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3Mammadov, Musa
2Dontchev, Asen
1Adam, Lukas
1Cibulka, Radek
1Evans, Robin
1Foumani, Mehdi
1Gao, David
1Grigoryev, Igor
1Gunawan, Indra
1Ibrahim, Yousef
1Ivanov, Anatoli
1Kruger, Alexander
1Mustafina, Svetlana
1Outrata, Jiri
1Shangareeva, Gulnaz
1Trofimchuk, Sergei
1Wu, Dan
1Zhu, Jinghao

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50102 Applied Mathematics
20101 Pure Mathematics
10103 Numerical and Computational Mathematics
10802 Computation Theory and Mathematics
10906 Electrical and Electronic Engineering
10913 Mechanical Engineering
10915 Interdisciplinary Engineering
1Asymptotic stability
1Blood
1Blood cell model
1Blood cells
1Canonical dual method
1Cells
1Climate change
1Coderivative
1Differential equations
1Diseases
1Economic applications
1Environment

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Applying the canonical dual theory in optimal control problems

- Zhu, Jinghao, Wu, Dan, Gao, David

**Authors:**Zhu, Jinghao , Wu, Dan , Gao, David**Date:**2012**Type:**Text , Journal article**Relation:**Journal of global optimization Vol. 54, no. 2 (2012), p. 221-233**Full Text:**false**Reviewed:****Description:**This paper presents some applications of the canonical dual theory in optimal control problems. The analytic solutions of several nonlinear and nonconvex problems are investigated by global optimizations. It turns out that the backward differential flow defined by the KKT equation may reach the globally optimal solution. The analytic solution to an optimal control problem is obtained via the expression of the co-state. Some examples are illustrated.

Predicting and controlling the dynamics of infectious diseases

- Evans, Robin, Mammadov, Musa

**Authors:**Evans, Robin , Mammadov, Musa**Date:**2015**Type:**Text , Conference proceedings**Relation:**54th IEEE Conference on Decision and Control, CDC 2015; Osaka, Japan; 15th-18th December 2015; Published in Proceedings of the IEEE Conference on Decision and Control; p. 5378-5383**Full Text:****Description:**This paper introduces a new optimal control model to describe and control the dynamics of infectious diseases. In the present model, the average time to isolation (i.e. hospitalization) of infectious population is the main time-dependent parameter that defines the spread of infection. All the preventive measures aim to decrease the average time to isolation under given constraints. The model suggested allows one to generate a small number of possible future scenarios and to determine corresponding trajectories of infected population in different regions. Then, this information is used to find an optimal distribution of bed capabilities across countries/regions according to each scenario. © 2015 IEEE.

**Authors:**Evans, Robin , Mammadov, Musa**Date:**2015**Type:**Text , Conference proceedings**Relation:**54th IEEE Conference on Decision and Control, CDC 2015; Osaka, Japan; 15th-18th December 2015; Published in Proceedings of the IEEE Conference on Decision and Control; p. 5378-5383**Full Text:****Description:**This paper introduces a new optimal control model to describe and control the dynamics of infectious diseases. In the present model, the average time to isolation (i.e. hospitalization) of infectious population is the main time-dependent parameter that defines the spread of infection. All the preventive measures aim to decrease the average time to isolation under given constraints. The model suggested allows one to generate a small number of possible future scenarios and to determine corresponding trajectories of infected population in different regions. Then, this information is used to find an optimal distribution of bed capabilities across countries/regions according to each scenario. © 2015 IEEE.

Turnpike theorem for an infinite horizon optimal control problem with time delay

**Authors:**Mammadov, Musa**Date:**2014**Type:**Text , Journal article**Relation:**SIAM Journal on Control and Optimization Vol. 52, no. 1 (2014), p. 420-438**Full Text:****Reviewed:****Description:**An optimal control problem for systems described by a special class of nonlinear differential equations with time delay is considered. The cost functional adopted could be considered as an analogue of the terminal functional defined over an infinite time horizon. The existence of optimal solutions as well as the asymptotic stability of optimal trajectories (that is, the turnpike property) are established under some quite mild restrictions on the nonlinearities of the functions involved in the description of the problem. Such mild restrictions on the nonlinearities allowed us to apply these results to a blood cell production model. Â© 2014 Society for Industrial and Applied Mathematics.

