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.

Local generalization of transversality conditions for optimal control problem

2019 ◽  
Vol 14 (3) ◽  
pp. 310
Author(s):  
Beyza Billur İskender Eroglu ◽  
Dіlara Yapişkan
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimal Control Problems ◽  
Hamiltonian Formalism ◽  
Variational Calculus ◽  
Control Problems ◽  
Conformable Derivative ◽  
Transversality Conditions ◽  
Control Functions

In this paper, we introduce the transversality conditions of optimal control problems formulated with the conformable derivative. Since the optimal control theory is based on variational calculus, the transversality conditions for variational calculus problems are first investigated and then supported by some illustrative examples. Utilizing from these formulations, the transversality conditions for optimal control problems are attained by using the Hamiltonian formalism and Lagrange multiplier technique. To illustrate the obtained results, the dynamical system on which optimal control problem constructed is taken as a diffusion process modeled in terms of the conformable derivative. The optimal control law is achieved by analytically solving the time dependent conformable differential equations occurring from the eigenfunction expansions of the state and the control functions. All figures are plotted using MATLAB.

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.

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 Mixed-integer optimal control problems with switching costs: a shortest path approach

2020 ◽  
Author(s):  
Felix Bestehorn ◽  
Christoph Hansknecht ◽  
Christian Kirches ◽  
Paul Manns
Keyword(s):  
Optimal Control ◽  
Shortest Path ◽  
Switching Costs ◽  
Optimal Control Problems ◽  
Directed Acyclic Graphs ◽  
Shortest Path Problem ◽  
Mixed Integer ◽  
Programming Approach ◽  
Control Problems ◽  
Decomposition Approach

Abstract We investigate an extension of Mixed-Integer Optimal Control Problems by adding switching costs, which enables the penalization of chattering and extends current modeling capabilities. The decomposition approach, consisting of solving a partial outer convexification to obtain a relaxed solution and using rounding schemes to obtain a discrete-valued control can still be applied, but the rounding turns out to be difficult in the presence of switching costs or switching constraints as the underlying problem is an Integer Program. We therefore reformulate the rounding problem into a shortest path problem on a parameterized family of directed acyclic graphs (DAGs). Solving the shortest path problem then allows to minimize switching costs and still maintain approximability with respect to the tunable DAG parameter $$\theta $$ θ . We provide a proof of a runtime bound on equidistant rounding grids, where the bound is linear in time discretization granularity and polynomial in $$\theta $$ θ . The efficacy of our approach is demonstrated by a comparison with an integer programming approach on a benchmark problem.

THE PROGRAM SYSTEM FOR SOLVING OPTIMAL CONTROL PROBLEMS WITH PHASE CONSTRAINTS

1993 ◽  
Vol 03 (04) ◽  
pp. 487-497 ◽  
Author(s):  
A.I. TYATUSHKIN ◽  
A.I. ZHOLUDEV ◽  
N.M. ERINCHEK
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Lagrange Function ◽  
Main Idea ◽  
Linear Constraints ◽  
Control Problems ◽  
Program System ◽  
Control Functions ◽  
Phase Constraints ◽  
Linearly Constrained

In this paper, we present a program system for solving optimal control problems with phase constraints. The main idea of the method realized in the program system consists in successive solving auxiliary problems, which minimizes a special constructed Lagrange function, subject to linearized phase constraints. The linearly constrained auxiliary problems are more simple than the original ones because linear constraints can be easily processed. We shall discuss different aspects connected with approximating control problems and using the program system for solving them. We shall then pay attention to optimal control problems with constraints on inertia of control functions. For illustrations, two control problems will be solved using the proposed software.

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.

Fractional-order Boubaker functions and their applications in solving delay fractional optimal control problems

2017 ◽  
Vol 24 (15) ◽  
pp. 3370-3383 ◽  
Author(s):  
Kobra Rabiei ◽  
Yadollah Ordokhani ◽  
Esmaeil Babolian
Keyword(s):  
Optimal Control ◽  
Fractional Order ◽  
Optimal Control Problems ◽  
Operational Matrix ◽  
Nonlinear Problems ◽  
Control Problems ◽  
Fractional Optimal Control ◽  
Newton’S Iterative Method ◽  
Quadratic Performance ◽  
Fractional Optimal Control Problems

In this paper, a new set of functions called fractional-order Boubaker functions is defined for solving the delay fractional optimal control problems with a quadratic performance index. To solve the problem, first we obtain the operational matrix of the Caputo fractional derivative of these functions and the operational matrix of multiplication to solve the nonlinear problems for the first time. Also, a general formulation for the delay operational matrix of these functions has been achieved. Then we utilized these matrices to solve delay fractional optimal control problems directly. In fact, the delay fractional optimal control problem converts to an optimization problem, which can then be easily solved with the aid of the Gauss–Legendre integration formula and Newton’s iterative method. Convergence of the algorithm is proved. The applicability of the method is shown by some examples; moreover, a comparison with the existing results shows the preference of this method.

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