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2012, 2(4): 383-398. doi: 10.3934/mcrf.2012.2.383

## Optimal syntheses for state constrained problems with application to optimization of cancer therapies

 1 Department of Mathematical Sciences, and Center for Computational and Integrative Biology, Rutgers University - Camden, 227 Penn Street, Camden NJ 08102, United States

Received  November 2011 Revised  June 2012 Published  October 2012

The use of combined therapies to treat cancer is common nowadays and some papers already addressed the relative optimization problems. In particular, it is natural to have state constraints, which usually correspond to bounds on feasible amounts of drugs to be used. The application of Pontryagin Maximum Principle is particularly difficult in such case. Therefore, we resort to sufficient conditions for optimality to achieve results more easily applicable to systems biology models. The approach is developed both for candidate value functions and optimal syntheses. Then it is shown at work on some specific problems in combined cancer therapy.
Citation: Benedetto Piccoli. Optimal syntheses for state constrained problems with application to optimization of cancer therapies. Mathematical Control & Related Fields, 2012, 2 (4) : 383-398. doi: 10.3934/mcrf.2012.2.383
##### References:
 [1] M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,", Birkhauser Boston, (1997). [2] A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control,", American Institute of Mathematical Sciences (AIMS), (2007). [3] T. Burden, J. Ernstberger and K. Renee Fister, Optimal control applied to immunotherapy,, Discrete Continuous Dynam. Systems - B, 4 (2004), 135. [4] P. Cannarsa, H. Frankowska and E. Marchini, On Bolza optimal control problems with constraints,, Discrete Continuous Dynam. Systems - B, 11 (2009), 629. doi: 10.3934/dcdsb.2009.11.629. [5] A. Cappuccio, F. Castiglione and B. Piccoli, Determination of the optimal therapeutic protocols in cancer immunotherapy,, Math. Biosci., 209 (2007), 1. doi: 10.1016/j.mbs.2007.02.009. [6] F. Castiglione and B. Piccoli, Optimal control in a model of dendritic cell transfection cancer immunotheraphy,, in, (2004), 585. [7] F. Castiglione and B. Piccoli, Optimal control in a model of dendritic cell transfection cancer immunotherapy,, Bull. Math. Biol., 68 (2006), 255. doi: 10.1007/s11538-005-9014-3. [8] F. Castiglione and B. Piccoli, Cancer immunotherapy, mathematical modeling and optimal control,, J. Theoret. Biol., 247 (2007), 723. doi: 10.1016/j.jtbi.2007.04.003. [9] S. Chareyron and M. Alamir, Mixed immunotherapy and chemotherapy of tumors: feedback design and model updating schemes,, J. Theor. Biol., 258 (2009), 444. doi: 10.1016/j.jtbi.2008.07.002. [10] L. G. de Pillis, K. Renee Fister, W. Gu, C. Collins, M. Daub, D. Gross, J. Moore and B. Preskill, Seeking bang-bang solutions of mixed immuno-chemotherapy of tumors,, Electron. J. Differential Equations, 171 (2007). [11] L. G. de Pillis, W. Gu and A. E. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: Modeling, applications and biological interpretations,, J. Theoret. Biol., 238 (2006), 841. doi: 10.1016/j.jtbi.2005.06.037. [12] A. D'Onofrio, U. Ledzewicz, H. Maurer and H. Schaettler, On optimal delivery of combination of therapy for tumors,, Math. Biosci., 222 (2009), 13. doi: 10.1016/j.mbs.2009.08.004. [13] K. Renee Fister and J. C. Panetta, Optimal control applied to competing chemotherapeutic cell-kill strategies,, SIAM J. Appl. Math., 63 (2003), 1954. doi: 10.1137/S0036139902413489. [14] R. A. Goldsby, T. J. Kindt and B. A. Osborne, "Kuby Immunology,", IV. eds. W. H. Freeman and Company, (2000). [15] R. F. Hartl, S. P. Sethi and R. G. Vickson, A survey of the maximum principles for optimal control problems with state constraints,, SIAM Rev., 37 (1995), 181. [16] D. Kirschner and J. C. Panetta, Modeling immunotherapy of teh tumor-immune interaction,, J. Math. Biol., 37 (1998), 235. doi: 10.1007/s002850050127. [17] U. Ledzewicz, J. Munden and H. Schaettler, Scheduling of angiogenic inhibitors for Gompertzian and logistic tumor growth models,, Discrete Continuous Dynam. Systems - B, 12 (2009), 415. doi: 10.3934/dcdsb.2009.12.415. [18] U. Ledzewicz, H. Schaettler and A. D'Onofrio, Optimal control for combination of therapy in cancer,, in, (2008), 1537. [19] B. Piccoli, Infinite time regular synthesis,, ESAIM Control Optim. Calc. Var., 3 (1998), 381. [20] B. Piccoli and H. J. Sussmann, Regular synthesis and sufficiency conditions for optimality,, SIAM J. Control Optim., 39 (2000), 359. doi: 10.1137/S0363012999322031. [21] R. Vinter, "Optimal Control,", Birkhauser Boston, (2000).