**Authors:**Mammadov, Musa**Date:**2014**Type:**Text , Journal article**Relation:**SIAM Journal on Control and Optimization Vol. 52, no. 1 (2014), p. 420-438**Full Text:****Reviewed:****Description:**An optimal control problem for systems described by a special class of nonlinear differential equations with time delay is considered. The cost functional adopted could be considered as an analogue of the terminal functional defined over an infinite time horizon. The existence of optimal solutions as well as the asymptotic stability of optimal trajectories (that is, the turnpike property) are established under some quite mild restrictions on the nonlinearities of the functions involved in the description of the problem. Such mild restrictions on the nonlinearities allowed us to apply these results to a blood cell production model. Â© 2014 Society for Industrial and Applied Mathematics.

Comparative analysis of numerical solution of optimal control problems

- Shangareeva, Gulnaz, Grigoryev, Igor, Mustafina, Svetlana

**Authors:**Shangareeva, Gulnaz , Grigoryev, Igor , Mustafina, Svetlana**Date:**2016**Type:**Text , Journal article**Relation:**International Journal of Pure and Applied Mathematics Vol. 110, no. 4 (2016), p. 645-649**Full Text:****Reviewed:****Description:**In this article step by step algorithms were developed for solving optimal control problems based on the method of successive approximations and the method of variations in the space of controls. The algorithm of the method of successive approximations requires details of the problem to the boundary problem of the maximum principle. In turn, the algorithm of the variations is more versatile because it is based on iterating state variables and control in the phase space. A numerical study and comparative analysis of the developed algorithms performed at different values of accuracy. © 2016 Academic Publications, Ltd.

**Authors:**Shangareeva, Gulnaz , Grigoryev, Igor , Mustafina, Svetlana**Date:**2016**Type:**Text , Journal article**Relation:**International Journal of Pure and Applied Mathematics Vol. 110, no. 4 (2016), p. 645-649**Full Text:****Reviewed:****Description:**In this article step by step algorithms were developed for solving optimal control problems based on the method of successive approximations and the method of variations in the space of controls. The algorithm of the method of successive approximations requires details of the problem to the boundary problem of the maximum principle. In turn, the algorithm of the variations is more versatile because it is based on iterating state variables and control in the phase space. A numerical study and comparative analysis of the developed algorithms performed at different values of accuracy. © 2016 Academic Publications, Ltd.

Strong metric subregularity of mappings in variational analysis and optimization

- Cibulka, Radek, Dontchev, Asen, Kruger, Alexander

**Authors:**Cibulka, Radek , Dontchev, Asen , Kruger, Alexander**Date:**2018**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 457, no. 2 (2018), p. 1247-1287**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:****Description:**Although the property of strong metric subregularity of set-valued mappings has been present in the literature under various names and with various (equivalent) definitions for more than two decades, it has attracted much less attention than its older “siblings”, the metric regularity and the strong (metric) regularity. The purpose of this paper is to show that the strong metric subregularity shares the main features of these two most popular regularity properties and is not less instrumental in applications. We show that the strong metric subregularity of a mapping F acting between metric spaces is stable under perturbations of the form f+F, where f is a function with a small calmness constant. This result is parallel to the Lyusternik–Graves theorem for metric regularity and to the Robinson theorem for strong regularity, where the perturbations are represented by a function f with a small Lipschitz constant. Then we study perturbation stability of the same kind for mappings acting between Banach spaces, where f is not necessarily differentiable but admits a set-valued derivative-like approximation. Strong metric q-subregularity is also considered, where q is a positive real constant appearing as exponent in the definition. Rockafellar's criterion for strong metric subregularity involving injectivity of the graphical derivative is extended to mappings acting in infinite-dimensional spaces. A sufficient condition for strong metric subregularity is established in terms of surjectivity of the Fréchet coderivative, and it is shown by a counterexample that surjectivity of the limiting coderivative is not a sufficient condition for this property, in general. Then various versions of Newton's method for solving generalized equations are considered including inexact and semismooth methods, for which superlinear convergence is shown under strong metric subregularity. As applications to optimization, a characterization of the strong metric subregularity of the KKT mapping is obtained, as well as a radius theorem for the optimality mapping of a nonlinear programming problem. Finally, an error estimate is derived for a discrete approximation in optimal control under strong metric subregularity of the mapping involved in the Pontryagin principle.

