# A Partial Solution of the Aizerman Problem for Second-Order Systems With Delays

TL;DR: It is proved that for retarded systems with a single delay the Aizerman conjecture is true, and for systems with multiple delays, a delay-dependent class of systems is found, for which the AIZerman conjectureIs true.

Abstract: This paper considers the Aizerman problem for second-order systems with delays. It is proved that for retarded systems with a single delay the Aizerman conjecture is true. For systems with multiple delays, a delay-dependent class of systems is found, for which the Aizerman conjecture is true. The proof is based on the Popov's frequency-domain criterion for absolute stability.

## Summary (1 min read)

### I. INTRODUCTION

- The Aizerman problem has a very long history.
- 1) The real parts of all the roots of the polynomial P (s) are negative.
- The next task is to compare these stability conditions with those for nonlinear systems.

### IV. DELAY INVOLVING THE FIRST DERIVATIVE

- For systems with a single delay involving first derivative, the situation is somewhat more complicated.
- If the first of the inequalities (11) holds, then the inequalities (12) are necessary and sufficient conditions for stability of the zero solution of (5) .
- Once again, the authors can define the function f(x) by ( 14 Therefore, the Aizerman conjecture is true for all second-order systems with a single delay involving the first derivative.

### VI. CONCLUSION

- The results obtained can be summarized as follows.
- For retarded systems with a single delay, the Aizerman problem is solved completely-the conjecture is proved to be true.
- For systems with multiple delays, the frequency-domain inequality yields a delay-dependent stability criterion.
- The problem is still open for neutral systems.
- Another open question is the possibility of improving the result in Section V since the estimate used in the derivation is rather coarse.

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##### Citations

20 citations

### Cites background from "A Partial Solution of the Aizerman ..."

...CTAC and CTKC with time delay systems [39], [40] remain open....

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11 citations

### Cites background from "A Partial Solution of the Aizerman ..."

...Illustrative examples Example 1: Let us consider the example given in Altshuller (2008) but in the case of time varying delay....

[...]

...This is true even for second-order systems with a constant delay and a single nonlinear coefficient (Altshuller, 2008). In addition, this system models the well-known Lurie Postnikov systems. However, some hypotheses are added for the stability study. In fact, in the work of Bliman (2000), Richard (2003), Han (2005) and Rãsvan et al. (2010), and the references therein, all the poles of the transfer function must have negative real parts....

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...Delaydependent results applicable to nonlinear systems are even fewer (see, for example, Altshuller, 2008; Bliman, 2000; ∗ Corresponding author....

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...This is true even for second-order systems with a constant delay and a single nonlinear coefficient (Altshuller, 2008). In addition, this system models the well-known Lurie Postnikov systems. However, some hypotheses are added for the stability study. In fact, in the work of Bliman (2000), Richard (2003), Han (2005) and Rãsvan et al....

[...]

...Illustrative examples Example 1: Let us consider the example given in Altshuller (2008) but in the case of time varying delay....

[...]

6 citations

5 citations

##### References

15 citations

### "A Partial Solution of the Aizerman ..." refers background or methods in this paper

... x(t) + a1 _ x(t) + '(x) + b1 _ x(t ) + bx(t ) = 0: (1)...

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...In addition to (1), we are also going to consider the linear equation...

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...The problem under investigation requires comparing the values of , for which the zero solution of (1) is GAS, with the values of a, for which such solution of (5) is GAS....

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...For the linear terms in (1), we can define a transfer function W (s) = s2 + a1s+ (b1s+ b)e s 1 : (3)...

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...In proving the results of this paper, we are going to rely extensively on the Popov’s frequency-domain stability criterion: the zero solution of (1) is globally asymptotically stable (GAS) if there exists a constant , such that for all values of !, including infinity, the following inequality holds: 1 +Re[(1 + i! )W (i!)] > 0: (4)...

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10 citations

### "A Partial Solution of the Aizerman ..." refers background in this paper

...The new nonlinearity f(x) satisfies the sector condition (2) with 1 = 0 and the transfer function of the linear terms becomes W (s) = s2 + a1s+ jbj+ be s 1 : (15)...

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