DESIGN OF LINEAR FEEDBACK FOR BILINEAR CONTROL SYSTEMS

DESIGN OF LINEAR FEEDBACK FOR BILINEAR CONTROL SYSTEMS

V

ASILIY

Y

E

. BELOZYOROV

^∗

∗Dnepropetrovsk National University Physical and Technical Department ul. Naukova, 13, 49050, Dnepropetrovsk, Ukraine

e-mail:belozvye@mail.ru

Sufficient conditions for the conditional stability of trivial solutions for quadratic systems of ordinary differential equations are obtained. These conditions are then used to design linear control laws on the output for a bilinear system of any order.

In the case of a homogeneous system, a domain of the conditional stability is also indicated (it is a cone). Some examples are given.

Keywords: system of ordinary quadratic differential equations, bilinear control system, linear control law, cone of stability, feedback, closed-loop system

1. Introduction

Consider a quadratic control system the state equation of which is

˙x(t) =

 A

₀

+

n

X

i=1

x

_i

(t)A

_i

+

m+n

X

i=n+1

u

_i−n

(t)A

_i



x(t), (1)

where

x(t) = x

1

(t), . . . , x

n

(t)

^T

∈ R

ⁿ

, u(t) = u

1

(t), . . . , u

m

(t)

T

∈ R

^m

, and the observation equation has the form

y(t) = Cx(t), y(t) = (y

1

(t), . . . , y

p

(t))

^T

∈ R

^p

, y(0) = (y

₁₀

, . . . , y

_p0

)

^T

. (2) Here R

ⁿ

, R

^m

, R

^p

are real vector spaces of column vec- tors, x(t), u(t), y(t) are vectors of states, inputs and outputs, respectively, y(0) is a vector of initial values, A

i

: R

ⁿ

→ R

ⁿ

and C : R

ⁿ

→ R

^p

are real linear map- pings of appropriate real spaces, i = 0, . . . , n + m. (If A

i

= 0, ∀ i ∈ {1, . . . , n}, then the system (1) is called a bilinear control system.)

In what follows, we shall continue to study the prob- lem, the research on which was started earlier. Therefore, for the reader’s convenience, we shall recall some results from the paper (Belozyorov, 2001).

Definition 1. If A

0

= 0, then the system (1) is called homogeneous. Otherwise, it is called non-homogeneous.

Fixing bases in spaces R

ⁿ

and R

^p

, we denote the matrices of operators A

i

and C in the selected bases as A

i

and C = (c

1

, . . . , c

n

), respectively. Here c

1

, . . . , c

n

are columns of the matrix C; i = 0, . . . , n + m. For arbi- trary column vectors a and b, we denote by (a, b) their scalar product; besides, we denote by kxk = p(x, x) the Euclidean norm of any vector x ∈ R

ⁿ

. Let us recall the definition of the conditional stability of solutions to a system of differential equations (Demidovich, 1967).

Definition 2. The trivial solution x(t) ≡ 0 of the system of differential equations

˙x(t) = F t, x(t)
,

with the vector of initial values x(0) = (x

10

, . . . , x

n0

)

^T

, where F(t, x) = (F

1

(t, x

1

, . . . , x

n

), . . . , F

n

(t, x

1

, . . . , x

n

))

^T

∈ R

ⁿ

is a vector function, is called conditionally stable if there exists a variety of initial values Θ ⊂ R

ⁿ

such that for any solution x(t) satisfying the conditions

x(0) ∈ Θ and kx(0)k < δ(), the inequality

kx(t)k <  is satisfied for t > 0. If also

t→∞

lim kx(t)k = 0,

then the solution x(t) ≡ 0 is called conditionally asymp-

totically stable. (Here  and δ are positive numbers,

where  is given and δ = δ() is a function of .)

In what follows, the structure of a variety Θ is not investigated. Note only that we shall deal with varieties of two types: it will be either an open sphere or an open cone with its top at the origin.

Now for the system (1), (2) let us formulate the fol- lowing problem of mathematical control theory.

