Controlled Lotka-Volterra systems in the modeling of biomedical processes

2021 ◽  
Author(s):  
Evgenii Khailov ◽  
Nikolai Grigorenko ◽  
Ellina Grigorieva ◽  
Anna Klimenkova
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Numerical Study ◽  
Treatment Strategies ◽  
Main Tool ◽  
Necessary Condition ◽  
Control Problems ◽  
Optimal Controls ◽  
Controlled Systems ◽  
Volterra Systems

This book is devoted to a consistent presentation of the recent results obtained by the authors related to controlled systems created based on the Lotka-Volterra competition model, as well as to theoretical and numerical study of the corresponding optimal control problems. These controlled systems describe various modern methods of treating blood cancers, and the optimal control problems stated for such systems, reflect the search for the optimal treatment strategies. The main tool of the theoretical analysis used in this book is the Pontryagin maximum principle - a necessary condition for optimality in optimal control problems. Possible types of the optimal blood cancer treatment - the optimal controls - are obtained as a result of analytical investigations and are confirmed by corresponding numerical calculations. This book can be used as a supplement text in courses of mathematical modeling for upper undergraduate and graduate students. It is our believe that this text will be of interest to all professors teaching such or similar courses as well as for everyone interested in modern optimal control theory and its biomedical applications.

scholarly journals A gradient technique for an optimal control problem governed by a system of nonlinear first order partial differential equations

1995 ◽  
Vol 36 (3) ◽  
pp. 261-273 ◽  
Author(s):  
Mohammad A. Kazemi
Keyword(s):  
Optimal Control ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Optimal Control Problems ◽  
Fréchet Derivative ◽  
Necessary Condition ◽  
Control Problems ◽  
Partial Differential ◽  
Frechet Derivative ◽  
First Order

AbstractIn this paper a class of optimal control problems with distributed parameters is considered. The governing equations are nonlinear first order partial differential equations that arise in the study of heterogeneous reactors and control of chemical processes. The main focus of the present paper is the mathematical theory underlying the algorithm. A conditional gradient method is used to devise an algorithm for solving such optimal control problems. A formula for the Fréchet derivative of the objective function is obtained, and its properties are studied. A necessary condition for optimality in terms of the Fréchet derivative is presented, and then it is shown that any accumulation point of the sequence of admissible controls generated by the algorithm satisfies this necessary condition for optimality.

Existence of optimal controls for viscous flow problems

1992 ◽  
Vol 439 (1905) ◽  
pp. 81-102 ◽  
Keyword(s):  
Optimal Control ◽  
Flow Control ◽  
Viscous Flow ◽  
Existence Theorem ◽  
General Class ◽  
Optimal Control Problems ◽  
Control Problems ◽  
Optimal Controls ◽  
Flow Problems ◽  
Existence Of Optimal Controls

A class of optimal control problems in viscous flow is studied. Main result is the existence theorem for optimal control. Three typical flow control problems are formulated within this general class.

scholarly journals On Two Optimal Control Problems for Magnetic Fields

2014 ◽  
Vol 14 (4) ◽  
pp. 555-573 ◽  
Author(s):  
Serge Nicaise ◽  
Simon Stingelin ◽  
Fredi Tröltzsch
Keyword(s):  
Optimal Control ◽  
Magnetic Fields ◽  
Optimal Control Problems ◽  
Necessary Optimality Conditions ◽  
Electrical Current ◽  
Induction Coil ◽  
Control Problems ◽  
Optimal Controls ◽  
Electrically Conducting ◽  
Electrical Voltage

AbstractTwo optimal control problems for instationary magnetization processes are considered in 3D spatial domains that include electrically conducting and nonconducting regions. The magnetic fields are generated by induction coils. In the first model, the induction coil is considered as part of the conducting region and the electrical current is taken as control. In the second, the coil is viewed as part of the nonconducting region and the electrical voltage is the control. Here, an integro-differential equation accounts for the magnetic induction law that couples the given electrical voltage with the induced electrical current in the induction coil. We derive first-order necessary optimality conditions for the optimal controls of both problems. Based on them, numerical methods of gradient type are applied. Moreover, we report on the application of model reduction by POD that lead to tremendous savings. Numerical tests are presented for academic 3D geometries but also for a real-world application.

