Pedro Gajardo

Minimal Time Sequential Batch Reactors with Bounded and Impulse Controls for One or More Species

We consider the optimal control problem of feeding in minimal time a tank where several species compete for a single resource, with the objective being to reach a given level of the resource. We allow controls to be bounded measurable functions of time plus possible impulses. For the one-species case, we show that the immediate one-impulse strategy (filling the whole reactor with one single impulse at the initial time) is optimal when the growth function is monotonic. For nonmonotonic growth functions with one maximum, we show that a particular singular arc strategy (precisely defined in section 3) is optimal. These results extend and improve former ones obtained for the class of measurable controls only. For the two-species case with monotonic growth functions, we give conditions under which the immediate one-impulse strategy is optimal. We also give optimality conditions for the singular arc strategy (at a level that depends on the initial condition) to be optimal. The possibility for the immediate one-impulse strategy to be nonoptimal while both growth functions are monotonic is a surprising result and is illustrated with the help of numerical simulations.

Optimal Synthesis for the Minimum Time Control Problems of Fed-Batch Bioprocesses for Growth Functions with Two Maxima

We address the problem of finding an optimal feedback control for feeding a fed-batch bioreactor with one species and one substrate from a given initial condition to a given target value in a minimal amount of time. Recently, the optimal synthesis (optimal feeding strategy) has been obtained in systems in which the microorganisms involved are represented by increasing growth functions or growth functions with one maxima, with either Monod or Haldane functions, respectively (widely used in bioprocesses modeling). In the present work, we allow impulsive controls corresponding to instantaneous dilutions, and we assume that the growth function of the microorganism present in the process has exactly two local maxima. This problem has been tackled from a numerical point of view without impulsive controls. In this article, we introduce two singular arc feeding strategies, and we define explicit regions of initial conditions in which the optimal strategy is either the first singular arc strategy or the second strategy.

Brief paper: Minimal time bioremediation of natural water resources

We study minimal time strategies for the treatment of pollution in large water volumes, such as lakes or natural reservoirs, with the help of an autonomous bioreactor. The control consists of feeding the bioreactor from the resource, with clean output returning to the resource with the same flow rate. We first characterize the optimal policies among constant and feedback controls under the assumption of a uniform concentration in the resource. In the second part, we study the influence of inhomogeneity in the resource, considering two measurement points. With the help of the Maximum Principle, we show that the optimal control law is non-monotonic and terminates with a constant phase, in contrast to the homogeneous case in which the optimal flow rate decreases with time. This study allows decision makers to identify situations in which the benefit of using non-constant flow rates is significant.

Reconstructing a matrix from a partial sampling of Pareto eigenvalues

AbstractLet Λ={λ1,…,λp} be a given set of distinct real numbers. This work deals with the problem of constructing a real matrix A of order n such that each element of Λ is a Pareto eigenvalue of A, that is to say, for all k∈{1,…,p} the complementarity system

Various Lipschitz-like Properties for Functions and Sets I: Directional Derivative and Tangential Characterizations

x\geq \mathbf{0}_n,\quad Ax-\lambda_k x\geq \mathbf{0}_n,\quad \langle x, Ax-\lambda_k x\rangle = 0

Equilibrium problems involving the Lorentz cone

admits a nonzero solution x∈ℝn.

Epsilon-Eigenvalues of Multivalued Operators

We study minimal time strategies for the treatment of pollution in large water volumes, such as lakes or natural reservoirs, with the help of an autonomous bioreactor. The control consists of feeding the bioreactor from the resource, with clean output returning to the resource with the same flow rate. We first recall recent characterizations of optimal policies among feedback controls under the assumption of a uniform concentration in the resource, in which the optimal flow rate decreases with time. Then, we introduce a dead zone in the resource, considering two measurement points. With the help of the Pontryagin Maximum Principle, we show that this variant has no influence on the optimal control law. This study advises decision makers to focus on simple models.

Viable states for monotone harvest models

In this work we introduce for extended real valued functions, defined on a Banach space