Problem of the synthesis of a static feedback law: Con- struct a matrix K = (k

^T₁

, . . . , k

_m^T

)

^T

∈ R

^m×p

of a linear control law u(t) = Ky(t), where k

1

, . . . , k

m

are row vectors, such that the trivial solution of the closed-loop system

˙x(t) =

 A

0

+

n

X

i=1

x

i

(t)A

i

+

n

X

i=1 n+m

X

j=n+1

x

i

(t)(k

j−n

, c

i

)A

j



x(t), (3)

with the vector of initial values x

₀

= {x

₁₀

, . . . , x

_n0

} ∈ Θ such that y

0

(t) = Cx

₀

, would be asymptotically sta- ble (at least conditionally).

Now, two practical examples of bilinear systems are given.

Control problem by the nuclear reactor on thermal neutrons: The kinetic equations of such a reactor can be presented in the following form (Bowen and Mas- ters,1959):



 

 

 

 

 

 

 

 

 

 

 

  dr

₁

dt = k

₂

β

₁

N

l − λ

1

r

₁

, .. .

dr

6

dt = k

₂

β

₆

N

l − λ

₆

r

₆

, dN

dt = k

1

N l − k

2

N l

6

X

i=1

β

i

+

6

X

i=1

λ

i

r

i

. (4)

Here N is the density of neutrons, λ

i

is the disintegra- tion constant for the nuclei of group i (there exist six such groups), r

i

is the density of the nuclei of group i, l is the average effective time of the life of neutrons, β

i

is part of lagging neutrons originating from a nucleus of group i, k

₁

is the excess reproduction coefficient, characterizing the affixed perturbation, and k

2

is the effective reproduc- tion coefficient.

Usually, it is considered that coefficients k

₁

and k

₂

are linear functions of the movements of graphite rods in the reactor, which play the role of controls. In other words, k

1

= b

1

v

1

+ · · · + b

s

v

s

, k

2

= d

1

v

1

+ · · · + d

s

v

s

, where s is the number of rods in the reactor, v

i

is the magni- tude of the movement of the i-th rod, b

i

and d

i

are some numerical coefficients, i = 1, . . . , s.

With the help of controls v

1

, . . . , v

s

it is required to stabilize the work of the reactor in a neighbourhood of some nominal values of variables N

0

, r

10

, . . . , r

60

.

Introduce the notation β = P

6

i=1

β

i

, k

1

= u

1

, k

2

= u

2

, N = x

7

, r

i

= x

i

, i = 1, . . . , 6. Then we will obtain the system (1), in which n = 7, m = 2 and

A

0

=













−λ

1

· · · 0 0 .. . . .. .. . .. . 0 · · · −λ

6

0 λ

1

· · · λ

6

0











 ,

A

₁

=













0 · · · 0 0 .. . . . . .. . .. . 0 · · · 0 0 0 · · · 0 1/l











 ,

A

2

=













0 · · · 0 −β

1

/l .. . . . . .. . .. . 0 · · · 0 −β

6

/l 0 · · · 0 −β/l











 .

Problem of a navigation officer: Any space curve γ in a fixed coordinate system OXYZ can be given by means of the variable radius vector r = r(s), where s is the magnitude of the movement along the curve from the ori- gin. (The representation of the radius vector in the form r = r(s) is called the natural parametrization of the curve γ.) Let P ∈ γ be any point on this curve with radius vector r. Let us denote by n, t and b the unit vectors which are normal, tangent and binormal to curve γ, out- going from the point P and having the same orientation as coordinate axes X, Y , Z, respectively. These vectors satisfy the differential equations

dt

ds = kn, dn

ds = −kt − τ b, db ds = τ n, which are known as Frenet’s formulae. Here k is the cur- vature of the curve γ at the point P , and τ is the torsion of the curve γ at the point P .

Let us look at the point P as at some flight vehicle, whose barycentre is located at the point P and the con- trol is realized in the plane (t, b) (the pitch) and in the plane (n, t) (the yaw). Set k = u

₁

, τ = u

₂

, x

₁

= n, x

2

= t, x

3

= b and x = (x

^T₁

, x

^T₂

, x

^T₃

)

^T

. Then Frenet’s equations will turn into the bilinear system (1), for which n = 9, m = 2 and

A

0

= 0, A

1

=







0 −I

3

0

I

₃

0 0

0 0 0





 ,

Fig. 1. Coordinate axes for the problem of the navigation officer.

