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# Comments on ‘Sweep algorithm for solving optimal control problem with multi-point boundary conditions’ by M Mutallimov, R Zulfuqarova, and L Amirova

*Advances in Difference Equations*
**volume 2016**, Article number: 131 (2016)

## Abstract

A counter example is given for the solution of the linear-quadratic optimization problem with three-point boundary conditions. The example shows that the solution obtained in (Mutallimov *et al.* in Adv. Differ. Equ. 2015:233, 2015) by using a sweep method is not optimal.

## Introduction

In [1] the linear-quadratic optimization problem with multi-point boundary conditions, both in the continuous and the discrete cases, are considered. The sweep method [2, 3], which generalizes the results [4] for the two-point boundary conditions is given in [5]. However, the results obtained for the discrete case [1] are not optimal.

Not passing to the illustration of an example, we form the problem of discrete optimal control with multi-point boundary conditions [1, 4]. Let the motion of an object be described by the following linear system of finite-difference equations:

with nonseparate boundary conditions

Here \(x(l)\) is an *n*-dimensional phase vector, \(u(i)\) an *m*-dimensional vector of control influences, \(\psi (i)\), \(\Gamma (i)\) (\(i = 0,1,\ldots,l - 1\)) matrices of the corresponding dimensions, being a controllability pair [4, 6], \(\Phi_{1},\Phi_{2},\Phi_{3}\) are constant matrices, such that the system (2) satisfies the Kronecker-Capelli condition [3, 4], \(0< s< l\).

It is required to find such a control \(u(i)\) as minimizes the following quadratic functional:

under the conditions (1), (2), where \(Q(i) = Q'(i) \ge 0\), \(C(i) = C'(i) \ge 0\) are the periodic matrices of the corresponding dimensions.

Let us illustrate this on the example from [4] in the one-dimensional case. Indeed, in the problem (23)-(25) from [1], let

Using the algorithm given in [1] we can see that the ‘optimal’ phase trajectory and control, respectively, have the form

Then it is easy to calculate [6, 7] that the ‘optimal’ value of the functional (25) of [1] will be \(J \approx 0.8\).

However, the algorithm as given in [4, 6] gives other results, *i.e.*

and the functional (25) of [1] takes the value

Thus, the above solution in [1] is not optimal.

## References

- 1.
Mutallimov, MM, Zulfugarova, RH, Amirova, LI: Sweep algorithm for solving optimal control problem with multi-point boundary conditions. Adv. Differ. Equ.

**2015**, 233 (2015) - 2.
Abramov, AA: On the transfer of boundary conditions for systems of ordinary linear differential equations (a variant of dispersive method). USSR Comput. Math. Math. Phys.

**1**(3), 617-622 (1962) - 3.
Aliev, FA, Larin, VB: On the algorithm for solving discrete periodic Riccati equation. Appl. Comput. Math.

**13**(1), 46-54 (2014) - 4.
Aliev, FA: Methods of Solution for the Application Problems of Optimization of the Dynamic Systems. Elm, Baku (1989)

- 5.
Tiwari, S, Kumar, M: An initial value technique solve two-point non-linear singularly perturbed boundary value problems. Appl. Comput. Math.

**14**(2), 150-157 (2015) - 6.
Gabasova, OR: On optimal control of linear hybrid systems with terminal constraints. Appl. Comput. Math.

**13**(2), 194-205 (2014) - 7.
Rashidinia, J, Khazaei, M, Nikmarvani, H: Spline collocation method for solution of higher order linear boundary value problems. TWMS J. Pure Appl. Math.

**6**(1), 38-47 (2015)

## Acknowledgements

The author thanks the reviewers of the comments and the editors for their instructive remarks.

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Aliev, F.A. Comments on ‘Sweep algorithm for solving optimal control problem with multi-point boundary conditions’ by M Mutallimov, R Zulfuqarova, and L Amirova.
*Adv Differ Equ* **2016, **131 (2016). https://doi.org/10.1186/s13662-016-0816-4

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### Keywords

- sweep algorithm
- optimization
- three-point boundary conditions