Feedback minimum principle for continuous, discrete and impulsive optimal control problems

2018 ◽  
Author(s):  
Vladimir Aleksandrovich Dykhta ◽  
Ol'ga Nikolaevna Samsonyuk ◽  
Stepan Pavlovich Sorokin
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Minimum Principle ◽  
Control Problems

On fractional optimal control problems with an application in fractional chaotic systems using a Legendre collocation-optimization technique

2020 ◽  
pp. 014233122096958
Author(s):  
Safiye Ghasemi ◽  
Alireza Nazemi ◽  
Raziye Tajik ◽  
Marziyeh Mortezaee
Keyword(s):  
Neural Network ◽  
Optimal Control ◽  
Optimal Control Problems ◽  
Minimum Principle ◽  
Control Problems ◽  
Back Propagation Algorithm ◽  
Functional Link ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems ◽  
Following Error

In this paper, an intelligence method based on single layer legendre neural network is proposed to solve fractional optimal control problems where the dynamic control system depends on Caputo fractional derivatives. First, with the help of an approximation, the Caputo derivative is replaced to integer order derivative. According to the Pontryagin minimum principle for optimal control problems and by constructing an error function, an unconstrained minimization problem is then defined. In the optimization problem, trial solutions are used for state, costate and control functions, where these trial solutions are constructed by using Legendre polynomial based functional link artificial neural network. In the following, error back propagation algorithm is used for updating the network parameters (weights). At the end, some illustrative examples are included to demonstrate the validity and capability of the proposed method. Three applicable examples about chaos control of Malkus waterwheel, finance fractional chaotic models and fractional-order geomagnetic field models are also considered.

Solving optimal control problems using the Picard’s iteration method

2020 ◽  
Vol 54 (5) ◽  
pp. 1419-1435
Author(s):  
Abderrahmane Akkouche ◽  
Mohamed Aidene
Keyword(s):  
Optimal Control ◽  
Iteration Method ◽  
Optimal Trajectory ◽  
Optimal Control Problems ◽  
Approximate Analytical Solution ◽  
Minimum Principle ◽  
Necessary Optimality Conditions ◽  
Control Law ◽  
Control Problems ◽  
Objective Functional

In this paper, the Picard’s iteration method is proposed to obtain an approximate analytical solution for linear and nonlinear optimal control problems with quadratic objective functional. It consists in deriving the necessary optimality conditions using the minimum principle of Pontryagin, which result in a two-point-boundary-value-problem (TPBVP). By applying the Picard’s iteration method to the resulting TPBVP, the optimal control law and the optimal trajectory are obtained in the form of a truncated series. The efficiency of the proposed technique for handling optimal control problems is illustrated by four numerical examples, and comparison with other methods is made.

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 BVPs Codes for Solving Optimal Control Problems

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2618
Author(s):  
Francesca Mazzia ◽  
Giuseppina Settanni
Keyword(s):  
Optimal Control ◽  
Numerical Methods ◽  
Boundary Value Problems ◽  
Optimal Control Problems ◽  
Boundary Value ◽  
Minimum Principle ◽  
General Purpose ◽  
Control Problems ◽  
Indirect Methods ◽  
Interesting Class

Optimal control problems arise in many applications and need suitable numerical methods to obtain a solution. The indirect methods are an interesting class of methods based on the Pontryagin’s minimum principle that generates Hamiltonian Boundary Value Problems (BVPs). In this paper, we review some general-purpose codes for the solution of BVPs and we show their efficiency in solving some challenging optimal control problems.

Differential Transformation and Its Application to Nonlinear Optimal Control

2009 ◽  
Vol 131 (5) ◽  
Author(s):  
Inseok Hwang ◽  
Jinhua Li ◽  
Dzung Du
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Optimal Control Problems ◽  
Minimum Principle ◽  
Nonlinear Optimal Control ◽  
Control Problems ◽  
Algebraic Equations ◽  
Differential Transformation ◽  
Nonlinear Algebraic Equations ◽  
Transformation Algorithm

A novel numerical method based on the differential transformation is proposed for solving nonlinear optimal control problems in this paper. The differential transformation is a linear operator that transforms a function from the original time and/or space domain into another domain in order to simplify the differential calculations. The optimality conditions for the optimal control problems can be represented by algebraic and differential equations. Using the differential transformation, these algebraic and differential equations with their boundary conditions are first converted into a system of nonlinear algebraic equations. Then the numerical optimal solutions are obtained in the form of finite-term Taylor series by solving the system of nonlinear algebraic equations. The differential transformation algorithm is similar to the spectral element methods in that the computational region splits into several subregions but it uses polynomials of high degrees by keeping a small number of subregions. The differential transformation algorithm could solve the finite- (or infinite-) time horizon optimal control problems formulated as either the algebraic and ordinary differential equations using Pontryagin’s minimum principle or the Hamilton–Jacobi–Bellman partial differential equation using dynamic programming in one unified framework. In addition, the differential transformation algorithm can efficiently solve optimal control problems with the piecewise continuous dynamics and/or nonsmooth control. The performance is demonstrated through illustrative examples.

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