A

₂

=







0 0 −I

3

0 0 0

I

₃

0 0





 . (Here I

3

is the identity matrix of the third order.)

At any time moment the orientation of axes n, t and b at the point P is assumed to be known: n = x

1

(s), t = x

2

(s) and b = x

3

(s). (Here s = s(t) is a known function of time.) It is necessary to stabilize the motion of a flight vehicle via linear feedback in a neighbourhood of the nominal values x

i0

of vectors x

i

(s), i = 1, 2, 3.

2. Some Generic Properties of Solutions of Homogeneous Quadratic Systems

It is obvious that the system of differential equations (3) can be rewritten as follows:



 

 

 

 

 

 

 

 

˙ x

1

(t) =

n

X

j=1

d

1j

x

j

(t) + x

^T

(t)B

1

x(t), .. .

˙ x

n

(t) =

n

X

j=1

d

nj

x

j

(t) + x

^T

(t)B

n

x(t).

(5)

Here D = (d

ij

), B

₁

, . . . , B

_n

∈ R

^n×n

are real matrixes and B

1

, . . . , B

_n

are also symmetric.

Definition 3. The system of equations (5) is called quadratic; if D = 0, then we call (5) homogeneous quadratic.

Definition 4. The homogeneous quadratic system (5) is called regular if there are no real constants τ

1

, . . . , τ

n

(at least one being non-zero) such that ∀ x ∈ R

ⁿ

x

^T

(τ

1

B

1

+

· · · + τ

n

B

n

)x = 0. Otherwise, (5) is called a singular or a special system.

In this section we will study regular homogeneous quadratic systems of order n:



 



 



˙

x

1

(t) = x

^T

(t)B

1

x(t), .. .

˙

x

_n

(t) = x

^T

(t)B

_n

x(t),

(6)

with the vector of initial values x

^T

(0) = (x

₁₀

, . . . , x

_n0

).

Consider the matrix ρ

1

B

₁

+ · · · + ρ

_n

B

_n

∈ R

^n×n

, where ρ

1

, . . . , ρ

_n

are arbitrary real parameters. Intro- duce basic symmetric functions for this matrix (Gant- macher, 1990): σ

₁

(ρ

₁

, . . . , ρ

_n

) = tr (ρ

1

B

₁

+ · · · + ρ

_n

B

_n

) = {it is the sum of all principal minors of the first order}, σ

2

(ρ

1

, . . . , ρ

n

) = {it is the sum of all prin- cipal minors of the second order}, . . . , σ

n

(ρ

1

, . . . , ρ

n

) = det(ρ

1

B

1

+ · · · + ρ

n

B

n

).

Consider the set of equations

σ

1

(ρ

1

, . . . , ρ

n

) = r, σ

2

(ρ

1

, . . . , ρ

n

) = 0, . . . , σ

n

(ρ

1

, . . . , ρ

n

) = 0, (7) with respect to the unknowns ρ

₁

, . . . , ρ

_n

and a known arbitrary non-zero constant r ∈ R.

It is easy to show (Gantmacher, 1990) that, for generic matrices B

1

, . . . , B

_n

, the system (7) has n lin- early independent solutions

f

₁

= (ρ

₁₁

, ρ

₁₂

, . . . , ρ

_1n

), f

₂

= (ρ

₂₁

, ρ

₂₂

, . . . , ρ

_2n

), . . . , f

_n

= (ρ

_n1

, ρ

_n2

, . . . , ρ

_nn

)

(generally speaking, they are complex).

Let us find these solutions and form the non-singular matrix F

⁻¹

= (f

1T

, . . . , f

nT

)

^T

∈ C

^n×n

, and then in- troduce into (6) the new variable v(t) = (v

1

(t), . . . , v

n

(t))

^T

∈ C

ⁿ

using the formula x(t) = F v(t). Then, as shown by Belozyorov (2001), the system (6) can be pre- sented as









˙v

1

(t) .. .

