2158 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 9, OCTOBER 2008
A Partial Solution of the Aizerman Problem for
Second-Order Systems With Delays
Dmitry A. Altshuller
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 conjec-
ture is true. The proof is based on the Popov’s frequency-domain criterion
for absolute stability.
Index Terms—Absolute stability, Aizerman problem, delay systems, fre-
quency-domain methods.
I. I
NTRODUCTION
The Aizerman problem has a very long history. For systems without
delays, the matter is completely settled: the Aizerman conjecture is
true for second-order systems and, generally, false for systems of order
three and higher [1]. For systems with delays, the problem is unsolved
[2] except that Rasvan himself proved that the Aizerman conjecture is
true for first-order systems with a single delay, independently of the
delay [3].
In this paper, we consider the second-order retarded system de-
scribed by the scalar delay-differential equation
x
(
t
)+
a
1
_
x
(
t
)+
'
(
x
)+
b
1
_
x
(
t
0
)+
bx
(
t
0
)=0
:
(1)
It is assumed that the function
'
(
x
)
, hereafter called the nonlinearity,
satisfies the sector condition
0
<
'
(
x
)
x
:
(2)
For the linear terms in (1), we can define a transfer function
W
(
s
)=
s
2
+
a
1
s
+(
b
1
s
+
b
)
e
0
s
0
1
:
(3)
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 in-
equality holds:
0
1
+Re[(1+
i!
)
W
(
i!
)]
>
0
:
(4)
In addition to (1), we are also going to consider the linear equation
x
(
t
)+
a
1
_
x
(
t
)+
ax
+
b
1
_
x
(
t
0
)+
bx
(
t
0
)=0
:
(5)
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. The Aizerman conjecture states that
these values are the same. The question is if this conjecture is true.
Manuscript received January 30, 2008; revised April 8, 2008 and July 31,
2008. Current version published October 8, 2008. Recommended by Associate
Editor G. Feng.
D. A. Altshuller is with the Crane Aerospace and Electronics, Burbank, CA
91510 USA (e-mail: altshuller@ieee.org).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TAC.2008.930193
The first step in answering this question is to determine stability con-
ditions for (5). This will be carried out in Section II. Delay not involving
derivatives
(
b
1
=0)
is considered in Section III. Delay involving
the first derivative is investigated in Section IV. Finally, in Section V,
we extend some of the results of Section III to systems with multiple
delays.
II. L
INEAR
SYSTEMS
For the system described by (5), we define the transfer function
W
L
(
s
)=
P
(
s
)+
Q
(
s
)
e
0
s
0
1
:
(6)
In this equation
P
(
s
)=
s
2
+
a
1
s
+
a
;
Q
(
s
)=
b
1
s
+
b:
(7)
It is well known that the zero solution of (5) is GAS if and only if
all the poles of
W
L
(
s
)
have negative real parts. An immediate conse-
quence of the results of Pontryagin [4] is that the following inequality
constitutes a necessary condition for this to be true:
j
b
j
<
j
a
j
:
(8)
If this inequality holds, then the necessary and sufficient condition
for the zero solution of (5) to be GAS for all nonnegative values of the
delay
is that the following two conditions are met [5].
1) The real parts of all the roots of the polynomial
P
(
s
)
are negative.
This is true if and only if both coefficients
a
1
and
a
are positive.
2) For any
!>
0
j
Q
(
i!
)
j
<
j
P
(
i!
)
j
:
(9)
These stability conditions can be reduced as follows. Both of the
following inequalities are the necessary conditions:
j
b
j
<a
;
j
b
1
j
<a
1
:
(10)
If (10) are satisfied, then the necessary and sufficient condition for
stability is that that one of the following inequalities is satisfied:
j
b
j
<
a
2
1
0
b
2
1
2
;
a>
4
b
2
+
a
2
1
0
b
2
1
2
4(
a
2
1
0
b
2
1
)
:
(11)
These conditions can be represented graphically in the plane of the
parameters
a
and
j
b
j
shown in Fig. 1.
The stability region is the shaded area, bounded by the abscissa axis,
the diagonal
j
b
j
=
a
, and the curve given by the equation
a
=
4
b
2
+
a
2
1
0
b
2
1
2
4(
a
2
1
0
b
2
1
)
:
(12)
at the point of tangency
j
b
j
=(
a
2
1
0
b
2
1
)
=
2
.
This provides a complete answer to the delay-independent stability
problem for the second-order linear systems with a single delay (with
the exception of neutral systems). The next task is to compare these
stability conditions with those for nonlinear systems.
III. D
ELAY NOT INVOLVING DERIVATIVES
In case of a single delay not involving derivatives, the inequalities
(11) simplify to
j
b
j
<
a
2
1
2
;
a>
4
b
2
+
a
4
1
4
a
2
1
:
(13)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 9, OCTOBER 2008 2159
Fig. 1. Stability boundaries for the linear system (5).
