 2012. Google Scholar

[16]

A. V. Rao, D. A. Benson, C. Darby, M. A. Patterson, C. Francolin, I. Sanders and G. T. Huntington, Algorithm 902: GPOPS, a MATLAB software for solving multiple-phase optimal control problems using the Gauss pseudospectral method, ACM Trans. Math. Software, 41 (2014), Art. 1, 37 pp. doi: 10.1145/1731022.1731032.  Google Scholar

[17]

R. Simon and L. Norton, The Norton-Simon hypothesis: Designing more effective and less toxic chemotherapeutic regimens, Nat. Clin. Pract. Oncol., 3 (2006), 406-407.  doi: 10.1038/ncponc0560.  Google Scholar

[18]

G. W. Swan, Role of optimal control theory in cancer chemotherapy, Math. Biosci., 101 (1990), 237-284.  doi: 10.1016/0025-5564(90)90021-P.  Google Scholar

[19]

A. SwierniakM. Kimmel and J. Smieja, Mathematical modeling as a tool for planning anticancer therapy, Eur. J. Pharmacol., 625 (2009), 108-121.  doi: 10.1016/j.ejphar.2009.08.041.  Google Scholar

[20]

T. A. TrainaU. DuganB. HigginsK. KolinskyM. TheodoulouC. A. Hudis and L. Norton, Optimizing chemotherapy dose and schedule by Norton-Simon mathematical modeling, Breas Dis., 31 (2010), 7-18.  doi: 10.3233/BD-2009-0290.  Google Scholar