˙v

_n

(t)









=







(p

₁₁

v

₁

+ · · · + p

_1n

v

_n

)

²

. . . . (p

_n1

v

₁

+ · · · + p

_nn

v

_n

)

²





 ,

where p

ij

are complex numbers; v(0) = F

⁻¹

x(0) = (v

10

, . . . , v

n0

)

^T

.

After the change of variables v(t) = P z(t), where P = (p

ij

) ∈ C

^n×n

, the last system takes the form













˙ z

1

(t)

˙ z

₂

(t)

.. .

˙ z

n

(t)













=











−β

11

β

12

. . . β

1n

β

21

−β

22

. . . β

2n

. . . . β

n1

β

n2

. . . −β

nn





















 z

²₁

(t) z

²₂

(t)

.. . z

_n²

(t)











 , (8)

where β

_ij

are complex numbers, z(0) = (F P )

⁻¹

x(0) =

(z

₁₀

, . . . , z

_n0

)

^T

.

Definition 5. The regular system (6) is called a sys- tem without invariant submanifolds (WIS system) if there exists no non-singular transformation S ∈ C

^n×n

such that, after the replacement x = Sw, where w = (w

1

, . . . , w

n

)

^T

, the system (6) takes the particular form (8):

˙ w =

























˙ w

1

.. .

˙ w

_k

˙ w

k+1

.. .

˙ w

n

























= S

⁻¹







w

^T

S

^T

B

₁

Sw . . . . w

^T

S

^T

B

_n

Sw







=







































n

P

j=1

α

_1j

w

²_j

.. .

n

P

j=1

α

kj

w

²_j

n

P

j=k+1

α

k+1,j

w

_j²

.. .

n

P

j=k+1

α

nj

w

_j²





































 .

Here α

11

, . . . , α

nn

are complex numbers.

Denote by Ψ the set of all homogeneous quadratic systems of order n. In Appendix it will be shown that the set of all WIS systems contains open subset everywhere dense in Ψ. Thus, the systems (6) being WIS systems are generic.

Denote by a

1

, a

₂

, . . . , a

_k

all real singular points such as the pole of some solution of the regular WIS system (8). Let d

_i

be the multiplicity of the point a

_i

, i = 1, . . . , k.

Theorem 1. Let (8) be a regular WIS system. Then all real singular points of any solution of such a system coincide with a

1

, a

₂

, . . . , a

_k

, where the multiplicity of the point a

i

is d

i

, i = 1, . . . , k.

Proof. For simplicity, assume that n = 2 and all poles of the solution z

1

(t) are equal to a

1

, . . . , a

_l

, and all poles of the solution z

2

(t) coincide with points a

l+1

, . . . , a

_k

.

Also assume that z

1

(t) = f

₁

(t)/(t − a

₁

)

^d

is the pole of multiplicity d, f

₁

(a

₁

) 6= 0 and the point a

1

is not a pole of z

₂

(t). Then, as t → a

1

, the second equa- tion of the system (8) can be rewritten as lim

_t→a1

(t − a

1

)

^2d

z ˙

2

(t) = β

21

f

₁²

(a

1

) − β

22

lim

t→a₁

(t − a

1

)

^2d

z

₂²

(t).

Since (8) is a WIS system, we have β

21

6= 0. Therefore from the last relation it follows that either f

1

(a

1

) = 0 or lim

t→a₁

(t − a

1

)

^2d

( ˙ z

2

(t) + β

22

z

₂²

(t)) = const 6= 0.

The first expression contradicts the assumption and the second is equivalent to the relation ˙z

2

(t) + β

22

z

²₂

(t) = g(t)/(t − a

1

)

^2d

, where g(a

1

) 6= 0. From this it follows that z

2

(t) = g

1

(t)/(t − a

1

)

^d

, where again g

1

(a

1

) 6= 0.

Repeating the same process for points a

₂

, . . . , a

_l

, we can prove that the points are poles corresponding to the ordinals of the function z

2

(t). It is obvious that a similar statement holds true for function z

1

(t), with poles at the points a

l+1

, . . . , a

k

.