show all references

##### References:
 [1] M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,", Birkhauser Boston, (1997). [2] A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control,", American Institute of Mathematical Sciences (AIMS), (2007). [3] T. Burden, J. Ernstberger and K. Renee Fister, Optimal control applied to immunotherapy,, Discrete Continuous Dynam. Systems - B, 4 (2004), 135. [4] P. Cannarsa, H. Frankowska and E. Marchini, On Bolza optimal control problems with constraints,, Discrete Continuous Dynam. Systems - B, 11 (2009), 629. doi: 10.3934/dcdsb.2009.11.629. [5] A. Cappuccio, F. Castiglione and B. Piccoli, Determination of the optimal therapeutic protocols in cancer immunotherapy,, Math. Biosci., 209 (2007), 1. doi: 10.1016/j.mbs.2007.02.009. [6] F. Castiglione and B. Piccoli, Optimal control in a model of dendritic cell transfection cancer immunotheraphy,, in, (2004), 585. [7] F. Castiglione and B. Piccoli, Optimal control in a model of dendritic cell transfection cancer immunotherapy,, Bull. Math. Biol., 68 (2006), 255. doi: 10.1007/s11538-005-9014-3. [8] F. Castiglione and B. Piccoli, Cancer immunotherapy, mathematical modeling and optimal control,, J. Theoret. Biol., 247 (2007), 723. doi: 10.1016/j.jtbi.2007.04.003. [9] S. Chareyron and M. Alamir, Mixed immunotherapy and chemotherapy of tumors: feedback design and model updating schemes,, J. Theor. Biol., 258 (2009), 444. doi: 10.1016/j.jtbi.2008.07.002. [10] L. G. de Pillis, K. Renee Fister, W. Gu, C. Collins, M. Daub, D. Gross, J. Moore and B. Preskill, Seeking bang-bang solutions of mixed immuno-chemotherapy of tumors,, Electron. J. Differential Equations, 171 (2007). [11] L. G. de Pillis, W. Gu and A. E. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: Modeling, applications and biological interpretations,, J. Theoret. Biol., 238 (2006), 841. doi: 10.1016/j.jtbi.2005.06.037. [12] A. D'Onofrio, U. Ledzewicz, H. Maurer and H. Schaettler, On optimal delivery of combination of therapy for tumors,, Math. Biosci., 222 (2009), 13. doi: 10.1016/j.mbs.2009.08.004. [13] K. Renee Fister and J. C. Panetta, Optimal control applied to competing chemotherapeutic cell-kill strategies,, SIAM J. Appl. Math., 63 (2003), 1954. doi: 10.1137/S0036139902413489. [14] R. A. Goldsby, T. J. Kindt and B. A. Osborne, "Kuby Immunology,", IV. eds. W. H. Freeman and Company, (2000). [15] R. F. Hartl, S. P. Sethi and R. G. Vickson, A survey of the maximum principles for optimal control problems with state constraints,, SIAM Rev., 37 (1995), 181. [16] D. Kirschner and J. C. Panetta, Modeling immunotherapy of teh tumor-immune interaction,, J. Math. Biol., 37 (1998), 235. doi: 10.1007/s002850050127. [17] U. Ledzewicz, J. Munden and H. Schaettler, Scheduling of angiogenic inhibitors for Gompertzian and logistic tumor growth models,, Discrete Continuous Dynam. Systems - B, 12 (2009), 415. doi: 10.3934/dcdsb.2009.12.415. [18] U. Ledzewicz, H. Schaettler and A. D'Onofrio, Optimal control for combination of therapy in cancer,, in, (2008), 1537. [19] B. Piccoli, Infinite time regular synthesis,, ESAIM Control Optim. Calc. Var., 3 (1998), 381. [20] B. Piccoli and H. J. Sussmann, Regular synthesis and sufficiency conditions for optimality,, SIAM J. Control Optim., 39 (2000), 359. doi: 10.1137/S0363012999322031. [21] R. Vinter, "Optimal Control,", Birkhauser Boston, (2000).
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