Figure 1.  Optimization results for Skipper model on the left (see Table 1) and for the $E_{max}$ model on the right (see Table 2)
Figure 2.  Some optimization results for $E_{max}$ model (see Table 2)
Figure 3.  Tumor volume with trivial controls for Skipper model (see Table 1)
Figure 4.  Tumor volume with trivial controls for $E_{max}$ model (see Table 2)
Table 3.  Switching times with $G = G_2$, $ k_1 = 4$ and $k_2 = 0.25$
Example Structure S_times $u_{sin}$
$ES_{52o2}$ $u_{max}/0$ 1.4e-01
$EM_{52o2}$ $u_{max}/0$ 2.3e+00
$EM_{52o3}$ $u_{max} / u_{sin}/$
$/0$
5.6e-01
2.1e+00
3.4e-01
$EL_{52o2}$ $u_{max}/0$ 2.9e+00
$EL_{52o3}$ $u_{max}/ u_{sin}/$
$/0$
6.8e-01
2.1e+00
4.0e-01
$ES_{62o2}$ $u_{max} / u_{sin}$ 1.6e-01 9.9e-02
$EM_{62o2}$ $u_{max} / u_{sin}$ 1.0e+00 5.6e-01
$EL_{62o2}$ $u_{max} / u_{sin}$ 1.4e+00 7.5e-01
Example Structure S_times $u_{sin}$
$ES_{52o2}$ $u_{max}/0$ 1.4e-01
$EM_{52o2}$ $u_{max}/0$ 2.3e+00
$EM_{52o3}$ $u_{max} / u_{sin}/$
$/0$
5.6e-01
2.1e+00
3.4e-01
$EL_{52o2}$ $u_{max}/0$ 2.9e+00
$EL_{52o3}$ $u_{max}/ u_{sin}/$
$/0$
6.8e-01
2.1e+00
4.0e-01
$ES_{62o2}$ $u_{max} / u_{sin}$ 1.6e-01 9.9e-02
$EM_{62o2}$ $u_{max} / u_{sin}$ 1.0e+00 5.6e-01
$EL_{62o2}$ $u_{max} / u_{sin}$ 1.4e+00 7.5e-01
Table 1.  Parameters and numerical results with $G = G_1$, $ k_1 = 4$
Example $\xi$ $\lambda$ T $\hat{\alpha}$ $\int_0^T \bar{u}(t) dt$ $\int_0^T \bar{c}(t) dt$ $L_0/\theta$ $L(T)/L_0$
$ES_{41a1}$ 0.1 1 2.3e-03 2.4e+00 5.0e-02 1.7e-02
$ES_{41a2}$ 0.1 1 3.0e-02 9.5e-01 5.0e-02 3.8e-01
$ES_{41a3}$ 0.1 1 7.3e-02 0 5.0e-02 1.3e+00
$ES_{51a1}$ 0.1 0.27 1 9.0e-05 2.4e+00 5.0e-02 3.0e-01
$ES_{51a2}$ 0.1 0.27 1 3.0e-02 7.0e-01 5.0e-02 7.0e-01
$ES_{51a3}$ 0.1 0.27 1 6.4e-02 0 5.0e-02 1.3e+00
$ES_{61a1}$ 0.1 0.27 1 2.5e-02 2.4e+00 1.1e+00 5.0e-02 3.0e-01
$ES_{61a2}$ 0.1 0.27 1 3.0e-02 1.8e+00 9.5e-01 5.0e-02 3.8e-01
$ES_{61a3}$ 0.1 0.27 1 7.3e-02 0 0 5.0e-02 1.3e+00
$EM_{41c1}$ 0.1 1 1.2e-01 2.4e+00 5.0e-01 3.9e-01
$EM_{41c2}$ 0.1 1 1.4e-01 1.9e+00 5.0e-01 5.2e-01
$EM_{41c3}$ 0.1 1 1.5e-01 0 5.0e-01 1.0e+00
$EM_{51a1}$ 0.1 0.27 1 0 2.4e+00 5.0e-01 7.6e-01
$EM_{51a2}$ 0.1 0.27 1 5.9e-02 1.4e+00 5.0e-01 8.2e-01
$EM_{61b1}$ 0.1 0.27 1 1.4e-01 2.4e+00 1.1e+00 5.0e-01 7.6e-01
$EL_{41b1}$ 0.1 1 1.1e-01 2.4e+00 7.5e-01 6.8e-01
$EL_{41b2}$ 0.1 1 1.2e-01 0 7.5e-01 1.0e+00
$EL_{51a1}$ 0.1 0.27 1 0 2.4e+00 7.5e-01 8.9e-01
$EL_{51a2}$ 0.1 0.27 1 3.5e-02 1.5e+00 7.5e-01 9.1e-01
$EL_{61b2}$ 0.1 0.27 1 9.3e-02 2.4e+00 1.1e+00 7.5e-01 8.9e-01
Example $\xi$ $\lambda$ T $\hat{\alpha}$ $\int_0^T \bar{u}(t) dt$ $\int_0^T \bar{c}(t) dt$ $L_0/\theta$ $L(T)/L_0$