Generally, let a be the pole of solutions z

t

, . . . , z

_n−1

(t). Then z

i

(t) = f

_i

(t)/(t − a)

^d

, where f

_i

(a) 6= 0, i = 1, . . . , n − 1. Substituting these relations into the last equation of the system (8) and passing to the limit as t → ∞, we obtain lim

t→a

(t − a)

^2d

( ˙ z

n

(t) + β

nn

z

²_n

(t)) = const 6= 0. The general case of n 6= 2 can be con- sidered in much the same way. The proof is thus com- pleted.

Rewrite the system of equations (8) in the following form:

˙

x(t) = BX(t)x(t), (9)

where

B =











−β

11

β

12

. . . β

1n

β

₂₁

−β

₂₂

. . . β

_2n

. . . . β

_n1

β

_n2

. . . −β

_nn









 ,

X(t) =











x

1

(t) 0 . . . 0 0 x

₂

(t) . . . 0 . . . .

0 0 . . . x

n

(t)









 .

Estimate the solution to the system (9), using the Taylor expansion. In the sequel, in order to denote the time derivative, symbols ‘ · ’ or ‘

⁰

’ will be used.

Note that ∀ k ∈ Z

⁺

, kX

^(k)

k = kx

^(k)

k. Then from (9) we have

kx

⁰

k ≤ kBkkxk

²

,

x

⁰⁰

= (BX)x + (BX) ˙ ˙ x = B ˙ Xx + (BX)(BX)x

= BX ˙ x + (BX)

²

x = 2!(BX)

²

x, kx

⁰⁰

k ≤ 2!kBk

²

kxk

³

,

x

⁰⁰⁰

= 2(B ˙ X)(BX)x + 2(BX)(B ˙ X)x + 2(BX)

²

x, ˙ kx

⁰⁰⁰

k ≤ 3!kBk

³

kxk

⁴

, . . . .

It is obvious that for arbitrary k ∈ Z

⁺

we have

kx

^(k)

k ≤ k!kBk

^k

kxk

^k+1

.

Represent formally the function kx(t)k as a Taylor series and estimate it:

kx(t)k ≤ kx(t

0

)k + kBkkx(t

0

)k

²

(t − t

0

) + · · · + kBk

^k

kx(t

0

)k

^k+1

(t − t

0

)

^k

+ · · ·

= (1 + kBkkx(t

₀

)k(t − t

₀

) + · · ·

+ kBk

^k

kx(t

0

)k

^k

(t−t

0

)

^k

+· · · )kx(t

0

)k. (10) As is well known, the series (10) converges for all t satisfying the condition kBkkx(t

0

)(t − t

0

)k < 1 or for any t ∈ [t

0

, t

0

+ (kBkkx(t

0

)k)

⁻¹

). If the last restriction is satisfied, then the series on the right-hand side of (10) converges; this sum is calculated using the formula for the geometric series and the estimate (10) takes the form

kx(t)k ≤ kx(t

0

)k

1 − kBkkx(t

₀

)kt . (11)

Theorem 2. Let (6) be a regular WIS system. Then one of the following statements holds true: (a) ∀ k ∈ {1, . . . , n} lim

t→∞

x

k

(t) = 0, (b) for all k ∈ {1, . . . , n}

lim

_t→a

|x

k

(t)| = ∞, where a is some positive pole.

Proof. (a) Assume that for any k ∈ {1, . . . , n} we have lim

_t→∞

x

_k

(t) = c

_k

, where at least one c

_k

6= 0. Then the system (6) can be rewritten as



 



 



x

^T

(∞)B

1

x(∞) = 0, .. . x

^T

(∞)B

_n

x(∞) = 0.

It is known (Fulton, 1984) that a system of equations which consists of linearly independent forms has only a trivial solution. Therefore we should have c

1

= · · · = c

_k

= 0, which proves the first statement of Theorem 2.

(b) Again, for simplicity, assume that n = 2 and let lim

_t→a

x

₁

(t) = ∞ and lim

t→a

x

₂

(t) = c

₂

= const.