scholarly journals Existence of Solutions for Optimal Control Problems on Time Scales whose States are Absolutely Continuous

2016 ◽  
Vol 17 (1) ◽  
pp. 81
Author(s):  
Iguer L D Santos
Keyword(s):  
Optimal Control ◽  
Time Scales ◽  
Optimal Control Problems ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Dynamic Equations ◽  
Control Problems ◽  
Optimal Controls ◽  
Absolutely Continuous

This paper considers a class of optimal control problems on time scales described by dynamic equations on time scales. We have established sufficient conditions for theexistence of optimal controls.

Regularization and discretization error estimates for optimal control of ODEs with group sparsity

2018 ◽  
Vol 24 (2) ◽  
pp. 811-834 ◽  
Author(s):  
Christopher Schneider ◽  
Gerd Wachsmuth
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Control Function ◽  
Semismooth Newton Method ◽  
Control Problems ◽  
Time Interval ◽  
Stability Estimates ◽  
Optimal Controls ◽  
Group Sparsity ◽  
Linear Quadratic Optimal Control

It is well known that optimal control problems with L1-control costs produce sparse solutions, i.e., the optimal control is zero on whole intervals. In this paper, we study a general class of convex linear-quadratic optimal control problems with a sparsity functional that promotes a so-called group sparsity structure of the optimal controls. In this case, the components of the control function take the value of zero on parts of the time interval, simultaneously. These problems are both theoretically interesting and practically relevant. After obtaining results about the structure of the optimal controls, we derive stability estimates for the solution of the problem w.r.t. perturbations and L2-regularization. These results are consequently applied to prove convergence of the Euler discretization. Finally, the usefulness of our approach is demonstrated by solving an illustrative example using a semismooth Newton method.

scholarly journals Sufficiency for Purely Essentially Bounded Optimal Controls

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 238
Author(s):  
Gerardo Sánchez Licea
Keyword(s):  
Nonlinear Dynamics ◽  
Optimal Control ◽  
Optimal Control Problems ◽  
Sufficient Conditions ◽  
Control Problems ◽  
Optimal Controls ◽  
Embedding Theorems ◽  
State Control ◽  
Discontinuous Solutions ◽  
Piecewise Continuous

For optimal control problems of Bolza with variable and free end-points, nonlinear dynamics, nonlinear isoperimetric inequality and equality restrictions, and nonlinear pointwise mixed time-state-control inequality and equality constraints, sufficient conditions for strong minima are derived. The algorithm used to prove the main theorem of the paper includes a crucial symmetric inequality, making this technique an independent self-contained method of classical concepts such as embedding theorems from ordinary differential equations, Mayer fields, Riccati equations, or Hamilton–Jacobi theory. Moreover, the sufficiency theory given in this article is able to detect discontinuous solutions, that is, solutions which need to be neither continuous nor piecewise continuous but only essentially bounded.

scholarly journals A numerical algorithm for optimal control problems with switching costs

1992 ◽  
Vol 34 (2) ◽  
pp. 212-228 ◽  
Author(s):  
David E. Stewart
Keyword(s):  
Optimal Control ◽  
Switching Costs ◽  
Optimal Control Problems ◽  
Nonlinear Problems ◽  
Control Problems ◽  
Optimal Controls ◽  
Linear Dynamics ◽  
Control Functions ◽  
Linear Problems ◽  
Chattering Controls

AbstractOptimal control problems with switching costs arise in a number of applications, and are particularly important when standard control theory gives “chattering controls”. A numerical method is given for finding optimal controls for linear problems (linear dynamics, linear plus switching cost). This is used to develop an algorithm for finding sub-optimal control functions for nonlinear problems with switching costs. Numerical results are presented for an implementation of this method.

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