Optimal synthesis for a minimum time problem in the plane with a triangle control set

This work is devoted to the study of a minimum time control problem where the state is governed by a two-dimensional affine system with two inputs taking values within a triangle, and describing a series of two interconnected chemostats. We show the existence of a subset of the invariant domain D associated to the system such that if the target is in this set, then it can be reached by any initial condition in D. For every target point in this subset, we provide an optimal synthesis of the problem by decomposing D into two subsets. In the first one, we give an explicit expression of the value function, and we show that there exist infinitely many optimal solutions whereas in the second one, we show that the optimal strategy is of singular type.

[1]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[2]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[3]  J. P. Lasalle,et al.  functional analysis and time Optimal Control , 1969 .

[4]  Johannes Tramper,et al.  Bioreactors in series : An overview of design procedures and practical applications , 1996 .

[5]  Jaime A. Moreno.  Optimal time control of bioreactors for the wastewater treatment , 1999 .

[6]  Paul Waltman,et al.  The Theory of the Chemostat , 1995 .

[7]  S. Grego,et al.  Eco-physiological Characterization of Soil Bacterial Populations in Different States of Growth , 2002, Microbial Ecology.

[8]  Alain Rapaport,et al.  Effects of spatial structure and diffusion on the performances of the chemostat. , 2010, Mathematical biosciences and engineering : MBE.

[9]  Campbell W. Robinson,et al.  Minimum tank volumes for CFST bioreactors in series , 1989 .

[10]  J W Wimpenny,et al.  The gradostat: a bidirectional compound chemostat and its application in microbiological research. , 1981, Journal of general microbiology.

[11]  Pauline M. Doran,et al.  Bioprocess Engineering Principles , 1995 .

[12]  Angelo Miele.  APPLICATION OF GREEN'S THEOREM TO THE EXTREMIZATION OF LINEAR INTEGRALS , 1961 .

[13]  Paul Waltman,et al.  The gradostat: A model of competition along a nutrient gradient , 1991, Microbial Ecology.

[14]  S. P. Hastings.  Differential Equations and Dynamical Systems (Lawrence Perko) , 1992, SIAM Rev..

[15]  Jérôme Harmand,et al.  Optimal design of interconnected bioreactors: New results , 2003 .

[16]  Eduardo D. Sontag,et al.  Mathematical control theory: deterministic finite dimensional systems (2nd ed.) , 1998 .

[17]  T. Clément,et al.  Use of a continuous multistage bioreactor to mimic winemaking fermentation. , 2011, International journal of food microbiology.

[18]  B. Piccoli,et al.  Optimal Syntheses for Control Systems on 2-D Manifolds , 2004 .

[19]  Richard B. Vinter,et al.  Optimal Control , 2000 .

[20]  J Tramper,et al.  Optimal design for continuous stirred tank reactors in series using Michaelis-Menten kinetics. , 1982, Biotechnology and bioengineering.

[21]  H. Robbins.  A generalized legendre-clebsch condition for the singular cases of optimal control , 1967 .

[22]  M. Chyba,et al.  Singular Trajectories and Their Role in Control Theory , 2003, IEEE Transactions on Automatic Control.

[23]  H el-Owaidy,et al.  Theoretical studies on extinction in the gradostat. , 1990, Mathematical biosciences.

[24]  Alain Rapaport,et al.  Minimal Time Sequential Batch Reactors with Bounded and Impulse Controls for One or More Species , 2008, SIAM J. Control. Optim..

[25]  Betty Tang,et al.  Mathematical investigations of growth of microorganisms in the gradostat , 1986 .

[26]  E B Lee,et al.  Foundations of optimal control theory , 1967 .

[27]  Mark I. Nelson,et al.  Evaluating the performance of a cascade of two bioreactors , 2006 .

[28]  Pedro Gajardo,et al.  Optimal Synthesis for the Minimum Time Control Problems of Fed-Batch Bioprocesses for Growth Functions with Two Maxima , 2013, J. Optim. Theory Appl..

[29]  Jérôme Harmand,et al.  Multiple steady state profiles in interconnected biological systems , 2006 .

[30]  V. Jurdjevic.  Geometric control theory , 1996 .