Since (6) is a WIS system, the second equation takes the form 0 = b

₁₁

x

²₁

(a) + b

₁₂

x

₁

(a)x

₂

(a) + b

₂₂

x

²₂

(a) as t → ∞, where b

11

6= 0 or b

12

6= 0. In this case x

1

(a)/x

2

(a) is a finite non-zero number. From the condition lim

t→a

x

2

(t) = c

2

6= ∞ it follows that

t→a

lim x

1

(t) = c

1

6= ∞ holds true. The last relation contra- dicts the assumption of the second part of Theorem 1.

Generally, the proof proceeds as follows. According to Theorem 1, any solution (6) can be represented as

x

i

(t) = f

i

(t)

(t − a

₁

)

^d1

· · · (t − a

_k

)

^dk

,

where poles a

1

, . . . , a

k

and their multiplicities d

1

, . . . , d

k

are the same for all solutions. These poles depend on an initial vector x

0

(the so-called moving poles).

It is obvious that if some equation t − a

i

(x

0

) = 0 has a solution in the interval [0, ∞), then ∀j, lim

t→∞

|x

j

(t)| = ∞. If none of these solutions belongs to the indicated interval, then from the formula (11) it fol- lows that lim

t→∞

x

k

(t) = 0.

In what follows, for an arbitrary non-negative integer k we write x(t

k

) = x

k

, X(t

k

) = X

k

. Construct a for- mal expansion of some vector function v(t) in the Taylor series in a neighbourhood of the time point t

k

:

v(t) = E + (BX

k

)(t − t

_k

)

+ · · · + (BX

k

)

ⁿ

(t − t

k

)

ⁿ

+ · · · x

k

. (12) Assume now that k = 1. Then x(t

1

) = x

₁

, where t

₁

is selected taking account of the unique restriction

t

0

≤ t

1

< t

0

+ 1 kBX

0

k .

It is obvious that in this case the series (12) converges for all t satisfying the condition

t

₁

≤ t < t

1

+ 1 kBX

1

k .

If we continue this procedure further, then, finally, we derive that ∀ k ∈ Z

⁺

, the series (12) converges ∀ t

∈ [t

k

, t

k

+ (kBX

k

k)

⁻¹

), and the next value t

k+1

is se- lected from the range

t

k

≤ t

k+1

< t

k

+ 1 kBX

k

k .

In the case of the convergence of the series (12), the sum of the series is computed using the following well-known formula from functional analysis:

v(t) = E − (BX

k

)(t − t

k

) 

−1

x

k

. (13) It is easy to check that in the case of the convergence for the function v(t), the same estimate (11) as for the function x(t) is correct:

kv(t)k ≤ kx(t

0

)k 1 − kBkkx(t

₀

)kt .

Again, we will search for the solution of the system (9) using the Taylor expansion in the vector form. For this purpose, we estimate the limit values of solutions of the system (9) at critical points a

₁

, . . . , a

_k

and as t → ∞.

Here the following result is required.

Theorem 3. Assume that a regular WIS system is re- duced to the form (9). Let ξ be one from singular points a

₁

, . . . , a

_k

or the symbol ±∞. Then

lim

t→ξ

X(t)BX(t) = lim ˙

t→ξ

X(t)B ˙ X(t).

Proof. It is obvious that the last equality is equivalent to the set of equations lim

t→ξ

( ˙ x

i

(t)x

j

(t)−x

i

(t) ˙ x

j

(t)) = 0, i, j = 1, . . . , n. According to Theorems 1 and 2, ei- ther ∀ i ∈ {1, . . . , n}, lim

t→ξ

x

i

(t) = 0, or ∀ i ∈ {1, . . . , n}, lim

t→ξ

|x

i

(t)| = ∞. In both these cases, ac- cording to the L’Hospital rule, we have

lim

t→ξ

˙ x

i

(t)

˙

x

_j

(t) = lim

t→ξ

x

i

(t)

x

_j

(t) = c

_ij

6= 0.

