Stabilization in a Chemostat with Sampled and Delayed Measurements Michael Malisoff Joint with Jerome Harmand and Frederic Mazenc 0/11 Models: Represent cell or microorganism growth, wastewater treatment, or natural environments like lakes.. States: Microorganism and substrate concentrations, prone to incomplete measurements and model uncertainties.. Our goals: Input-to-state stabilization of equilibria with uncertain uptake functions using only delayed sampled substrate values O. Bernard, D. Dochain, J. Gouze, J. Harmand, J. Monod, .... Background and Motivation Chemostat: Laboratory apparatus for continuous culture of microorganisms, many biotechnological applications.. 1/11 States: Microorganism and substrate concentrations, prone to incomplete measurements and model uncertainties.. Our goals: Input-to-state stabilization of equilibria with uncertain uptake functions using only delayed sampled substrate values O. Bernard, D. Dochain, J. Gouze, J. Harmand, J. Monod, .... Background and Motivation Chemostat: Laboratory apparatus for continuous culture of microorganisms, many biotechnological applications.. Models: Represent cell or microorganism growth, wastewater treatment, or natural environments like lakes.. 1/11 Our goals: Input-to-state stabilization of equilibria with uncertain uptake functions using only delayed sampled substrate values O. Bernard, D. Dochain, J. Gouze, J. Harmand, J. Monod, .... Background and Motivation Chemostat: Laboratory apparatus for continuous culture of microorganisms, many biotechnological applications.. Models: Represent cell or microorganism growth, wastewater treatment, or natural environments like lakes.. States: Microorganism and substrate concentrations, prone to incomplete measurements and model uncertainties.. 1/11 O. Bernard, D. Dochain, J. Gouze, J. Harmand, J. Monod, .... Background and Motivation Chemostat: Laboratory apparatus for continuous culture of microorganisms, many biotechnological applications.. Models: Represent cell or microorganism growth, wastewater treatment, or natural environments like lakes.. States: Microorganism and substrate concentrations, prone to incomplete measurements and model uncertainties.. Our goals: Input-to-state stabilization of equilibria with uncertain uptake functions using only delayed sampled substrate values 1/11 Background and Motivation Chemostat: Laboratory apparatus for continuous culture of microorganisms, many biotechnological applications.. Models: Represent cell or microorganism growth, wastewater treatment, or natural environments like lakes.. States: Microorganism and substrate concentrations, prone to incomplete measurements and model uncertainties.. Our goals: Input-to-state stabilization of equilibria with uncertain uptake functions using only delayed sampled substrate values O. Bernard, D. Dochain, J. Gouze, J. Harmand, J. Monod, .... 1/11 Background and Motivation Chemostat: Laboratory apparatus for continuous culture of microorganisms, many biotechnological applications.. Models: Represent cell or microorganism growth, wastewater treatment, or natural environments like lakes.. States: Microorganism and substrate concentrations, prone to incomplete measurements and model uncertainties.. Our goals: Input-to-state stabilization of equilibria with uncertain uptake functions using only delayed sampled substrate values S. Pilyugin, G. Robledo, H. Smith, P. Waltman, G. Wolkowicz, ... 1/11 Background and Motivation Chemostat: Laboratory apparatus for continuous culture of microorganisms, many biotechnological applications.. Models: Represent cell or microorganism growth, wastewater treatment, or natural environments like lakes.. States: Microorganism and substrate concentrations, prone to incomplete measurements and model uncertainties.. Our goals: Input-to-state stabilization of equilibria with uncertain uptake functions using only delayed sampled substrate values S. Pilyugin, G. Robledo, H. Smith, P. Waltman, G. Wolkowicz, ... 1/11 Background and Motivation Chemostat: Laboratory apparatus for continuous culture of microorganisms, many biotechnological applications.. Models: Represent cell or microorganism growth, wastewater treatment, or natural environments like lakes.. States: Microorganism and substrate concentrations, prone to incomplete measurements and model uncertainties.. Our goals: Input-to-state stabilization of equilibria with uncertain uptake functions using only delayed sampled substrate values S. Pilyugin, G. Robledo, H. Smith, P. Waltman, G. Wolkowicz, ... 