Two-Sided Guaranteed Estimates of the Cost Functional for Optimal Control Problems with Elliptic State Equations

2014 ◽  
pp. 325-342
Author(s):  
Pekka Neittaanmäki ◽  
Sergey Repin
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Control Problems ◽  
State Equations ◽  
Cost Functional ◽  
The Cost

scholarly journals Linear quadratic stochastic optimal control problems with operator coefficients: open-loop solutions

2019 ◽  
Vol 25 ◽  
pp. 17 ◽  
Author(s):  
Qingmeng Wei ◽  
Jiongmin Yong ◽  
Zhiyong Yu
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Optimal Control Problems ◽  
Mean Field ◽  
Open Loop ◽  
Control Problems ◽  
Linear Quadratic ◽  
Cost Functional ◽  
The Cost

An optimal control problem is considered for linear stochastic differential equations with quadratic cost functional. The coefficients of the state equation and the weights in the cost functional are bounded operators on the spaces of square integrable random variables. The main motivation of our study is linear quadratic (LQ, for short) optimal control problems for mean-field stochastic differential equations. Open-loop solvability of the problem is characterized as the solvability of a system of linear coupled forward-backward stochastic differential equations (FBSDE, for short) with operator coefficients, together with a convexity condition for the cost functional. Under proper conditions, the well-posedness of such an FBSDE, which leads to the existence of an open-loop optimal control, is established. Finally, as applications of our main results, a general mean-field LQ control problem and a concrete mean-variance portfolio selection problem in the open-loop case are solved.

High-order relaxation approaches for adjoint-based optimal control problems governed by nonlinear hyperbolic systems of conservation laws

2016 ◽  
Vol 24 (1) ◽  
Author(s):  
Elimboto M. Yohana ◽  
Mapundi K. Banda
Keyword(s):  
Optimal Control ◽  
Conservation Laws ◽  
Optimal Control Problems ◽  
Initial Conditions ◽  
Hyperbolic Systems ◽  
Control Problems ◽  
Computational Investigation ◽  
Backward Problem ◽  
Systems Of Conservation Laws ◽  
The Cost

AbstractA computational investigation of optimal control problems which are constrained by hyperbolic systems of conservation laws is presented. The general framework is to employ the adjoint-based optimization to minimize the cost functional of matching-type between the optimal and the target solution. Extension of the numerical schemes to second-order accuracy for systems for the forward and backward problem are applied. In addition a comparative study of two relaxation approaches as solvers for hyperbolic systems is undertaken. In particular optimal control of the 1-D Riemann problem of Euler equations of gas dynamics is studied. The initial values are used as control parameters. The numerical flow obtained by optimal initial conditions matches accurately with observations.

Time-optimal control of multiple unmanned aerial vehicles with human control input

2019 ◽  
Vol 12 (1) ◽  
pp. 138-152 ◽  
Author(s):  
Tao Han ◽  
Bo Xiao ◽  
Xi-Sheng Zhan ◽  
Jie Wu ◽  
Hongling Gao
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Minimum Principle ◽  
Time Optimal Control ◽  
Control Problems ◽  
Control Input ◽  
Content Type ◽  
Human Control ◽  
Time Optimal ◽  
The Cost

Purpose The purpose of this paper is to investigate time-optimal control problems for multiple unmanned aerial vehicle (UAV) systems to achieve predefined flying shape. Design/methodology/approach Two time-optimal protocols are proposed for the situations with or without human control input, respectively. Then, Pontryagin’s minimum principle approach is applied to deal with the time-optimal control problems for UAV systems, where the cost function, the initial and terminal conditions are given in advance. Moreover, necessary conditions are derived to ensure that the given performance index is optimal. Findings The effectiveness of the obtained time-optimal control protocols is verified by two contrastive numerical simulation examples. Consequently, the proposed protocols can successfully achieve the prescribed flying shape. Originality/value This paper proposes a solution to solve the time-optimal control problems for multiple UAV systems to achieve predefined flying shape.

scholarly journals Solution to Singular Optimal Control by Backward Differential Flow

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Jinghao Zhu ◽  
Shangrui Zhao ◽  
Guohua Liu
Keyword(s):  
Optimal Control ◽  
Global Optimization ◽  
Optimal Control Problems ◽  
Optimization Problems ◽  
Variational Problems ◽  
Equivalent Transformation ◽  
Control Problems ◽  
Singular Optimal Control ◽  
Differential Flow ◽  
The Cost

This paper presents a backward differential flow for solving singular optimal control problems. By using Krotov equivalent transformation, the cost functional is converted to a class of global optimization problems. Some properties of the flow are given to reveal the significant relationship between the dynamic of the flow and the geometry of the feasible set. The proposed method is also used in solving a class of variational problems. Some examples are illustrated.

Numerical solution of time-delay optimal control problems by the operational matrix based on Hartley series

2021 ◽  
pp. 014233122110533
Author(s):  
Mahmood Dadkhah ◽  
Kamal Mamehrashi
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Main Idea ◽  
Control Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
The Cost ◽  
And Control

In this paper, a numerical technique based on the Hartley series for solving a class of time-delayed optimal control problems (TDOCPs) is introduced. The main idea is converting such TDOCPs into a system of algebraic equations. Thus, we first expand the state and control variables in terms of the Hartley series with undetermined coefficients. The delay terms in the problem under consideration are expanded in terms of the Hartley series. Applying the operational matrices of the Hartley series including integration, differentiation, dual, product, delay, and substituting the estimated functions into the cost function, the given TDOCP is reduced to a system of algebraic equations to be solved. The convergence of the proposed method is extensively investigated. At last, the precision and applicability of the proposed method is studied through different types of numerical examples.

scholarly journals Stability Analysis of Optimal Trajectory for Nonlinear Optimal Control Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Hongyong Deng ◽  
Wei Zhang ◽  
Changchun Shen
Keyword(s):  
Optimal Control ◽  
Optimal Trajectory ◽  
Optimal Control Problems ◽  
Baire Category ◽  
The State ◽  
Optimal Trajectories ◽  
Control Problems ◽  
State Equations ◽  
Right Hand ◽  
The Right

Due to the need for numerical calculation and mathematical modelling, this paper focuses on the stability of optimal trajectories for optimal control problems. The basic ideas and techniques are based on the compactness of the optimal trajectory set and set-valued mapping theorem. Through lack of optimal control stability, the result of generic stability for optimal trajectories is obtained under the perturbations of the right-hand side functions of the state equations; in the sense of Baire category, the right-hand side functions of the state equations of optimal control can be approximated by other functions.

Optimal control of nonlinear systems with dynamic programming

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Isaac Tawiah ◽  
Yinglei Song
Keyword(s):  
Optimal Control ◽  
Optimization Problem ◽  
Optimal Control Problems ◽  
Optimal Solution ◽  
Functional Approach ◽  
Equality Constraints ◽  
Control Problems ◽  
Iterative Approach ◽  
Simulation Results ◽  
The Cost

Abstract In this paper, a generalized technique for solving a class of nonlinear optimal control problems is proposed. The optimization problem is formulated based on the cost-to-go functional approach and the optimal solution can be obtained by Bellman’s technique. Specifically, a continuous nonlinear system is first discretized and a set of equality constraints can be obtained from the discretization. We show that, under a certain condition, the optimal solution of a problem in this class can be approximated by a solution of the set of equality constraints within any precision and the system is guaranteed to be stable under a control signal obtained from the solution. An iterative approach is then applied to numerically solve the set of equality constraints. The technique is tested on a nonlinear control problem from the class and simulation results show that the approach is not only effective but also leads to a fast convergence and accurate optimal solution.

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