scholarly journals Control and the Analysis of Cancer Growth Models

2018 ◽  
Author(s):  
Allen Tannenbaum ◽  
Tryphon Georgiou ◽  
Joseph Deasy ◽  
Larry Norton
Keyword(s):  
Tumor Cell ◽  
Growth Models ◽  
Point Of View ◽  
Cancer Growth ◽  
Volterra Equations ◽  
Linear Quadratic Control ◽  
Dynamical Models ◽  
Linear Quadratic ◽  
Tumor Cell Population ◽  
Theoretic Point

AbstractIn this note, we analyze two cancer dynamical models from a system-theoretic point of view. The first model is based upon stochastic controlled versions of the classical Lotka-Volterra equations. Here we consider from a controls point of view the utility of employing ultrahigh dose flashes in radiotherapy. The second is based on work of Norton-Simon-Massagué growth model that takes into account the heterogeneity of a tumor cell population. We indicate an optimal strategy based on linear quadratic control applied to a linear transformed model.

scholarly journals Optimal Control with Partial Information for Stochastic Volterra Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-25 ◽  
Author(s):  
Bernt øksendal ◽  
Tusheng Zhang
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Control Problem ◽  
Partial Information ◽  
Volterra Equations ◽  
Linear Quadratic Control ◽  
Linear Quadratic ◽  
Stochastic Delay ◽  
Stochastic Delay Equations ◽  
Linear Quadratic Control Problem

In the first part of the paper we obtain existence and characterizations of an optimal control for a linear quadratic control problem of linear stochastic Volterra equations. In the second part, using the Malliavin calculus approach, we deduce a general maximum principle for optimal control of general stochastic Volterra equations. The result is applied to solve some stochastic control problem for some stochastic delay equations.

On the validity of hilbert's nullstellensatz, artin's theorem, and related results in grothendieck toposes

1988 ◽  
Vol 53 (4) ◽  
pp. 1177-1187
Author(s):  
W. A. MacCaull
Keyword(s):  
Intuitionistic Logic ◽  
Main Idea ◽  
Rational Functions ◽  
Completeness Theorem ◽  
Point Of View ◽  
Polynomial Rings ◽  
Von Neumann ◽  
Ordered Fields ◽  
Von Neumann Regular ◽  
Theoretic Point

Using formally intuitionistic logic coupled with infinitary logic and the completeness theorem for coherent logic, we establish the validity, in Grothendieck toposes, of a number of well-known, classically valid theorems about fields and ordered fields. Classically, these theorems have proofs by contradiction and most involve higher order notions. Here, the theorems are each given a first-order formulation, and this form of the theorem is then deduced using coherent or formally intuitionistic logic. This immediately implies their validity in arbitrary Grothendieck toposes. The main idea throughout is to use coherent theories and, whenever possible, find coherent formulations of formulas which then allow us to call upon the completeness theorem of coherent logic. In one place, the positive model-completeness of the relevant theory is used to find the necessary coherent formulas.The theorems here deal with polynomials or rational functions (in s indeterminates) over fields. A polynomial over a field can, of course, be represented by a finite string of field elements, and a rational function can be represented by a pair of strings of field elements. We chose the approach whereby results on polynomial rings are reduced to results about the base field, because the theory of polynomial rings in s indeterminates over fields, although coherent, is less desirable from a model-theoretic point of view. Ultimately we are interested in the models.This research was originally motivated by the works of Saracino and Weispfenning [SW], van den Dries [Dr], and Bunge [Bu], each of whom generalized some theorems from algebraic geometry or ordered fields to (commutative, von Neumann) regular rings (with unity).

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