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, 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.

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[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.

[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.

[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

[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.

[12]

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

[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.

[14]

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

[15]

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

[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.

[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.

[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.

[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.

[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.

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.

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[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.

[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.

[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

[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.

[12]

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

[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.

[14]

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

[15]

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

[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.

[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.

[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.

[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.

[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.

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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