scholarly journals Controlled singular evolution equations and Pontryagin type maximum principle with applications

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xiao-Li Ding ◽  
Iván Area ◽  
Juan J. Nieto
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Control Problem ◽  
Evolution Equations ◽  
Optimal Control Problems ◽  
Type Theorem ◽  
Control Problems ◽  
Optimal Controls ◽  
Special Cases ◽  
The Impact

<p style='text-indent:20px;'>Due to the propagation of new coronavirus (COVID-19) on the community, global researchers are concerned with how to minimize the impact of COVID-19 on the world. Mathematical models are effective tools that help to prevent and control this disease. This paper mainly focuses on the optimal control problems of an epidemic system governed by a class of singular evolution equations. The mild solutions of such equations of Riemann-Liouville or Caputo types are special cases of the proposed equations. We firstly discuss well-posedness in an appropriate functional space for such equations. In order to reduce the cost caused by control process and vaccines, and minimize the total number of susceptible people and infected people as much as possible, an optimal control problem of an epidemic system is presented. And then for associated control problem, we use a generalized Liapunov type theorem and the spike perturbation technique to obtain a Pontryagin type maximum principle for its optimal controls. In order to derive the maximum principle for an optimal control problems, some techniques from analytical semigroups are employed to overcome the difficulties. Finally, we discuss the potential applications.</p>

scholarly journals Numerical Optimal Control of HIV Transmission in Octave/MATLAB

2019 ◽  
Vol 25 (1) ◽  
pp. 1 ◽  
Author(s):  
Carlos Campos ◽  
Cristiana J. Silva ◽  
Delfim F. M. Torres
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Differential Equations ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Optimal Control Problems ◽  
Control Problems ◽  
Initial Value

We provide easy and readable GNU Octave/MATLAB code for the simulation of mathematical models described by ordinary differential equations and for the solution of optimal control problems through Pontryagin’s maximum principle. For that, we consider a normalized HIV/AIDS transmission dynamics model based on the one proposed in our recent contribution (Silva, C.J.; Torres, D.F.M. A SICA compartmental model in epidemiology with application to HIV/AIDS in Cape Verde. Ecol. Complex. 2017, 30, 70–75), given by a system of four ordinary differential equations. An HIV initial value problem is solved numerically using the ode45 GNU Octave function and three standard methods implemented by us in Octave/MATLAB: Euler method and second-order and fourth-order Runge–Kutta methods. Afterwards, a control function is introduced into the normalized HIV model and an optimal control problem is formulated, where the goal is to find the optimal HIV prevention strategy that maximizes the fraction of uninfected HIV individuals with the least HIV new infections and cost associated with the control measures. The optimal control problem is characterized analytically using the Pontryagin Maximum Principle, and the extremals are computed numerically by implementing a forward-backward fourth-order Runge–Kutta method. Complete algorithms, for both uncontrolled initial value and optimal control problems, developed under the free GNU Octave software and compatible with MATLAB are provided along the article.

scholarly journals CONSTRAINT ALGORITHM FOR EXTREMALS IN OPTIMAL CONTROL PROBLEMS

2009 ◽  
Vol 06 (07) ◽  
pp. 1221-1233 ◽  
Author(s):  
MARÍA BARBERO-LIÑÁN ◽  
MIGUEL C. MUÑOZ-LECANDA
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Optimal Control Problem ◽  
Control Theory ◽  
Control Problem ◽  
Optimal Control Theory ◽  
Optimal Control Problems ◽  
Necessary Conditions ◽  
Control Problems ◽  
Affine System

A geometric method is described to characterize the different kinds of extremals in optimal control theory. This comes from the use of a presymplectic constraint algorithm starting from the necessary conditions given by Pontryagin's Maximum Principle. The algorithm must be run twice so as to obtain suitable sets that once projected must be compared. Apart from the design of this general algorithm useful for any optimal control problem, it is shown how to classify the set of extremals and, in particular, how to characterize the strict abnormality. An example of strict abnormal extremal for a particular control-affine system is also given.

scholarly journals REQUIRED OPTIMALITY CONDITIONS IN ONE OPTIMAL CONTROL PROBLEM WITH MULTIPOINT FUNCTIONAL

2020 ◽  
pp. 7-26
Author(s):  
Shahla Rasulzade ◽  
◽  
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Differential Equations ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimality Conditions ◽  
Optimal Control Problems ◽  
Control Problems ◽  
The Maximum Principle ◽  
Distributed Parameters