If the first of these inequalities holds, then
a>
j
b
j
is a necessary and
sufficient condition for stability of the zero solution of (5). Therefore,
the nonlinearity
'
(
x
)
must lie in the sector
(
j
b
j
;
+
1
)
and it makes
sense to define the function
f
(
x
)
by
f
(
x
)=
'
(
x
)
0j
b
j
x:
(14)
The new nonlinearity
f
(
x
)
satisfies the sector condition (2) with
0
1
=0
and the transfer function of the linear terms becomes
W
(
s
)=
s
2
+
a
1
s
+
j
b
j
+
be
0
s
0
1
:
(15)
Expansion of (4) shows that it holds for all
>
0
if
j
b
j
+(
a
1
0
1)
!
2
0
b!
sin
!
+
b
cos
!
0
:
(16)
Using a well-known trigonometric identity, this inequality can be
rewritten in the form
j
b
j
+(
a
1
0
1)
!
2
+ (
b!
)
2
+
b
2
sin(
!
+
)
0
:
(17)
It is easy to show that as long as the first of the inequalities (13) holds,
(17) holds for all values of
!
if the constant
is chosen to satisfy
a
1
j
b
j
0
a
2
1
0
2
j
b
j
b
2
<<
a
1
j
b
j
+
a
2
1
0
2
j
b
j
b
2
:
(18)
This shows that the Aizerman conjecture is true in this case.
Of course, if
b
=0
, we have a system without delays, and the Aiz-
erman conjecture is known to be true.
Let us turn our attention to the case when
a>
4
b
2
+
a
4
1
4
a
2
1
:
(19)
Instead of (14), we now have
f
(
x
)=
'
(
x
)
0
4
b
2
+
a
2
1
4
a
2
1
x:
(20)
Similarly, (15) is replaced with
W
(
s
)=
s
2
+
a
1
s
+
4
b
2
+
a
4
1
4
a
2
1
+
be
0
s
0
1
:
(21)
Expansion of (4) shows that it holds for all
if
a
4
1
+16
b
2
+4
a
2
1
(
a
1
0
1)
!
2
+4
a
2
1
b
(cos
!
0
!
sin
!
)
0
:
(22)
Following the same procedure as in the previous case, it can be
shown that this inequality holds for all values of
!
if
is chosen to
satisfy both of the following inequalities:
16
a
1
+
128
a
2
1
+
a
6
1
b
4
+
24
a
2
1
b
2
+
a
3
1
b
2
>
4
(23)
128
a
2
1
+
a
6
1
b
4
+
24
a
2
1
b
2
+4
>
16
a
1
+
a
3
1
b
2
:
(24)
Therefore, in this case the Aizerman conjecture is true as well. This
proves that it is true for all second-order systems with a single delay
not involving derivatives.
IV. D
ELAY INVOLVING THE
FIRST DERIVATIVE
For systems with a single delay involving first derivative, the sit-
uation is somewhat more complicated. If the first of the inequalities
(11) holds, then the inequalities (12) are necessary and sufficient con-
ditions for stability of the zero solution of (5). Once again, we can de-
fine the function
f
(
x
)
by (14). The transfer function of the linear terms
becomes
W
(
s
)=
s
2
+
a
1
s
+
j
b
j
+(
b
+
b
1
s
)
e
0
s
0
1
:
(25)
Expansion of (4) shows that it holds for all
>
0
if
j
b
j
+(
a
1
0
1)
!
2
+(
b
1
0
b
)
!
sin
!
+
b
+
b
1
!
2
cos
!
0
:
(26)
It can be shown by the same process as in the previous section that as
long as the inequalities (10) and the first of the inequalities (11) hold,
(26) holds for all values of
!
if the constant
is chosen to satisfy
On the other hand, if
b>
0
, we choose
to satisfy
a
1
j
b
j
0
a
2
1
0
2
j
b
j0
b
2
1
b
2
<<
a
1
j
b
j
+
a
2
1
0
2
j
b
j0
b
2
1
b
2
:
(27)
If
b
=0
, we replace (26) with
(
a
1
0
1)
!
2
+
b
1
!
sin
!
+
b
1
!
2
cos
!
0
:
(28)
Following the same procedure as before, we find that (28) holds for
all values of
!
if we choose
to satisfy
1+
b
2
1
0
a
2
1
>
0
:
(29)
This proves the Aizerman conjecture for the case of a single delay
involving the first derivative if the first of the inequalities (11) holds.
If the second of the inequalities (11) holds, then we define
B
=
4
b
2
+
a
2
1
0
b
2
1
2
4(
a
2
1
0
b
2
1
)
;
(30)
Instead of (26), we have
B
+(
a
1
0
1)
!
2
+(
b
1
0
b
)
!
sin
!
+
b
+
b
1
!
2
cos
!
0
:
(31)
It is easy to see that
B
b
. Therefore, if (26) holds, then (31) holds
as well.
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2160 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 9, OCTOBER 2008
Therefore, the Aizerman conjecture is true for all second-order sys-
tems with a single delay involving the first derivative.
