On an optimal control problem with an integral functional of a rational control function

2009 ◽  
Vol 45 (11) ◽  
pp. 1621-1635 ◽  
Author(s):  
N. L. Grigorenko ◽  
D. V. Kamzolkin ◽  
L. N. Luk’yanova ◽  
D. G. Pivovarchuk
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Control Function ◽  
Integral Functional ◽  
Rational Control

Optimal Control Problem with an Integral Functional of an Exponential Control Function

2014 ◽  
Vol 26 (1) ◽  
pp. 1-13 ◽  
Author(s):  
N. L. Grigorenko ◽  
D. V. Kamzolkin ◽  
L. N. Luk’yanova ◽  
D. G. Pivovarchuk
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Control Function ◽  
Integral Functional

Solving the Problem of the Optimal Control System General Synthesis Based on Approximation of a Set of Extremals using the Symbol Regression Method

2020 ◽  
pp. 59-74
Author(s):  
S.V. Konstantinov ◽  
A.I. Diveev
Keyword(s):  
Optimal Control ◽  
Control System ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Control Function ◽  
Symbolic Regression ◽  
Regression Method ◽  
Synthesis Problem ◽  
Optimal Control System ◽  
Initial States

A new approach is considered to solving the problem of synthesizing an optimal control system based on the extremals' set approximation. At the first stage, the optimal control problem for various initial states out of a given domain is being numerically sold. Evolutionary algorithms are used to solve the optimal control problem numerically. At the second stage, the problem of approximating the found set of extremals by the method of symbolic regression is solved. Approach considered in the work makes it possible to eliminate the main drawback of the known approach to solving the control synthesis problem using the symbolic regression method, which consists in the fact that the genetic algorithm used in solving the synthesis problem does not provide information about proximity of the found solution to the optimal one. Here, control function is built on the basis of a set of extremals; therefore, any particular solution should be close to the optimal trajectory. Computational experiment is presented for solving the applied problem of synthesizing the four-wheel robot optimal control system in the presence of phase constraints. It is experimentally demonstrated that the synthesized control function makes it possible for any initial state from a given domain to obtain trajectories close to optimal in the quality functional. Initial states were considered during the experiment, both included in the approximating set of optimal trajectories and others from the same given domain. Approximation of the extremals set was carried out by the network operator method

scholarly journals On Optimal Retirement

2014 ◽  
Vol 51 (02) ◽  
pp. 333-345 ◽  
Author(s):  
Philip A. Ernst ◽  
Dean P. Foster ◽  
Larry A. Shepp
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Control Function ◽  
Time Interval ◽  
Investment Strategies ◽  
Net Worth ◽  
New Model ◽  
Ito Process

We pose an optimal control problem arising in a perhaps new model for retirement investing. Given a control function f and our current net worth X(t) for any t, we invest an amount f(X(t)) in the market. We need a fortune of M ‘superdollars’ to retire and want to retire as early as possible. We model our change in net worth over each infinitesimal time interval by the Itô process dX(t) = (1 + f(X(t)))dt + f(X(t))dW(t). We show how to choose the optimal f = f 0 and show that the choice of f 0 is optimal among all nonanticipative investment strategies, not just among Markovian ones.

scholarly journals The properties of the solution for a class of switched systems with internally forced switching

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Huanting Li ◽  
Xiankang Chen
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Switched Systems ◽  
Special Class ◽  
Continuous Dependence ◽  
Control Function ◽  
The Past ◽  
Controlled Systems ◽  
Affine System

AbstractIn this paper, the dynamic behavior of a class of switched systems with internally forced switching (IFS) is investigated. By introducing the definitions of continuous dependence and differentiability, the continuous dependence and differentiability of the solution relative to the control function are obtained. In the past studies, the optimal control problem given by IFS mainly focused on a special class of controlled systems (the piece affine system). Our results lay a good foundation for studying the more general internally forced switching problem.

scholarly journals Optimal Control of HIV Dynamic Using Embedding Method

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
H. Zarei ◽  
A. V. Kamyad ◽  
M. H. Farahi
Keyword(s):  
Optimal Control ◽  
Immune System ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Control Function ◽  
Treatment Protocol ◽  
Optimal Solution ◽  
Embedding Method ◽  
Piecewise Constant Control ◽  
The Cost

This present study proposes an optimal control problem, with the final goal of implementing an optimal treatment protocol which could maximize the survival time of patients and minimize the cost of drug utilizing a system of ordinary differential equations which describes the interaction of the immune system with the human immunodeficiency virus (HIV). Optimal control problem transfers into a modified problem in measure space using an embedding method in which the existence of optimal solution is guaranteed by compactness of the space. Then the metamorphosed problem is approximated by a linear programming (LP) problem, and by solving this LP problem a suboptimal piecewise constant control function, which is more practical from the clinical viewpoint, is achieved. The comparison between the immune system dynamics in treated and untreated patients is introduced. Finally, the relationships between the healthy cells and virus are shown.

