On Abnormal Optimal Control Problems with Mixed Equality and Inequality Constraints

AbstractIn this paper an optimal control problem with mixed equality and inequality constraints is considered in the so-called abnormal case, i.e., in the case when the classical Pontryagin-type Maximum Principle has a degenerate form which does not depend on the minimized functional. Some extension of the Dubovitskii-Milyutin method to the case of nonregular operator Constraints obtained by using Avakov*s generalization of the Lusternik theorem is applied and the extension of the local Maximum Principle (LMP) for the optimal control problem with mixed equality and inequality constraints is proved. The extended version of LMP presented here has a nondegenerate form for abnormal optimal controls and reduces to the classical LMP under classical regularity conditions

Optimal control strategies for tuberculosis treatment: a case study in Angola

We apply optimal control theory to a tuberculosis model given by a system of ordinary differential equations. Optimal control strategies are proposed to minimize the cost of interventions. Numerical simulations are given using data from Angola.Comment: This is a preprint of a paper whose final and definite form will appear in the international journal Numerical Algebra, Control and Optimization (NACO). Paper accepted for publication 15-March-201

Mathematical models for immunology:current state of the art and future research directions

The advances in genetics and biochemistry that have taken place over the last 10 years led to significant advances in experimental and clinical immunology. In turn, this has led to the development of new mathematical models to investigate qualitatively and quantitatively various open questions in immunology. In this study we present a review of some research areas in mathematical immunology that evolved over the last 10 years. To this end, we take a step-by-step approach in discussing a range of models derived to study the dynamics of both the innate and immune responses at the molecular, cellular and tissue scales. To emphasise the use of mathematics in modelling in this area, we also review some of the mathematical tools used to investigate these models. Finally, we discuss some future trends in both experimental immunology and mathematical immunology for the upcoming years

Singular controls and chattering arcs in optimal control problems arising in biomedicine

We consider an optimal control problem of the Mayer-type for a single-input, control affine, nonlinear system in small dimension. In this paper, we analyze effects that a modeling extension has on the optimality of singular controls when the control is replaced with the output of a first-order, time-invariant linear system driven by a new control. This analysis is motivated by an optimal control problem for a novel cancer treatment method, tumor anti-angiogenesis, when such a linear differential equation, which represents the pharmacokinetics of the therapeutic agent, is added to the model. We show that formulas that define a singular control of order 1 and its associated singular arc carry over verbatim under this model extension, albeit with a different interpretation. But the intrinsic order of the singular control increases to 2. As a, consequence, optimal concatenation sequences with the singular control change and the possibility of optimal chattering arcs arises

A Review of Optimal Chemotherapy Protocols: From MTD towards Metronomic Therapy

We review mathematical results about the qualitative structure of chemotherapy protocols that were obtained with the methods of optimal control. As increasingly more complex features are incorporated into the mathematical model—progressing from models for homogeneous, chemotherapeutically sensitive tumor cell populations to models for heterogeneous agglomerations of subpopulations of various sensitivities to models that include tumor immune-system interactions—the structures of optimal controls change from bang-bang solutions (which correspond to maximum dose rate chemotherapy with restperiods) to solutions that favor singular controls (representing reduced dose rates). Medically, this corresponds to a transition from standard MTD (maximum tolerated dose) type protocols to chemo-switch strategies towards metronomic dosing

Optimal Control for a Class of Compartmental Models in Cancer Chemotherapy

We consider a general class of mathematical models P for cancer chemotherapy described as optimal control problems over a fixed horizon with dynamics given by a bilinear system and an objective which is linear in the control. Several two- and three-compartment models considered earlier fall into this class. While a killing agent which is active during cell division constitutes the only control considered in the two-compartment model, Model A, also two three-compartment models, Models B and C, are analyzed, which consider a blocking agent and a recruiting agent, respectively. In Model B a blocking agent which slows down cell growth during the synthesis allowing in consequence the synchronization of the neoplastic population is added. In Model C the recruitment of dormant cells from the quiescent phase to enable their efficient treatment by a cytotoxic drug is included. In all models the cumulative effect of the killing agent is used to model the negative effect of the treatment on healthy cells. For each model it is shown that singular controls are not optimal. Then sharp necessary and sufficient optimality conditions for bang-bang controls are given for the general class of models P and illustrated with numerical examples