One specific optimal control problem with distributed parameters of the Moskalenko type with a multipoint quality functional is considered. To date, the theory of necessary first-order optimality conditions such as the Pontryagin maximum principle or its consequences has been sufficiently developed for various optimal control problems described by ordinary differential equations, i.e. for optimal control problems with lumped parameters. Many controlled processes are described by various partial differential equations (processes with distributed parameters). Some features are inherent in optimal control problems with distributed parameters, and therefore, when studying the optimal control problem with distributed parameters, in particular, when deriving various necessary optimality conditions, non-trivial difficulties arise. In particular, in the study of cases of degeneracy of the established necessary optimality conditions, fundamental difficulties arise. In the present work, we study one optimal control problem described by a system of first-order partial differential equations with a controlled initial condition under the assumption that the initial function is a solution to the Cauchy problem for ordinary differential equations. The objective function (quality criterion) is multi-point. Therefore, it becomes necessary to introduce an unconventional conjugate equation, not in differential (classical), but in integral form. In the work, using one version of the increment method, using the explicit linearization method of the original system, the necessary optimality condition is proved in the form of an analog of the maximum principle of L.S. Pontryagin. It is known that the maximum principle of L.S. Pontryagin for various optimal control problems is the strongest necessary condition for optimality. But the principle of a maximum of L.S. Pontryagin, being a necessary condition of the first order, often degenerates. Such cases are called special, and the corresponding management, special management. Based on these considerations, in the considered problem, we study the case of degeneration of the maximum principle of L.S. Pontryagin for the problem under consideration. For this purpose, a formula for incrementing the quality functional of the second order is constructed. By introducing auxiliary matrix functions, it was possible to obtain a second-order increment formula that is constructive in nature. The necessary optimality condition for special controls in the sense of the maximum principle of L.S. Pontryagin is proved. The proved necessary optimality conditions are explicit.

scholarly journals Maximum Principle for Optimal Control Problems of Forward-Backward Regime-Switching Systems Involving Impulse Controls

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Shujun Wang ◽  
Zhen Wu
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Markov Chains ◽  
Regime Switching ◽  
Optimal Control Problems ◽  
Sufficient Conditions ◽  
Control Problems ◽  
Optimal Controls ◽  
Switching Systems ◽  
Impulse Controls

This paper is concerned with optimal control problems of forward-backward Markovian regime-switching systems involving impulse controls. Here the Markov chains are continuous-time and finite-state. We derive the stochastic maximum principle for this kind of systems. Besides the Markov chains, the most distinguishing features of our problem are that the control variables consist of regular and impulsive controls, and that the domain of regular control is not necessarily convex. We obtain the necessary and sufficient conditions for optimal controls. Thereafter, we apply the theoretical results to a financial problem and get the optimal consumption strategies.

A Computational Method in Ship Routing Using the Concept of Limited Manoeuvrability

2004 ◽  
Vol 57 (3) ◽  
pp. 357-369 ◽  
Author(s):  
S. J. Bijlsma
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Control Problem ◽  
Optimal Control Problems ◽  
Computational Method ◽  
Control Problems ◽  
Wave Direction ◽  
Ship Routing ◽  
Additional Constraints ◽  
Problem Of Bolza

Under some circumstances, dependent on a ship's velocity, the wave period and wave direction, certain courses induce heavy rolling and must be avoided. This paper proposes a computational method for the solution of optimal control problems in ship routing for ships with such limited manoeuvrability. Known results for the control problem of Bolza with additional constraints are interpreted in terms of this new problem. This approach is equivalent to the application of Pontryagin's maximum principle. The method is an extension of an earlier method dealing with the meteorological navigation of ships with unrestricted manoeuvrability and gives a more realistic picture of what really could happen in practice.

Dynamic Path Synthesis and Optimal Control

1974 ◽  
Vol 96 (1) ◽  
pp. 19-24
Author(s):  
P. J. Starr
Keyword(s):  
Optimal Control ◽  
Control Problem ◽  
Fourth Order ◽  
Optimal Control Problems ◽  
Singular Control ◽  
Physical Nature ◽  
Linear Control ◽  
Control Problems ◽  
Dynamic Path ◽  
Path Synthesis

Dynamic Path Synthesis refers to a class of linkage synthesis problems in which constraint paths between specified positions are determined in such a way as to optimize some measure of the resulting dynamic behavior. These problems can be transformed into nonlinear optimal control problems which are generally non-autonomous. The physical nature of the system allows general comments to be made regarding uniqueness, controllability, and singular control. The ideas are developed in the context of a two-link device yielding a fourth order non-linear control problem, for which a numerical example is presented.

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