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Approximation of systems with delay and their application

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Mykhaylo Petryk, Igor CHEREVKO, Svitlana ILIKA

Yuriy Fedkovich Chernivtsi National University

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XX International Conference "Dynamical System Modeling and Stability Investigations" (DSMSI-2023)

December 19-21, 2023 (Kiev, Ukraine)

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Plan

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I. INTRODUCTION

II. APPROXIMATION SCHEMES

III. APPLICATION

IV. EXAMPLES

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I. INTRODUCTION

(2)

In mathematical modeling of physical and technical processes, the evolution of which depends on prehistory, we arrive at differential equations with a delay. With the help of such equations, it was possible to identify and describe new effects and phenomena in physics, biology, technology [1-5].

1. Gopalsamy K. Stability and oscillations in delay differential equations of population dynamics. The Netherlands. Kluwer Academic Pub. Dordrecht. 1992. – 512 p.
2. Yang Kuang. Delay Differential Equations: With Applications in Population Dynamics. – Academic Press, 2012. – 412 p.
3. Corduneanu С., Li Y., Mahdavi M. Functional Differential Equations: Advances and Applications. – John Wiley & Sons, 2016. – 368 p.
4. Schiesser W.E. Time Delay ODE/PDE Models. Applications in Biomedical Science and Engineering. – Boca Rona, 2019. – 250 p.
5. Rodrıguez F., Carlos J., Lopez C., Castro M. Models of Delay Differential Equations . – MDPI: Basel , 2021. – 248 p.

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Schemes for approximating differential-difference equations by special systems of ordinary differential equations are proposed in the works [6-7]. Further research was found in I. M. Cherevko, L. A. Piddubna, O. V. Matwiy, S. A. Ilika works [8-11] in various functional spaces. The study of approximation of linear stationary systems with a delay allowed us to construct algorithms for approximate detection of nonasymptotic roots of quasi-polynomials.

The obtained algorithms for finding non-asymptotic quasipolynomial roots and constructing regions of stability of linear delay differential equations can be used to study applied problems of optimal control, modeling of dynamic processes in economics, ecology and others.

6. Krasovsky N.N. Approximation of a problem of analytical design of controllers in the system with delay // J. AMM. - 1964. - V. 28, № 4. - P.716-725.

7. Halanay A. Approximations of delays by ordinary differential equations. // Recent advances in differential equations. – New York : Academic Press, 1981. – P. 155–197.

8. Cherevko I., Piddubna L. Approximations of differential difference equations and calculation of nonasymptotic roots of quasipolynomials // Revue D’Analyse numerique et de theorie de l’approximations.- 1999.-28, №1 .- P. 15-21.

9. Matviy O.V., Cherevko I.M. About approximation of system with delay and them stability // Nonlinear oscilations.-2004.-7, №2.- P. 208-216.

10.  Ilika S.A., Matviy O.V., Piddubna L.A., Cherevko I.M. Approximation of differential-functional equations and their application // Bukovinian Mathematical Journal. – 2014.-- 2, № 2-3. - P. 92-96.

11. Cherevko I., Tuzyk I., Ilika S., Pertsov A. Approximation of Systems with Delay and Algorithms for Modeling Their Stability. 2021 11th International Conference on Advanced Computer Information Technologies ACIT’2021, Deggendorf, Germany, 15-17 September 2021. P. 49-52.

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II. Approximation schemes�

Let us consider the problem

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We shall assign a system of ordinary differential equations [1-2] to the initial problem (1.1)-(1.2)

(1.3)

with the following initial conditions

(1.4)

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1. Krasovsky N.N. Approximation of a problem of analytical design of controllers in the system with delay // J. AMM. - 1964. - V. 28, № 4. - P.716-725.
2. Halanay A. Approximations of delays by ordinary differential equations // Recent advances in differential equations. – New York : Academic Press, 1981. – P. 155–197.

The system (1.3) is called the approximation system of (1.1) due to the following approximation results:

3. Cherevko I., Piddubna L. An approximation of solutions of differential-difference equations // Mathematychni Studii. – 1998. –V9,№2. – p.187-192.

4. Cherevko I.M., Piddubna L.A. Approximation of differential-difference equations systems of ordinary differential equations / / Nonlinear Oscillations. - 1999. - № 1. - P. 42-50.

