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October  2018, 15(5): 1137-1154. doi: 10.3934/mbe.2018051

## Optimal control problems with time delays: Two case studies in biomedicine

 1 Münster University of Applied Sciences, Department of Mechanical Engineering, Stegerwaldstr. 39, 48565 Steinfurt, Germany 2 University of Münster, Applied Mathematics: Institute of Analysis and Numerics, Einsteinstr. 62, 49143 Münster, Germany

* Corresponding author: H. Maurer, maurer@math.uni-muenster.de

Received  April 30, 2017 Accepted  March 18, 2018 Published  May 2018

There exists an extensive literature on delay differential models in biology and biomedicine, but only a few papers study such models in the framework of optimal control theory. In this paper, we consider optimal control problems with multiple time delays in state and control variables and present two applications in biomedicine. After discussing the necessary optimality conditions for delayed optimal control problems with control-state constraints, we propose discretization methods by which the delayed optimal control problem is transformed into a large-scale nonlinear programming problem. The first case study is concerned with the delay differential model in [21] describing the tumour-immune response to a chemo-immuno-therapy. Assuming $L^1$-type objectives, which are linear in control, we obtain optimal controls of bang-bang type. In the second case study, we introduce a control variable in the delay differential model of Hepatitis B virus infection developed in [7]. For $L^1$-type objectives we obtain extremal controls of bang-bang type.

Citation: Laurenz Göllmann, Helmut Maurer. Optimal control problems with time delays: Two case studies in biomedicine. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1137-1154. doi: 10.3934/mbe.2018051
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Optimal solution of the non-delayed control problem with $\tau_1 = \tau_2 = 0$ and weights $B_1 = 5, B_2 = 10$. Top row: (a) dose control $u_1(t)$ of chemotherapy, (b) effector cells $E(t)$, (c) tumour cells $T(t)$. Bottom row: (a) dose control $u_2(t)$ of immune therapy, (b) healthy cells $N(t)$, (c) cytostatic agent $U(t)$.
Optimal controls $u_k(t)$ and switching functions $\phi_k(t), \,(k = 1,2)$ in a neighborhood of the switching times $t_k$ illustrating the control-law (35) and the strict bang-bang property (37).
Optimal controls $u_1(t)$ and $u_2(t)$ for functionals $J_p(p,u), p = 1,2,$ with weights $B_1 = 5, B_2 = 10$.
Optimal solution of the delayed control problem with state delay $\tau_1 = 1.5$, control delay $\tau_2 = 3.0$ and weights $B_1 = 5, B_2 = 10$. Top row: (a) dose control $u_1(t)$ of chemotherapy, (b) effector cells $E(t)$, (c) tumour cells $T(t)$. Bottom row: (a) dose control $u_2(t)$ of immune therapy, (b) healthy cells $N(t)$, (c) cytostatic agent $U(t)$.
Delayed solution with $\tau_1 = 1.5$ and $\tau_2 = 3.0$: controls $u_k(t)$ and switching functions $\phi_k(t), \,(k = 1,2)$ in a neighborhood of the switching times $t_k$ illustrating the control-law (35) and the strict bang-bang property (37).
Optimal controls $u_1(t)$ and $u_2(t)$ for functionals $J_1(x,u)$ and $J_2(x,u)$ with delays $\tau_1 = 1.5, \tau_2 = 3.0$ and weights $B_1 = 5, B_2 = 10$.
Optimal solution of the delayed control problem with state delay $\tau_1 = 1.5$, control delay $\tau_2 = 3.0$ and mixed control-state constraint $U(t) + u_2(t) \leq 3$. Top row: (a) dose control $u_1(t)$ of chemotherapy, (b) function $U(t)+u_2(t)$, (c) effector cells $E(t)$. Bottom row: (a) dose control $u_2(t)$ of immune therapy, (b) multiplier $\mu(t)$ for mixed constraint, (c) tumour cells $T(t)$.
Controls and switching functions (51) for delays $\tau = 0$, $\tau = 10$ and $\tau = 15$. For all delays the control law (52) is satisfied and the strict bang-bang property holds.
Comparison of state variables for delays $\tau = 0, 10, 15$. Top row: (a) healthy cells $x$, (b) exposed cells $p$. Bottom row: (a) infected cells $y$, (b) free virions $v$.
Parameters in the control problem of chemo-immunotherapy [21].
 Parameter Description Value $t_f$ final time $30$ d (days) $\tau_1$ state delay $1.5$ d $\tau_2$ control delay $3.0$ d $(u_{k,\min},u_{k,\max})$ control bounds $(0, 1)$ for $\,k=1,2$ $(a_1,\,a_2,\,a_3)$ cell kill rate response $(0.2,\,0.4,\,0.1)$ $(\beta,\, \beta_2)$ reciprocal carrying capacities of tumour and host cells $(0.002,\,1.0)$ $(c_1,\, c_2)$ scaling parameters $(3\times 10^{-5},\,3\times 10^{-8})$ $d_1$ drug decay rate $0.01$ $\delta$ immune cell death rate $0.2$ $\eta$ steepness of immune response $0.3$ $\mu_e$ uninfected effector cell decrease rate $0.003611$ $(\sigma,\,\rho)$ immune cell influx and decay rate resp. $(0.2,\,0.2)$ $(s_1,\, r_2,\, r_3)$ cell growth rates $(0.3,\,1.03,\,1.0)$ $n_T$ immune effector cell decrease rate $1.0$ $(B_1,B_2)$ weights $(5,\,10)$
 Parameter Description Value $t_f$ final time $30$ d (days) $\tau_1$ state delay $1.5$ d $\tau_2$ control delay $3.0$ d $(u_{k,\min},u_{k,\max})$ control bounds $(0, 1)$ for $\,k=1,2$ $(a_1,\,a_2,\,a_3)$ cell kill rate response $(0.2,\,0.4,\,0.1)$ $(\beta,\, \beta_2)$ reciprocal carrying capacities of tumour and host cells $(0.002,\,1.0)$ $(c_1,\, c_2)$ scaling parameters $(3\times 10^{-5},\,3\times 10^{-8})$ $d_1$ drug decay rate $0.01$ $\delta$ immune cell death rate $0.2$ $\eta$ steepness of immune response $0.3$ $\mu_e$ uninfected effector cell decrease rate $0.003611$ $(\sigma,\,\rho)$ immune cell influx and decay rate resp. $(0.2,\,0.2)$ $(s_1,\, r_2,\, r_3)$ cell growth rates $(0.3,\,1.03,\,1.0)$ $n_T$ immune effector cell decrease rate $1.0$ $(B_1,B_2)$ weights $(5,\,10)$
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