On optimal control of a sweeping process coupled with an ordinary differential equation

**Authors:**Adam, Lukas , Outrata, Jiri**Date:**2014**Type:**Text , Journal article**Relation:**Discrete and Continuous Dynamical Systems - Series B Vol. 19, no. 9 (November 2014 2014), p. 2709-2738**Full Text:**false**Reviewed:****Description:**We study a special case of an optimal control problem governed by a differential equation and a differential rate{independent variational inequality, both with given initial conditions. Under certain conditions, the variational inequality can be reformulated as a differential inclusion with discontinuous right-hand side. This inclusion is known as sweeping process. We perform a discretization scheme and prove the convergence of optimal solutions of the discretized problems to the optimal solution of the original problem. For the discretized problems we study the properties of the solution map and compute its coderivative. Employing an appropriate chain rule, this enables us to compute the subdifferential of the objective function and to apply a suitable optimization technique to solve the discretized problems. The investigated problem is used to model a situation arising in the area of queuing theory.

On some open problems in optimal control

**Authors:**Dontchev, Asen**Date:**2018**Type:**Text , Book chapter**Relation:**Control Systems and Mathematical Methods in Economics : Essays in Honor of Vladimir M. Veliov (part of the Lecture Notes in Economics and Mathematical Systems book series) p. 3-13**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:****Description:**Several open problems are presented concerning regularity properties of solutions of optimal control problems with constraints.

Global stabilization in nonlinear discrete systems with time-delay

- Ivanov, Anatoli, Mammadov, Musa, Trofimchuk, Sergei

**Authors:**Ivanov, Anatoli , Mammadov, Musa , Trofimchuk, Sergei**Date:**2012**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol.56, no. 2 (2012), p. 1-13**Full Text:**false**Reviewed:****Description:**A class of scalar nonlinear difference equations with delay is considered. Sufficient conditions for the global asymptotic stability of a unique equilibrium are given. Applications in economics and other fields lead to consideration of associated optimal control problems. An optimal control problem of maximizing a consumption functional is stated. The existence of optimal solutions is established and their stability (the turnpike property) is proved. © 2012 Springer Science+Business Media, LLC.

Cyclic production for robotic cells served by multi-function robots with resumable processing regime

- Foumani, Mehdi, Ibrahim, Yousef, Gunawan, Indra

**Authors:**Foumani, Mehdi , Ibrahim, Yousef , Gunawan, Indra**Date:**2013**Type:**Text , Conference paper**Relation:**2013 IEEE International Conference on Industrial Engineering and Engineering Management, Bangkok, (10 - 13 December 2013) p. 551-555**Full Text:**false**Reviewed:****Description:**This paper addresses the problem of finding the optimal robot move cycle to minimise the cycle time of two-machine cells. The earlier robot's function was mainly moving parts between machines in a manufacturing process. We lift this assumption on robot tasks and assumed a special robot, namely multi-function, which performs a unique operation in transit. This robot starts performing this operation after unloading a part from input buffer and finishes it before loading the part to the output buffer. The processing mode on robot is “stop resume”. Thus, regardless of the gap interrupts during the operation, the robot continues processing on part when it is reloaded to the robot without any loss in time. The focus of this study is on one-unit cycles since they are very popular in industry. The cycle time of two possible one-unit cycles is obtained, and the optimality condition of them is determined.

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