Optimal Control Problem of Positive Solutions to Second Order Impulsive Differential Equations

2012 ◽  
pp. 237-250 ◽  
Author(s):  
Lingling Zhang ◽  
Noriaki Yamazaki ◽  
Chengbo Zhai
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Positive Solutions ◽  
Impulsive Differential Equations ◽  
Second Order

scholarly journals Positive Solutions for Impulsive Differential Equations with Mixed Monotonicity and Optimal Control

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Lingling Zhang ◽  
Noriaki Yamazaki ◽  
Rui Guo
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Control Problem ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Impulsive Differential Equation ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Initial Value ◽  
Mixed Monotone Operator

We consider positive solutions and optimal control problem for a second order impulsive differential equation with mixed monotone terms. Firstly, by using a fixed point theorem of mixed monotone operator, we study positive solutions of the boundary value problem for impulsive differential equations with mixed monotone terms, and sufficient conditions for existence and uniqueness of positive solutions will be established. Also, we study positive solutions of the initial value problem for our system. Moreover, we investigate the control problem of positive solutions to our equations, and then, we prove the existence of an optimal control and its stability. In addition, related examples will be given for illustrations.

scholarly journals An Optimal Control Problem by a Hybrid System of Hyperbolic and Ordinary Differential Equations

Games ◽  
2021 ◽  
Vol 12 (1) ◽  
pp. 23
Author(s):  
Alexander Arguchintsev ◽  
Vasilisa Poplevko
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Ordinary Differential Equations ◽  
Hyperbolic Equations ◽  
Control Methods ◽  
The Right ◽  
The Cost ◽  
Optimal Control Methods

This paper deals with an optimal control problem for a linear system of first-order hyperbolic equations with a function on the right-hand side determined from controlled bilinear ordinary differential equations. These ordinary differential equations are linear with respect to state functions with controlled coefficients. Such problems arise in the simulation of some processes of chemical technology and population dynamics. Normally, general optimal control methods are used for these problems because of bilinear ordinary differential equations. In this paper, the problem is reduced to an optimal control problem for a system of ordinary differential equations. The reduction is based on non-classic exact increment formulas for the cost-functional. This treatment allows to use a number of efficient optimal control methods for the problem. An example illustrates the approach.

Pontryagin maximum principle and second order optimality conditions for optimal control problems governed by 2D nonlocal Cahn–Hilliard–Navier–Stokes equations

Analysis ◽  
2020 ◽  
Vol 40 (3) ◽  
pp. 127-150
Author(s):  
Tania Biswas ◽  
Sheetal Dharmatti ◽  
Manil T. Mohan
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Stokes Equations ◽  
Minimum Principle ◽  
Immiscible Fluids ◽  
Second Order ◽  
Navier Stokes ◽  
Distributed Optimal Control ◽  
Navier Stokes Equations

AbstractIn this paper, we formulate a distributed optimal control problem related to the evolution of two isothermal, incompressible, immiscible fluids in a two-dimensional bounded domain. The distributed optimal control problem is framed as the minimization of a suitable cost functional subject to the controlled nonlocal Cahn–Hilliard–Navier–Stokes equations. We describe the first order necessary conditions of optimality via the Pontryagin minimum principle and prove second order necessary and sufficient conditions of optimality for the problem.

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.

Trajectory Optimization for DDE Models of Supercavitating Underwater Vehicles

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
Carlo L. Bottasso ◽  
Francesco Scorcelletti ◽  
Massimo Ruzzene ◽  
Seong S. Ahn
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Trajectory Optimization ◽  
Past History ◽  
Flight Mechanics ◽  
Multiple Shooting ◽  
Supercavitating Vehicles ◽  
Delay Differential

In this study we first develop a flight mechanics model for supercavitating vehicles, which is formulated to account for the dependence of the cavity shape from the past history of the system. This mathematical model is governed by a particular class of delay differential equations, featuring time delays on the states of the system. Next, flight trajectories and maneuvering strategies for supercavitating vehicles are obtained by solving an optimal control problem, whose solution, given a cost function and general constraints and bounds on states and controls, yields the control time histories that maneuver the vehicle according to a desired strategy, together with the associated flight path. The optimal control problem is solved using a novel direct multiple shooting approach, which is formulated to properly handle conditions dictated by the delay differential equation formulation governing the dynamic behavior of the vehicle. Specifically, the new formulation enforces the state continuity line conditions in a least-squares sense using local interpolations, which supports local time stepping and drastically reduces the number of optimization unknowns. Examples of maneuvers and resulting trajectories demonstrate the effectiveness of the proposed methodology and the generality of the formulation. The results are also compared with those obtained from a previously developed model governed by ordinary differential equations to highlight the differences and demonstrate the need for the current formulation.

Solution for fractional distributed optimal control problem by hybrid meshless method

2016 ◽  
Vol 24 (11) ◽  
pp. 2149-2164 ◽  
Author(s):  
Majid Darehmiraki ◽  
Mohammad Hadi Farahi ◽  
Sohrab Effati
Keyword(s):  
Optimal Control ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Least Squares ◽  
Meshless Method ◽  
Basis Functions ◽  
Moving Least Squares ◽  
Distributed Optimal Control

We use a hybrid local meshless method to solve the distributed optimal control problem of a system governed by parabolic partial differential equations with Caputo fractional time derivatives of order α ∈ (0, 1]. The presented meshless method is based on the linear combination of moving least squares and radial basis functions in the same compact support, this method will change between interpolation and approximation. The aim of this paper is to solve the system of coupled fractional partial differential equations, with necessary and sufficient conditions, for fractional distributed optimal control problems using a combination of moving least squares and radial basis functions. To keep matters simple, the problem has been considered in the one-dimensional case, however the techniques can be employed for both the two- and three-dimensional cases. Several test problems are employed and results of numerical experiments are presented. The obtained results confirm the acceptable accuracy of the proposed method.

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