1/11 Background and Motivation Chemostat: Laboratory apparatus for continuous culture of microorganisms, many biotechnological applications.. Models: Represent cell or microorganism growth, wastewater treatment, or natural environments like lakes.. States: Microorganism and substrate concentrations, prone to incomplete measurements and model uncertainties.. Our goals: Input-to-state stabilization of equilibria with uncertain uptake functions using only delayed sampled substrate values S. Pilyugin, G. Robledo, H. Smith, P. Waltman, G. Wolkowicz, ... 1/11 Contents go in and out of culture vessel at constant rate F (l3=t). Constant culture volume V (l3). 3 Constant input nutrient concentration sin (mass/l ). rate of change of nutrient = input - washout - consumption. rate of change of organism = growth - washout. Review of Simple Chemostat Three vessels: feed bottle, culture vessel, collection vessel. 2/11 Constant culture volume V (l3). 3 Constant input nutrient concentration sin (mass/l ). rate of change of nutrient = input - washout - consumption. rate of change of organism = growth - washout. Review of Simple Chemostat Three vessels: feed bottle, culture vessel, collection vessel. Contents go in and out of culture vessel at constant rate F (l3=t). 2/11 3 Constant input nutrient concentration sin (mass/l ). rate of change of nutrient = input - washout - consumption. rate of change of organism = growth - washout. Review of Simple Chemostat Three vessels: feed bottle, culture vessel, collection vessel. Contents go in and out of culture vessel at constant rate F (l3=t). Constant culture volume V (l3). 2/11 rate of change of nutrient = input - washout - consumption. rate of change of organism = growth - washout. Review of Simple Chemostat Three vessels: feed bottle, culture vessel, collection vessel. Contents go in and out of culture vessel at constant rate F (l3=t). Constant culture volume V (l3). 3 Constant input nutrient concentration sin (mass/l ). 2/11 rate of change of organism = growth - washout. Review of Simple Chemostat Three vessels: feed bottle, culture vessel, collection vessel. Contents go in and out of culture vessel at constant rate F (l3=t). Constant culture volume V (l3). 3 Constant input nutrient concentration sin (mass/l ). rate of change of nutrient = input - washout - consumption. 2/11 Review of Simple Chemostat Three vessels: feed bottle, culture vessel, collection vessel. Contents go in and out of culture vessel at constant rate F (l3=t). Constant culture volume V (l3). 3 Constant input nutrient concentration sin (mass/l ). rate of change of nutrient = input - washout - consumption. rate of change of organism = growth - washout. 2/11 s = concentration of nutrient in culture vessel. msx 3 Consumption: a+s , x = concentration of organism (mass=l ). m = maximum growth rate (1=t). a = half-saturation constant. ( 0 ms x s = (sin − s)D − a+s γ 0 ms (SC) x = x a+s − D Review of Simple Chemostat 0 Without organisms or consumption, (Vs) (t) = sinF − s(t)F. 3/11 msx 3 Consumption: a+s , x = concentration of organism (mass=l ). m = maximum growth rate (1=t). a = half-saturation constant. ( 0 ms x s = (sin − s)D − a+s γ 0 ms (SC) x = x a+s − D Review of Simple Chemostat 0 Without organisms or consumption, (Vs) (t) = sinF − s(t)F. s = concentration of nutrient in culture vessel. 3/11 m = maximum growth rate (1=t). a = half-saturation constant. ( 0 ms x s = (sin − s)D − a+s γ 0 ms (SC) x = x a+s − D Review of Simple Chemostat 0 Without organisms or consumption, (Vs) (t) = sinF − s(t)F. s = concentration of nutrient in culture vessel. msx 3 Consumption: a+s , x = concentration of organism (mass=l ). 3/11 ( 0 ms x s = (sin − s)D − a+s γ 0 ms (SC) x = x a+s − D Review of Simple Chemostat 0 Without organisms or consumption, (Vs) (t) = sinF − s(t)F. s = concentration of nutrient in culture vessel. msx 3 Consumption: a+s , x = concentration of organism (mass=l ). m = maximum growth rate (1=t). a = half-saturation constant. 3/11 Review of Simple Chemostat 0 Without organisms or consumption, (Vs) (t) = sinF − s(t)F. s = concentration of nutrient in culture vessel. msx 3 Consumption: a+s , x = concentration of organism (mass=l ). m = maximum growth rate (1=t). a = half-saturation constant. ( 0 ms x s = (sin − s)D − a+s γ 0 ms (SC) x = x a+s − D 3/11 Review of Simple Chemostat 0 Without organisms or consumption, (Vs) (t) = sinF − s(t)F. s = concentration of nutrient in culture vessel. msx 3 Consumption: a+s , x = concentration of organism (mass=l ). m = maximum growth rate (1=t). a = half-saturation constant. ( 0 ms x s = (sin − s)D − a+s γ 0 ms (SC) x = x a+s − D 3/11 X 0(t) = G(t; X(t); X(t − τ(t))); X(t) 2 X (Σ) t0−t jX(t)j ≤ γ1 e γ2(jXj[t0−τ¯;t0]) (UGAS) γi 2 K1: 0 at 0, strictly increasing, unbounded. supt≥0 τ(t) ≤ τ¯. 0 X (t) = G t; X(t); X(t − τ(t));δ (t) ; X(t) 2 X (Σpert) t0−t jX(t)j ≤ γ1 e γ2(jXj[t0−τ¯;t0]) + γ3(jδj[t0;t]) (ISS) Find γi ’s by building Lyapunov-Krasovskii functionals (LKFs). Equivalent to KL formulation; see Sontag 1998