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- Open Access
Sweep algorithm for solving optimal control problem with multi-point boundary conditions
- Mutallim Mutallimov^{1},
- Rena T Zulfugarova^{1}Email author and
- Leyla I Amirova^{1}
https://doi.org/10.1186/s13662-015-0569-5
© Mutallimov et al. 2015
Received: 4 December 2014
Accepted: 10 July 2015
Published: 28 July 2015
Abstract
The sweep algorithm for solving optimal control problem with multi-point boundary conditions is offered. According to this algorithm, the search of initial conditions is reduced to the solution of the corresponding system of linear algebraic equations. A similar algorithm is proposed for the discrete case.
Keywords
- sweep algorithm
- multi-point boundary conditions
- initial conditions
- system of linear algebraic equations
MSC
- 49J15
- 49M25
- 49N10
1 Introduction
To solve the control problem with two-point boundary conditions, different computational algorithms have been developed: those increasing the initial system dimension [1] and based on solutions of the corresponding Hamiltonian equations [2] as well as the ones not increasing the initial problem dimensions [3, 4].
On the example of construction of the program trajectory and control for biped apparatus (PA) [3], some difficulties are demonstrated in application of algorithms suggested in [1]. These difficulties are mainly related to bad conditionality of the Hamiltonian matrix of the corresponding linear algebraic equations.
That is why in [2] other methods as well as sweep method are suggested for solving the problem under consideration.
Different situation arises in the optimization problems with multi-point boundary conditions. Namely, in this case the Lagrange multiplier becomes discontinuous at the internal points, and direct application of methods of two-point boundary value problems to optimization problems with multi-point boundary conditions is not possible.
In the paper the modified sweep method is used for solving of the optimal control problem with multi-point boundary conditions. Both the continuous and discrete cases are considered.
2 Statement of the problem (the continuous case)
Sweep method
These equations mean that the functions \(S (t )\), \(\omega (t )\) are continuous and the function \(N (t )\) is discontinuous at the points \(t=t_{i} \).
It should be noted that the coefficients for \(x(t)\), \(u(t)\) determined from (21), (22) coincide with the coefficients of the corresponding optimal stabilization problem. This fact significantly simplifies solution of the general optimal control problem (construction of the program trajectories and control, optimal stabilization).
3 Statement of the problem (the discrete case)
4 The numerical algorithm
- (1)
\(F(x)\), \(G(t)\), \(v(t)\), \(\Phi_{1}\), \(\Phi_{2}\), \(\Phi_{3}\), q, \(Q(t)\), \(C(t)\) are formed.
- (2)
- (3)
The solutions of equation (47) - the functions \(n(t)\) and \(W(t)\) - are determined.
- (4)
\(x_{0} =x(0)\) and γ are determined by solving the system of linear algebraic equations (47).
- (5)
\(u(t)\) is determined by formula (51).
- (6)
The solution \(x(t)\) is obtained from equation (51).
This algorithm implies that to determine \(x(t)\) one should solve the Riccati equation and the system of differential equations. To demonstrate the performance of this algorithm, we use Runge-Kutta method.
In analogous way we can transform the other equations of (48) and (50).
Note that condition (55) somehow complicates the application of Runge-Kutta method in solving equation (54). To overcome the difficulties that occurred, we divide the interval \([0,T ]\) into n equal parts so that the point \(t=\tau\) would coincide with one of the nodal points \(t_{i} \). Then, considering the initial condition \(\bar{N}(0)=\Phi'_{3} \), we find the solution on the interval \((0,\tau+0)\), and using this solution as the initial conditions we obtain the next solution on the entire interval \((0,T)\). To find the functions \(\bar{n}(t)\) and \(\bar{W}(t)\), we also apply Runge-Kutta method, but in this case on each interval \((0,\tau)\) and \((\tau,T)\) the corresponding differential equations are solved independently.
5 Conclusion
Declarations
Acknowledgements
The authors are grateful to prof. FA Aliev and prof. YS Gasimov for their valuable advice. The authors also are grateful to the anonymous referees for their constructive comments and helpful suggestions to improve this paper greatly.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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