scholarly journals Infinite Horizon Optimal Control of Stochastic Delay Evolution Equations in Hilbert Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Xueping Zhu ◽  
Jianjun Zhou
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Hilbert Spaces ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Infinite Horizon ◽  
Stochastic Delay ◽  
Delay Partial Differential Equations

The aim of the present paper is to study an infinite horizon optimal control problem in which the controlled state dynamics is governed by a stochastic delay evolution equation in Hilbert spaces. The existence and uniqueness of the optimal control are obtained by means of associated infinite horizon backward stochastic differential equations without assuming the Gâteaux differentiability of the drift coefficient and the diffusion coefficient. An optimal control problem of stochastic delay partial differential equations is also given as an example to illustrate our results.

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.

scholarly journals Intrinsic Optimal Control for Mechanical Systems on Lie Group

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Chao Liu ◽  
Shengjing Tang ◽  
Jie Guo
Keyword(s):  
Optimal Control ◽  
Analytical Solution ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Lie Group ◽  
Control Method ◽  
Geodesic Distance ◽  
Infinite Horizon ◽  
Mechanical Systems ◽  
Programming Approach

The intrinsic infinite horizon optimal control problem of mechanical systems on Lie group is investigated. The geometric optimal control problem is built on the intrinsic coordinate-free model, which is provided with Levi-Civita connection. In order to obtain an analytical solution of the optimal problem in the geometric viewpoint, a simplified nominal system on Lie group with an extra feedback loop is presented. With geodesic distance and Riemann metric on Lie group integrated into the cost function, a dynamic programming approach is employed and an analytical solution of the optimal problem on Lie group is obtained via the Hamilton-Jacobi-Bellman equation. For a special case on SO(3), the intrinsic optimal control method is used for a quadrotor rotation control problem and simulation results are provided to show the control performance.

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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