V. M
ULTIPLE DELAYS
Let us extend some of our results to systems with multiple delays. We
are only going to consider the case of delays not involving derivatives
x
(
t
)+
a
1
_
x
(
t
)+
'
(
x
)+
m
j
=1
b
j
x
(
t
0
j
)=0
:
(32)
We are not going to investigate in depth the stability of the corre-
sponding linear system except to note that the necessary condition (8)
now becomes [5]
m
j
=1
j
b
j
j
<
j
a
j
:
(33)
Therefore, instead of (14), we have
f
(
x
)=
'
(
x
)
0
x
m
j
=1
j
b
j
j
:
(34)
Instead of (16), we now obtain
m
j
=1
j
b
j
j
+(
a
1
0
1)
!
2
0
!
m
j
=1
b
j
sin
!
j
+
m
j
=1
b
j
cos
!
j
0
:
(35)
We can take advantage of the easily verified estimate
sin
2
, valid for
>
0
, and state that (35) holds for all values of
!
if
there exists
>
0
such that the following inequality holds for all values
of
!
:
m
j
=1
(
j
b
j
j
+
b
j
cos
!
j
)
+
a
1
0
m
j
=1
j
b
j
j
j
0
1
!
2
0
:
(36)
This can be assured by choosing
>
0
to satisfy
a
1
0
m
j
=1
j
b
j
j
j
>
1
:
(37)
Clearly, this can be done if and only if the following inequality is
true
a
1
>
m
j
=1
j
b
j
j
j
:
(38)
Thus, in this case we have identified a delay-dependent class of sys-
tems, for which the Aizerman conjecture is true.
VI. C
ONCLUSION
The results obtained can be summarized as follows. For re-
tarded 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-de-
pendent 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. Indeed, if we set
m
=1
, the
resulting stability criterion is much weaker than the one obtained in
Section III.
REFERENCES
[1] V. A. Pliss, Certain Problems in Theory of Stability of Motion in the
Whole. Washington, DC: National Aeronautics and Space Adminis-
tration, 1958.
[2] V. Rasvan, “Problem 6.6: delay-independent and delay-dependent Aiz-
erman problem,” in Unsolved Problems in Mathematical Systems and
Control Theory, V. D. Blondel and A. Megretski, Eds. Princeton, NJ:
Princeton Univ. Press, 2004, pp. 212–220.
[3] V. Rasvan, Absolute Stability of Automatic Control Systems With Time
Delay (in Russian). Moscow, Russia: Nauka Publishing House, 1975.
[4] L. S. Pontryagin, “On zeros of some elementary transcendental func-
tions,” in L. S. Pontryagin. Selected Scholarly Works (in Russian), R.
V. Gamkrelidze, Ed. Moscow: Nauka Publishing House, 1988, vol.
2, pp. 35–50.
[5] L. E. El’sgol’ts and S. B. Norkin, Introduction to the Theory and Ap-
plications of Differential Equations With Deviating Arguments.New
York: Academic, 1973, p. 176.
Reversibility and Poincaré Recurrence in
Linear Dynamical Systems
Sergey G. Nersesov and Wassim M. Haddad
Abstract—In this paper, we study the Poincaré recurrence phenomenon
for linear dynamical systems, that is, linear systems whose trajectories re-
turn infinitely often to neighborhoods of their initial condition. Specifically,
we provide several equivalent notions of Poincaré recurrence and review
sufficient conditions for nonlinear dynamical systems that ensure that the
system exhibits Poincaré recurrence. Furthermore, we establish necessary
and sufficient conditions for Poincaré recurrence in linear dynamical sys-
tems. In addition, we show that in the case of linear systems the absence
of volume-preservation is equivalent to the absence of Poincaré recurrence
implying irreversibility of a dynamical system. Finally, we introduce the
notion of output reversibility and show that in the case of linear systems,
Poincaré recurrence is a sufficient condition for output reversibility.
Index Terms—Irreversibility, Lagrangian and Hamiltonian systems,
output reversibility, Poincaré recurrence, volume-preserving flows.
I. I
NTRODUCTION
The Poincaré recurrence theorem states that every finite-dimen-
sional, isolated dynamical system with volume-preserving flow and
bounded trajectories will return arbitrarily close to its initial state
infinitely many times. This theorem was proven by Poincaré [1] and
further studied by Birkhoff [2] for Lagrangian systems and Halmos [3]
for ergodic systems. Poincaré recurrence has been the main source for
the long and fierce debate between the microscopic and macroscopic
points of view of thermodynamics [4]. In thermodynamic models
predicated on statistical mechanics, an isolated dynamical system will
return arbitrarily close to its initial state of molecular positions and
velocities infinitely often. If the system entropy is determined by the
Manuscript received November 14, 2006; revised February 11, 2008. Current
version published October 8, 2008. This work was supported in part by the Air
Force Office of Scientific Research under Grant FA9550-06-1-0240. Recom-
mended by Associate Editor M. A. Demetriou.
S. G. Nersesov is with the Department of Mechanical Engineering,
Villanova University, Villanova, PA 19085 USA (e-mail: sergey.nersesov@vil-
lanova.edu).
W. M. Haddad is with the School of Aerospace Engineering, Georgia Insti-
tute of Technology, Atlanta, GA 30332 USA (e-mail: wm.haddad@aerospace.
gatech.edu).
Digital Object Identifier 10.1109/TAC.2008.930194
0018-9286/$25.00 © 2008 IEEE
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