Optimal control problem for the equation of vibrations of an elastic plate

2018 ◽  
Vol 25 (3) ◽  
pp. 371-379 ◽  
Author(s):  
Hamlet F. Guliyev ◽  
Khayala I. Seyfullaeva
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Elastic Plate ◽  
Control Function ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Necessary Condition ◽  
Necessary Condition Of Optimality ◽  
Initial Boundary Value

AbstractAn optimal control problem for the vibration equation of an elastic plate is considered when the control function is included in the coefficient of the highest order derivative and the right-hand side of the equation. The solvability of the initial boundary value problem is shown, the theorem on the existence of an optimal control is proved and a necessary condition of optimality in the form of an integral equation is obtained.

scholarly journals Optimal Control for Multistage Nonlinear Dynamic System of Microbial Bioconversion in Batch Culture

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Lei Wang ◽  
Zhilong Xiu ◽  
Yuduo Zhang ◽  
Enmin Feng
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Batch Culture ◽  
Control Function ◽  
Nonlinear Dynamical System ◽  
Control Model ◽  
Optimization Techniques ◽  
Terminal Time ◽  
Continuous State

In batch culture of glycerol biodissimilation to 1,3-propanediol (1,3-PD), the aim of adding glycerol is to obtain as much 1,3-PD as possible. Taking the yield intensity of 1,3-PD as the performance index and the initial concentration of biomass, glycerol, and terminal time as the control vector, we propose an optimal control model subject to a multistage nonlinear dynamical system and constraints of continuous state. A computational approach is constructed to seek the solution of the above model. Firstly, we transform the optimal control problem into the one with fixed terminal time. Secondly, we transcribe the optimal control model into an unconstrained one based on the penalty functions and an extension of the state space. Finally, by approximating the control function with simple functions, we transform the unconstrained optimal control problem into a sequence of nonlinear programming problems, which can be solved using gradient-based optimization techniques. The convergence analysis and optimality function of the algorithm are also investigated. Numerical results show that, by employing the optimal control, the concentration of 1,3-PD at the terminal time can be increased, compared with the previous results.

scholarly journals The Dubovitskii and Milyutin Methodology Applied to an Optimal Control Problem Originating in an Ecological System

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 479
Author(s):  
Aníbal Coronel ◽  
Fernando Huancas ◽  
Esperanza Lozada ◽  
Marko Rojas-Medar
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Cost Function ◽  
Ecological Model ◽  
Control Function ◽  
Semigroup Theory ◽  
Reaction Diffusion ◽  
Functional Responses ◽  
Adjoint State

We research a control problem for an ecological model given by a reaction–diffusion system. The ecological model is given by a nonlinear parabolic PDE system of three equations modelling the interaction of three species by considering the standard Lotka-Volterra assumptions. The optimal control problem consists of the determination of a coefficient such that the population density of predator decreases. We reformulate the control problem as an optimal control problem by introducing an appropriate cost function. Then, we introduce and prove three types of results. A first contribution of the paper is the well-posedness framework of the mathematical model by considering that the interaction of the species is given by a general functional responses. Second, we study the differentiability properties of a cost function. The third result is the existence of optimal solutions, the existence of an adjoint state, and a characterization of the control function. The first result is proved by the application of semigroup theory and the second and third result are proved by the application of Dubovitskii and Milyutin formalism.

scholarly journals On Optimal Retirement

2014 ◽  
Vol 51 (2) ◽  
pp. 333-345 ◽  
Author(s):  
Philip A. Ernst ◽  
Dean P. Foster ◽  
Larry A. Shepp
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Control Function ◽  
Time Interval ◽  
Investment Strategies ◽  
Net Worth ◽  
New Model ◽  
Ito Process

We pose an optimal control problem arising in a perhaps new model for retirement investing. Given a control function f and our current net worth X(t) for any t, we invest an amount f(X(t)) in the market. We need a fortune of M ‘superdollars’ to retire and want to retire as early as possible. We model our change in net worth over each infinitesimal time interval by the Itô process dX(t) = (1 + f(X(t)))dt + f(X(t))dW(t). We show how to choose the optimal f = f0 and show that the choice of f0 is optimal among all nonanticipative investment strategies, not just among Markovian ones.

RELAXATION OF A WEAKLY DISCONTINUOUS FUNCTIONAL DEPENDING ON ONE CONTROL FUNCTION

2008 ◽  
Vol 13 (1) ◽  
pp. 79-85
Author(s):  
U. Raitums
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Control Function ◽  
Rank One ◽  
Measurable Vector ◽  
Admissible Controls

The paper considers an optimal control problem of the typewhere the set M of admissible controls consists of all measurable vector‐functions h, which can take only two values h1 or h2. It is shown that the relaxation of this problem can be explicitly computed by rank‐one laminates.

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