5. Cherevko I.M, Matvij O.V. On the approximation of systems with delay and their stability / / Nonlinear Oscillations. - 2004. - 7, № 2. - P.2008-2016

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3) In the case of difference-differential equation system ( ) the following inequality is true [5]

Approximation scheme of differential-difference equations applies to the case of the initial-value problem for linear functional differential equations of the form

(2.1)

We shall put in accordance with the initial problem (2.1) the Cauchy problem for a system of ordinary differential equations

(2.2)

(2.3)

where are uniquely determined by the inequalities

Approximation of functional differential equations

А) Linear functional differential equations with delay.

Theorem 1. If - continuous matrix functions and

then the following correlations are true

for

(2.4)

В) Nonlinear differential-functional equations.

Let us consider the initial problem for the functional differential equation

(2.5)

where T is positive constant, –

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a linear functional which is most commonly applied as

– matrix functions having continuous components if

– matrix function with continuous components

on the set of variables function on

is a continuous function which satisfies the Lipschitz condition

We shall put a system of ordinary differential equations in conformity with (2.5)

(2.6)

(2.7)

where

Theorem 2. If – continuous matrix functions,

function is continuous and holds the Lipschitz condition for then the following correlations for the initial problem solution (2.11) and the Cauchy problem solution (2.6)-(2.7) are true

6. Cherevko I., Ilika S. Approximation nonlinear differential-functional equations / / Math. methods and fiz. meh. field. - 2012. - 55, № 1. - P. 39-48.

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Example.

Consider the initial problem

Corresponding approximating system of ordinary differential equations has the form

Figure 1 contains results of numerical experiments of calculating

when m = 8, 16, 32, using the first order Gear difference scheme. To compare the obtained values we provide a solution x(t) of the initial problem, found in [7] using the block method of the fourth order.

7. Cryer C.W. Numerical methods for functional differential equations // Delay andF unctional Differential Equations and their Applications. – Academic Press, New York, 1972. – P. 17-101.

Fig.1

III. Application.

А) Finding quasi-polynomial roots.

Consider the system of differential equations with a delay

(3.1)

where – matrices of

A quasi-polynom for (3.1) has the form

(3.2)

According to the scheme (1.3), an approximating system of ordinary differential equations corresponds to the system (3.1)

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where

Characteristic equation of the approximating system (3.3) has the form

(3.4)

Lemma 1 [8]. The following equality for the characteristic equation of the approximating system is true

(3.5)

Lemma 2. If is fixed then a sequence of functions

(3.6)

converges to the quasi-polynom (3.2), as

Remark : approximates the quasi-polynomial (3.2), as . This property can be used to approximate the non-asymptotic roots of (3.2). Since zeros of functions and converge, according to (3.6), roots of the characteristic polynomial (3.5) can be used as approximate values of non-asymptotic roots of (3.2).

8. L.A.Piddubna, I.M.Cherevko Approximations of differential-differencee quations and calculations of nonasymptotic roots of quasipolynomials // Revue d’-analyse numerique et de theorie de l’approximations. – 1999. – Vol. 28,N1. – P. 15 -21.

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Consider the linear system with delay

(3.7)

where – matrices of ,

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Quasi-polynomial for the system (3.7) has the form

(3.8)

We shall put in correspondence to (3.7) a system of ordinary differential equations [9-10]

(3.9)

There is a valid relation to the characteristic equation of (3.9) [9]

(3.10)

9. Cherevko I.M, Matvij O.V. On the approximation of systems with delay and their stability / / Nonlinear Oscillations. - 2004. - 7, № 2. - P.2008-2016

10. Ilika S.A., Piddubna L.A., Tuzyk I. I., Cherevko I.M. Approximation of linear differential-difference equations and their application// Bukovinian Mathematical Journal. -2018.- 6, № 3-4. - P. 80-83.

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converges to (3.8) while . This property can be used to approximate the non-asymptotic roots of (3.8). Since zeros of functions and converge, according to (3.11), roots of the characteristic polynomial (3.10) can be used as approximate values ​​of non-asymptotic roots of the quasi-polynomial (3.8).

In [9] it was shown that the sequence of functions

(3.11)

In practice, the application of these results proved to be difficult, because the satisfactory accuracy of the approximation of non-asymptotic roots of quasi-polynomials was achieved if the dimension of the approximating system and above, which means that you need to find the roots of polynomials of this degree.

We present a scheme for approximating the non-asymptotic roots of quasi-polynomials of high accuracy, which allows to achieve the same accuracy as in the scheme (3.9) by approximating systems of a much smaller dimension.

High-precision approximation scheme

We introduce a function similar to that in the classical schemes (1.3)

we will decompose

in a row Taylor near the point

up to and including the third member, we obsessed

Then the scheme of approximation of increased accuracy for equation (3.7) has the form

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For a fixed

a sequence of functions

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coincides with the quasi-polynomial when .

