# Biomedical and Biotechnological Control Applications

Mathematical control theory can help us choose between different interventions in biomedical and biotechnological systems. The interventions are represented as manipulated parameters in mathematical models for the systems. Two approaches to applying mathematical control theory to biological models are (a) optimal control, where the manipulated parameters are functions of time that minimize cost functions and (b) feedback controls, which are state-dependent choices of the manipulated parameters that are chosen to ensure convergence of the states of the systems towards desired steady states, or to ensure that the states remain in a prescribed set of values. Examples of manipulated parameters in biological models include levels of nutrients that are supplied to a bioreactor, stimulation of skeletal muscles by electrodes in neuromuscular electrical stimulation, and vaccine doses in pandemic models. Feedback control for biological systems often uses Lyapunov function methods, where the feedback controls are chosen such that a Lyapunov function for the system has a nonpositive derivative along all trajectories of the system that are outside of the desired steady state, which can be useful for concluding that the Lyapunov function values converge to zero over time, using the LaSalle Invariance Principle. Since Lyapunov functions can quantify how far the systems are from a desired operating mode, the convergence of Lyapunov function values towards zero over time can be used to ensure desired behavior of biological systems.

One significant application of feedback control is to models of the COVID-19 pandemic, whose manipulated parameters include levels of vaccination or quarantining. A benefit of using strict Lyapunov functions and feedback control is that strict Lyapunov functions can facilitate quantifying the effects of input delays, sampling, and uncertain immigration into a population, which are useful for modeling delayed responses to treatment protocols in diseased populations or incomplete knowledge of the numbers of susceptible individuals. Quantifying these effects can sometimes be done by transforming strict Lyapunov functions for systems without delays into Lyapunov-Krasovskii functionals that are amenable to ensuring convergence to desired steady states on the infinite dimensional state spaces that arise from delayed systems. The Lyapunov-Krasovskii functionals are defined on infinite dimensional sets of functions instead of being defined on subsets of finite dimensional Euclidean spaces. The strictness property calls for the time derivative of the Lyapunov function to be negative at all points outside the equilibrium when the uncertainties are zero. By constrast, nonstrict Lyapunov functions are usually not conducive to quantifying the effects of sampling and uncertainty. Other feedback control applications involve mathematical models of bioreactors, whose feedback controls determine the amount of nutrients to be provided and must be computed from incomplete information about the state of the bioreactor, such as the total population of multiple microorganisms instead of the individual populations of each type of microorganism. Below are references on applications of feedback control to biomedical and biotechnological systems. Click on the titles below to see the presentation or paper.

References:

1. Atkins, S., and M. Malisoff, "Robustness of feedback control for SIQR epidemic model under measurement uncertainty," Mathematical Control and Related Fields, 2023, to appear, 33 pages.
2. Ito, H., M. Malisoff, and F. Mazenc, "Feedback control of isolation and contact for SIQR epidemic model via strict Lyapunov function," Mathematical Control and Related Fields, Volume 13, Issue 4, 2023, pp. 1438-1465.
3. Ito, H., M. Malisoff, and F. Mazenc, "Strict Lyapunov functions and feedback controls for SIR models with quarantine and vaccination," Discrete and Continuous Dynamical Systems Series B, Volume 27, Issue 12, 2022, pp. 6969-6988.
4. Karafyllis, I., M. Malisoff, M. de Queiroz, M. Krstic, and R. Yang, "Predictor-based tracking for neuromuscular electrical stimulation," International Journal of Robust and Nonlinear Control, Volume 25, Issue 14, 2015, pp. 2391-2419.
5. Mazenc, F., J. Harmand, and M. Malisoff, "Stabilization in a chemostat with sampled and delayed measurements and uncertain growth functions," Automatica, Volume 78, April 2017, pp. 241-249.
6. Mazenc, F., and M. Malisoff, "Stabilization of a chemostat model with Haldane growth functions and a delay in the measurements," Automatica, Volume 46, Issue 9, September 2010, pp. 1428-1436.
7. Mazenc, F., and M. Malisoff, "Stability and stabilization for models of chemostats with multiple limiting substrates," , Volume 6, Issue 2, 2012, pp. 612-627.
8. Mazenc, F., M. Malisoff, and O. Bernard, "A simplified design for strict Lyapunov functions under Matrosov conditions," IEEE Transactions on Automatic Control, Volume 54, Issue 1, 2009, pp. 177-183.
9. Mazenc, F., M. Malisoff, and P. De Leenheer, "On the stability of periodic solutions in the perturbed chemostat," Mathematical Biosciences and Engineering, Volume 4, Number 2, 2007, pp. 319-338.
10. Mazenc, F., M. Malisoff, and M. de Queiroz, "Tracking control and robustness analysis for a nonlinear model of human heart rate during exercise," Automatica, Volume 47, Issue 5, May 2011, pp. 968-974.
11. Mazenc, F., M. Malisoff, and J. Harmand, "Further results on stabilization of periodic trajectories for a chemostat with two species," IEEE Transactions on Automatic Control, Volume 53, Special Issue on Systems Biology, January 2008, pp. 66-74.
12. Mazenc, F., M. Malisoff, and J. Harmand, "Stabilization in a two-species chemostat with Monod growth functions," IEEE Transactions on Automatic Control, Volume 54, Issue 4, April 2009, pp. 855-861.
13. Mazenc, F., G. Robledo, and M. Malisoff, "Stability and robustness analysis for a multispecies chemostat model with delays in the growth rates and uncertainties," Discrete and Continuous Dynamical Systems Series B, Volume 23, Issue 4, June 2018, pp. 1851-1872.

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