Low-Regret and No-Regret Control of Tumor Development to Fill in Some Limitations of Classical Optimal Control Theory

Abstract

This paper is about a Cauchy problem for a parabolic type linear operator. The main system describes the spread and development of a tumor in an organism. From the classical optimal control theory, we show some results of variation calculations. And an optimality system for the considered control problem is established.It is known that the classical techniques of optimal control theory are ineffective for certain evolutionary parabolic systems type with missing data.

Keywords

• low-regret control
• no-regret control
• optimal control
• estimations
• strategic zone control.

References

1. [1] H.P. Greenspan. On the growth on cell culture and solid tumors, Theoretical Biology 56 (1976), 229-242.
2. [2] H.P. Greenspan. Models for the growth of a solid tumor by diffusion, Studies Appl. Math. 52 (1972), 317-340.
3. [3] M. Kimmel and A. Swierniak. Control Theory Approach to Cancer Chemotherapy: Benefiting from Phase Dependence and Overcoming Drug Resistance, Lect. Notes Math. Vol. 1872, 2006 pp. 185-221.
4. [4] U. Ledzewicz and H. Sachattlerl. Drug resistance in cancer chemotherapy as an optimal control problem, Discrete and continuous dynamical systemsseries- B Volume 6, Number 1, January 2006,pp. 129-150.
5. [5] M.Ngom, I.Ly and D.Seck. Chemotherapy of a tumor by optimal control approach. Mathematica Aeterna, Vol. 2, 2012, no. 9, 779 - 803.
6. [6] M.Ngom, I.Ly and D.Seck. Study of a tumor by shape and topological optimization. Applied Mathematical Sciences, Vol. 5, 2011, no. 1, 1-21.
7. [7] A.Friedman. Free boundary problems arising in tumor models Mat. Acc. Lincei (2004) s.9, v.15 : 161-168
8. [8] A.Friedman and F.Reitich Analysis of a mathematical model for the growth of tumors J. Math. Biol. (1999) 38: 262-284.

9. [9] M.A. J. Chaplain. The development of a spatial pattern in a model for cancer growth, Experimental and Theoretical Advances in Biological Pattern Formation (H. G. Othmer, P. K. Maini, and J. D. Murray, eds.), Plenum Press, 1993, pp. 45-60.
10. [10] J. L. Lions. Controle a moindres regrets des systemes distribues. C.R.Acad. Sci. Paris, Ser. I Math.315, (1992), 125312.
11. [11] M. L. L. Ane, C. Seck and A. S`ene Boundary Exact Controllability of the Heat Equation in 1D by Strategic Actuators and a Linear Surjective Compact Operator Applied Mathematics,(2020), 991-999.
12. [12] E. A. Jai. Quelques problemes de controle propres aux systemes distribues. Annals of University of Craiova. Math. Comp. Sci. Ser. 2003;30:137-153.
13. [13] O. Nakoulima, A. Omrane, J. Velin. Perturbations a moindres regrets dans les systemes distribues a donnees manquantes. C. R. Acad. Sci. Ser. I Math. (Paris), (2000);330:801 806.
14. [14] G. W. Swan, B. Dumitru, J. Claire M. Gisele. Low-regret control for a fractional wave equation with incomplete data, Baleanu et al. Advances in Difference Equations, 2016, DOI 10.1186/s13662-016-0970-8.
15. [15] S. Cui, A. Friedman. Analysis of a mathematical of the effect inhibitors on the growth of tumors. Mathematical Biosciences 164 (2000), 103-137.
16. [16] C. Seck, M. L. L. Ane and A. Sene Boundary Exact Controllability of the Heat Equation in 1D by Strategic Actuators and a Linear Surjective Compact Operator Applied Mathematics, 2020, 11, 991-999: Applied Mathematics , 11, 991-999, https://doi.org/10.4236/am.2020.1110065.
17. [17] S.Cui and A.Friedman. Analysis of a mathematical model of the growth of necrotic tumors Journal of Mathematical Analysis and Applications 255, 636-677 (2001).
18. [18] J. D. Djida, G. Mophou and I. Area. Optimal Control of Diffusion Equation with Fractional Time Derivative with Nonlocal and Nonsingular Mittag-Leffler Kernel Springer, 2018.
19. [19] B.V. Bazaly and A.Friedman A Free Boundary Probleme for an Elliptic-Parabolic System : Application to a Model of Tumor Growth Communication in Partial Differential Equations Vol. 28 Nos 3 and 4 text(2003), pp. 517-560.
20. [20] C. Seck, M. Ngom, L. Ndiaye A study on non-classical optimal control and dymamic regional controllability by scalability of tumor evolution .International Journal of Numerical Methods and Applications © Pushpa Publishing House, Prayagraj, India, http://www.pphmj.com http://dx.doi.org/10.17654/0975045223004 Volume 23, Number 1, (2023), Pages 67-85 P-ISSN: 0975-0452.