Lemma 3. The following relation holds for the characteristic polynomial of approximation system

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B) Investigation of the stability of systems with a delay

It is known that a necessary and sufficient conditions for stability of solutions of linear systems with delays is placing its quasi-polynomial roots in the left half-plane of the complex variable .

The direct test of this criterion in practice is difficult, so it is better to consider various other sufficient conditions for stability, which are easier to check.

Consider the approximation schemes [1-2] to study the stability of the system

(3.12)

We shall put in correspondence to (3.12) an approximating system of ordinary differential equations

(3.13)

where

STABILITY STUDY ALGORITHM

For quasi-polynomial approximation of the second order difference-differential equation system (3.12) we obtain a characteristic polynomial

(3.14)

which is convenient for numerical root finding on the computer.

Calculating the approximate roots of the quasi-polynomial (3.14) with different , we can estimate the upper value of the delay , for which the system (3.12) is stable.

C) Building stability region

Consider the linear differential-difference equation with delays

(3.15)

whеre

Theorem 4 [10]. The region of stability of equation (3.16) is bounded.

10. Cherevko I., Klevchuk I., Pernay S. Building of stability regions of linear differential-difference equations / / Adv. Academy of Sciences of Ukraine. - 2012. - № 7. - P. 28-34.

Let in the equation (3.15) delays are rational positive numbers. If we make linear substitution of independent value, then it is possible to achieve another linear equation which corresponds to the characteristic equation of the form:

(3.16)

Remark. The region of stability of equation (3.15), where are positive rational numbers is bounded

Lemma 3 [10]. If the vector belongs to the region of stability

eguation (3.16), then

As an example, consider a linear differential equation with two delays

where are natural numbers

(3.17)

From Theorem 4 and Lemma 3 we get that the stability region of equation (3.17) in the plane of the parameters a1, a2 is contained in a limited polygon

We cover this area with a net of points For every point we have a linear equation with a delay of the form (3.17).

The results of numerical experiments for m = 1, n = 2, are shown on the fig.2, where stability regions is the shaded part of the plane.

Fig. 2

IV. Examples

For automation of the roots of quasi-polynoms using the algorithms described in Section 2 we used the mathematical package MATHCAD. Using function Polyroots in this package we define the root of polynoms and thus obtain the approximation of nonasymptotic roots with corresponding equations.�  

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Example 1.

(4.1)

The characteristic quasi-polynom has the form

(4.2)

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We provide an approximation of equation (4.1) by a system of ordinary differential equations, roots of the characteristic equation whose roots are approximate quasi-polynom (4.2).�   Table 2 shows the approximate values ​​of the roots of equation (4.2)�  for different m.

Example 2.

(4.5)

where

A characteristic polynom is of the form

(4.6)

Table 4 contains the approximate values ​​of the roots of equation (4.6), which are arranged by decreasing of real parts of the roots for different m, the dimension of the approximating system to equation (4.5).

Theorem 5 [11]. The system (3.1) is stable for any delay if the following conditions are correct:

1. the matrix A + B is stable;
2. performed matrix inequality:

In [11] it was described a simple sufficient condition for stability of (3.1)

for all

In [11] the upper limit of the delay for which the stored stability system (4.7) according to Theorem 5 is given by the relation

Applying the approximation scheme of differential-difference equations and Theorem 3 we obtain [12] that the maximum delay in which to stored the system stability (4.7)

11. Barkin A. Stability of linear delay systems / / Automation and telemehanіka. - 2006. - № 3. - P.3-7.

12. Matvij O., Pernay S., Cherevko I. Stability of linear systems with delay / / Science. Visn. Czern. Univ: Mathematics . - 2008. -P.61-65.

Example 3.

(4.7)

(4.8)

Example 4.

Consider a linear differential equation with two delays

where

The results of numerical experiments at different steps of approximation grid h were obtained in Mathcad and are shown on figures 3 and 4. The region of stability is a shaded part of the plane.

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13.  Ilika S.A., Matviy O.V., Piddubna L.A., Cherevko I.M. Approximation of differential-functional equations and their application // Bukovinian Mathematical Journal. – 2014.-- 2, № 2-3. - P. 92-96.

14. Cherevko I., Tuzyk I., Ilika S., Pertsov A. Approximation of Systems with Delay and Algorithms for Modeling Their Stability. 2021 11th International Conference on Advanced Computer Information Technologies ACIT’2021, Deggendorf, Germany, 15-17 September 2021. P. 49-52.

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Fig. 3 (h=0.1)

Fig. 4 (h=0.02)

Thank you for your attention!