$ES_{41a1}$ 0.1 1 2.3e-03 2.4e+00 5.0e-02 1.7e-02
$ES_{41a2}$ 0.1 1 3.0e-02 9.5e-01 5.0e-02 3.8e-01
$ES_{41a3}$ 0.1 1 7.3e-02 0 5.0e-02 1.3e+00
$ES_{51a1}$ 0.1 0.27 1 9.0e-05 2.4e+00 5.0e-02 3.0e-01
$ES_{51a2}$ 0.1 0.27 1 3.0e-02 7.0e-01 5.0e-02 7.0e-01
$ES_{51a3}$ 0.1 0.27 1 6.4e-02 0 5.0e-02 1.3e+00
$ES_{61a1}$ 0.1 0.27 1 2.5e-02 2.4e+00 1.1e+00 5.0e-02 3.0e-01
$ES_{61a2}$ 0.1 0.27 1 3.0e-02 1.8e+00 9.5e-01 5.0e-02 3.8e-01
$ES_{61a3}$ 0.1 0.27 1 7.3e-02 0 0 5.0e-02 1.3e+00
$EM_{41c1}$ 0.1 1 1.2e-01 2.4e+00 5.0e-01 3.9e-01
$EM_{41c2}$ 0.1 1 1.4e-01 1.9e+00 5.0e-01 5.2e-01
$EM_{41c3}$ 0.1 1 1.5e-01 0 5.0e-01 1.0e+00
$EM_{51a1}$ 0.1 0.27 1 0 2.4e+00 5.0e-01 7.6e-01
$EM_{51a2}$ 0.1 0.27 1 5.9e-02 1.4e+00 5.0e-01 8.2e-01
$EM_{61b1}$ 0.1 0.27 1 1.4e-01 2.4e+00 1.1e+00 5.0e-01 7.6e-01
$EL_{41b1}$ 0.1 1 1.1e-01 2.4e+00 7.5e-01 6.8e-01
$EL_{41b2}$ 0.1 1 1.2e-01 0 7.5e-01 1.0e+00
$EL_{51a1}$ 0.1 0.27 1 0 2.4e+00 7.5e-01 8.9e-01
$EL_{51a2}$ 0.1 0.27 1 3.5e-02 1.5e+00 7.5e-01 9.1e-01
$EL_{61b2}$ 0.1 0.27 1 9.3e-02 2.4e+00 1.1e+00 7.5e-01 8.9e-01
Table 2.  Parameters and numerical results with $G = G_2$, $ k_1 = 4$ and $k_2 = 0.25$
Example $\xi$ $\lambda$ T $\hat{\alpha}$ $\int_0^T \bar{u}(t) dt$ $\int_0^T \bar{c}(t) dt$ $L_0/\theta$ $ L(T)/L_0$
$ES_{42a1}$ 0.1 1 1.1e-03 2.4e+00 5.0e-02 4.1e-01
$ES_{42a2}$ 0.1 1 3.0e-02 3.6e-01 5.0e-02 6.5e-01
$ES_{42a3}$ 0.1 1 2.9e-01 0 5.0e-02 1.3e+00
$ES_{52o1}$ 0.1 0.27 1 5.2e-07 2.4e+00 5.0e-02 5.2e-01
$ES_{52o2}$ 0.1 0.27 1 3.0e-02 3.4e-01 5.0e-02 7.1e-01
$ES_{52o3}$ 0.1 0.27 1 2.4e-1 0 5.0e-02 1.3e+00
$ES_{62o1}$ 0.1 0.27 1 1.7e-03 2.4e+00 1.1e+00 5.0e-02 5.2e-01
$ES_{62o2}$ 0.1 0.27 1 3.0e-02 4.6e-01 3.4e-01 5.0e-02 6.7e-01
$ES_{62o3}$ 0.1 0.27 1 2.9e-01 0 0 5.0e-02 1.3e+00
$EM_{42a1}$ 0.1 5 2.7e-03 1.2e+01 5.0e-01 1.5e-01
$EM_{42a2}$ 0.1 5 4.0e-03 1.0e+01 5.0e-01 1.7e-01
$EM_{42a3}$ 0.1 5 4.5e-01 0 5.0e-01 1.3e+00
$EM_{52o1}$ 0.1 0.27 5 4.4e-10 1.2e+01 5.0e-01 1.5e-01
$EM_{52o2}$ 0.1 0.27 5 4.0e-03 5.6e+00 5.0e-01 1.7e-01
$EM_{52o3}$ 0.1 0.27 5 4.5e-02 1.9e+00 5.0e-01 2.8e-01
$EM_{62o1}$ 0.1 0.27 5 4.2e-04 1.2e+01 2.0e+01 5.0e-01 1.5e-01
$EM_{62o2}$ 0.1 0.27 5 4.0e-03 4.7e+00 9.5e+00 5.0e-01 1.9e-01
$EL_{42a1}$ 0.1 5 5.2e-03 1.2e+01 7.5e-01 4.6e-01
$EL_{42a3}$ 0.1 5 2.4e-01 0 7.5e-01 1.1e+00
$EL_{52o1}$ 0.1 0.27 5 8.8e-10 1.2e+01 7.5e-01 4.6e-01
$EL_{52o2}$ 0.1 0.27 5 4.0e-03 6.9e+00 7.5e-01 4.7e-01
$EL_{52o3}$ 0.1 0.27 5 4.5e-02 2.2e+00 7.5e-01 5.6e-01
$EL_{62o1}$ 0.1 0.27 5 7.8e-4 1.2e+01 2.0e+01 7.5e-01 4.6e-01
$EL_{62o2}$ 0.1 0.27 5 4.0e-03 6.0e+00 1.2e+01 7.5e-01 4.8e-01
Example $\xi$ $\lambda$ T $\hat{\alpha}$ $\int_0^T \bar{u}(t) dt$ $\int_0^T \bar{c}(t) dt$ $L_0/\theta$ $ L(T)/L_0$
$ES_{42a1}$ 0.1 1 1.1e-03 2.4e+00 5.0e-02 4.1e-01
$ES_{42a2}$ 0.1 1 3.0e-02 3.6e-01 5.0e-02 6.5e-01
$ES_{42a3}$ 0.1 1 2.9e-01 0 5.0e-02 1.3e+00
$ES_{52o1}$ 0.1 0.27 1 5.2e-07 2.4e+00 5.0e-02 5.2e-01
$ES_{52o2}$ 0.1 0.27 1 3.0e-02 3.4e-01 5.0e-02 7.1e-01
$ES_{52o3}$ 0.1 0.27 1 2.4e-1 0 5.0e-02 1.3e+00
$ES_{62o1}$ 0.1 0.27 1 1.7e-03 2.4e+00 1.1e+00 5.0e-02 5.2e-01
$ES_{62o2}$ 0.1 0.27 1 3.0e-02 4.6e-01 3.4e-01 5.0e-02 6.7e-01
$ES_{62o3}$ 0.1 0.27 1 2.9e-01 0 0 5.0e-02 1.3e+00
$EM_{42a1}$ 0.1 5 2.7e-03 1.2e+01 5.0e-01 1.5e-01
$EM_{42a2}$ 0.1 5 4.0e-03 1.0e+01 5.0e-01 1.7e-01
$EM_{42a3}$ 0.1 5 4.5e-01 0 5.0e-01 1.3e+00
$EM_{52o1}$ 0.1 0.27 5 4.4e-10 1.2e+01 5.0e-01 1.5e-01
$EM_{52o2}$ 0.1 0.27 5 4.0e-03 5.6e+00 5.0e-01 1.7e-01
$EM_{52o3}$ 0.1 0.27 5 4.5e-02 1.9e+00 5.0e-01 2.8e-01
$EM_{62o1}$ 0.1 0.27 5 4.2e-04 1.2e+01 2.0e+01 5.0e-01 1.5e-01
$EM_{62o2}$ 0.1 0.27 5 4.0e-03 4.7e+00 9.5e+00 5.0e-01 1.9e-01
$EL_{42a1}$ 0.1 5 5.2e-03 1.2e+01 7.5e-01 4.6e-01
$EL_{42a3}$ 0.1 5 2.4e-01 0 7.5e-01 1.1e+00
$EL_{52o1}$ 0.1 0.27 5 8.8e-10 1.2e+01 7.5e-01 4.6e-01
$EL_{52o2}$ 0.1 0.27 5 4.0e-03 6.9e+00 7.5e-01 4.7e-01
$EL_{52o3}$ 0.1 0.27 5 4.5e-02 2.2e+00 7.5e-01 5.6e-01
$EL_{62o1}$ 0.1 0.27 5 7.8e-4 1.2e+01 2.0e+01 7.5e-01 4.6e-01
$EL_{62o2}$ 0.1 0.27 5 4.0e-03 6.0e+00 1.2e+01 7.5e-01 4.8e-01
Table 4.  Comparison Min, Max therapies with $G = G_1$, $ k_1 = 4$, $\lambda = 0.27$, $\xi = 0.1$, $T = 2$ and $L_0 = 0.5\theta$
Problem $\hat{\alpha}$ $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ Problem $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$
$(\widehat{OP}_5)$ 7.e-2 2.3e-01 2.9e+00 $(\widehat{OP}_{2MAX})$ 6.9e-01 2.9e+00
$(\widehat{OP}_6)$ 7.e-2 1.3e-01 4.7e+00 $(\widehat{OP}_{3MAX})$ 1.3e-01 4.7e+00
$(\widehat{OP}_5)$ 9.e-2 2.8e-01 2.6e+00 $(\widehat{OP}_{2MAX})$ 7.7e-01 2.6e+00
$(\widehat{OP}_6)$ 9.e-2 1.9e-01 3.8e+00 $(\widehat{OP}_{3MAX})$ 1.9e-01 4.5e+00
Problem $\hat{\alpha}$ $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ Problem $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$
$(\widehat{OP}_5)$ 7.e-2 2.3e-01 2.9e+00 $(\widehat{OP}_{2MAX})$ 6.9e-01 2.9e+00
$(\widehat{OP}_6)$ 7.e-2 1.3e-01 4.7e+00 $(\widehat{OP}_{3MAX})$ 1.3e-01 4.7e+00
$(\widehat{OP}_5)$ 9.e-2 2.8e-01 2.6e+00 $(\widehat{OP}_{2MAX})$ 7.7e-01 2.6e+00
$(\widehat{OP}_6)$ 9.e-2 1.9e-01 3.8e+00 $(\widehat{OP}_{3MAX})$ 1.9e-01 4.5e+00
Table 5.  Comparison Min, Max therapies with $G = G_2$, $ k_1 = 4$, $ k_2 = 4$, $\lambda = 0.27$, $\xi = 0.1$, $T = 4$ and$L_0 = 0.05\theta$
Problem $\hat{\alpha}$ $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ Problem $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$
$(\widehat{OP}_5)$ 1.e-3 3.8e-01 7.9e+00 $(\widehat{OP}_{2MAX})$ 6.7e-01 7.9e+00
$(\widehat{OP}_6)$ 1.e-3 3.7e-01 9.6e+00 $(\widehat{OP}_{3MAX})$ 3.7e-01 9.6e+00
$(\widehat{OP}_5)$ 5.e-3 5.4e-01 4.9e+00 $(\widehat{OP}_{2MAX})$ 1.5e+00 4.9e+00
$(\widehat{OP}_6)$ 5.e-3 7.2e-01 4.3e+00 $(\widehat{OP}_{3MAX})$ 9.3e-01 6.8e+00
Problem $\hat{\alpha}$ $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ Problem $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$
$(\widehat{OP}_5)$ 1.e-3 3.8e-01 7.9e+00 $(\widehat{OP}_{2MAX})$ 6.7e-01 7.9e+00
$(\widehat{OP}_6)$ 1.e-3 3.7e-01 9.6e+00 $(\widehat{OP}_{3MAX})$ 3.7e-01 9.6e+00
$(\widehat{OP}_5)$ 5.e-3 5.4e-01 4.9e+00 $(\widehat{OP}_{2MAX})$ 1.5e+00 4.9e+00
$(\widehat{OP}_6)$ 5.e-3 7.2e-01 4.3e+00 $(\widehat{OP}_{3MAX})$ 9.3e-01 6.8e+00
Table 6.  Comparison Min, Max therapies with $G = G_2$, $ k_1 = 4$, $ k_2 = 4$, $\lambda = 0.27$, $\xi = 0.1$, $T = 8$ and $L_0 = 0.5\theta$
Problem $\hat{\alpha}$ $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ Problem $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$
$(\widehat{OP}_5)$ 1.e-3 3.6e-01 1.9e+01 $(\widehat{OP}_{2MAX})$ 4.0e-01 1.9e+01
$(\widehat{OP}_6)$ 1.e-3 3.6e-01 1.9e+01 $(\widehat{OP}_{3MAX})$ 3.6e-01 1.9e+01
$(\widehat{OP}_5)$ 1.e-2 4.1e-01 1.4e+01 $(\widehat{OP}_{2MAX})$ 7.6e-01 1.4e+01
$(\widehat{OP}_6)$ 1.e-2 5.8e-01 1.1e+01 $(\widehat{OP}_{3MAX})$ 7.6e-01 1.4e+01
Problem $\hat{\alpha}$ $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ Problem $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$
$(\widehat{OP}_5)$ 1.e-3 3.6e-01 1.9e+01 $(\widehat{OP}_{2MAX})$ 4.0e-01 1.9e+01
$(\widehat{OP}_6)$ 1.e-3 3.6e-01 1.9e+01 $(\widehat{OP}_{3MAX})$ 3.6e-01 1.9e+01
$(\widehat{OP}_5)$ 1.e-2 4.1e-01 1.4e+01 $(\widehat{OP}_{2MAX})$ 7.6e-01 1.4e+01
$(\widehat{OP}_6)$ 1.e-2 5.8e-01 1.1e+01 $(\widehat{OP}_{3MAX})$ 7.6e-01 1.4e+01
Table 7.  Catalog for Skipper model ($G = G_1$) under $({\bf{H}}_{\bf{1}}))$
Problem Optimal control
$(OP_1)$ $u_{max}/0$ (*)
$(OP_2)$ $u_{max}/0$
$(OP_3)$ $u_{max}/0$ (*)
Problem Optimal control
$(OP_1)$ $u_{max}/0$ (*)
$(OP_2)$ $u_{max}/0$
$(OP_3)$ $u_{max}/0$ (*)
Table 8.  Catalog for $E_{max}$ model ($G = G_2$) under $({\bf{H}}_{\bf{1}}))$
Problem Optimal control
$(OP_1)$ $u_{sin}$
$(OP_2)$ $u_{max}/0$   $u_{max}/u_{sin}/0$
$(OP_3)$ $u_{max}/u_{sin}$
Problem Optimal control
$(OP_1)$ $u_{sin}$
$(OP_2)$ $u_{max}/0$   $u_{max}/u_{sin}/0$
$(OP_3)$ $u_{max}/u_{sin}$
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