But then we have lim

t→ξ

x ˙

i

(t) = c

ij

lim

t→ξ

x ˙

j

(t), lim

t→ξ

x

i

(t) = c

ij

lim

t→ξ

x

j

(t), and the limit system lim

t→ξ

( ˙ x

i

(t)x

j

(t) − x

i

(t) ˙ x

j

(t)) = 0, i, j = 1, . . . , n, is satisfied.

Starting from Theorem 3, we have lim t → ξx

⁰⁰

= lim

_t→ξ

( (BX)x + (BX) ˙ ˙ x) = lim

_t→ξ

(B ˙ Xx + (BX)(BX)x) = lim

_t→ξ

(BX ˙ x + (BX)

²

x) = 2! lim

t→ξ

(BX)

²

x; lim

t→ξ

x

⁰⁰⁰

= lim

t→ξ

(4(BX)B ˙ Xx +2(BX)

²

x) = lim ˙

t→ξ

(4(BX)

³

x + 2(BX)

²

(BX)x) = 3! lim

t→ξ

(BX)

³

x, . . . . It is obvious that for any n ∈ Z

⁺

we have lim

t→ξ

x

⁽ⁿ⁾

= n! lim

t→ξ

(BX)

ⁿ

x.

It is clear that the formal expansion of the solution x(t) in the Talor series, in a neighbourhood of the point t

_k

= ξ, has the form (12) and the convergence of this series is guaranteed by the above-mentioned conditions.

Thus we have lim

_t→ξ

kv(t) − x(t)k = 0 and the function x(t) is asymptotically equivalent to the function v(t) (Demidovich, 1967). Therefore it is possible to study the behaviour of x(t) for t → ξ via the function v(t).

From the above deliberations one can conclude that 0 ≤ t

0

< t

1

< · · · < t

m

< · · · and hence the sequence {t

m

, m = 0, 1, . . . } is monotonically increasing. There- fore there exists a (finite or infinite) limit lim

t→ξ

t

m

= t

s

of this sequence. It is obvious that if t

s

= a

i

for some i ∈ {1, . . . , k}, then t

s

is a singular point of the solution of the system (9), so that lim

t→t_s

kx(t)k = ∞. Other- wise, if t

s

= ∞, then lim

t→ts

kx(t)k = 0. Indeed, it is possible to show that the values of the function v(t

_m

) at the point t

m

can be calculated using the formula

v(t

m

) =

m

Y

i=1

E − (BX

i−1

)(t

i

− t

i−1

) 

−1

x(0).

Then, from the definition of the inverse matrix, it follows that the degree of the numerator in (13) with respect to the variable t is less than the degree of the denominator. It also reduces to the last limit.

Theorem 4. Let B ∈ R

^n×n

and let all the coordinates of the vector of the initial data x

₀

= (x

₁₀

, . . . , x

_n0

)

^T

be positive. Then for the conditional asymptotic stability of the system (9) it is sufficient that for ∀ λ ≥ 0 all the elements of the inverse matrix [E −(BX

0

)(λ)]

⁻¹

be non- negative.

Corollary 1. For the conditional asymptotic stability of the system (9) it is necessary that the polynomial f (λ) = det(E − (BX

0

)λ) have only negative real roots.

Proof. Let us investigate the behaviour of solutions to the system (9) as t → ∞. So, form positive differences

∆t

k

= t

k+1

− t

k

, k = 0, 1, . . . . Then the formula (13) shows that each term of the sequence x

k

is a rational function with the denominator

f (λ) = 1 + (β

11

x

k1

+ · · · + β

nn

x

kn

)∆t

k

+ · · · + (−1)

ⁿ

(det B)(x

_k1

· · · x

kn

)(∆t

_k

)

ⁿ

. It is obvious that to satisfy conditions of Theo- rem 4 it is sufficient that the function f (λ) be posi- tive simultaneously with all the cofactors of the matrix [E − (BX

0

)(λ)]

⁻¹

. If these conditions are fulfilled, then the proof of the stability of solutions follows from Theo- rem 2 in (Belozyorov, 2001). The proof of the corollary is then straightforward.

3. Construction of Domains of Conditional Stability for Homogeneous Quadratic Systems of the Second Order

Theorem 4 can be strengthened for n = 2. For that pur- pose we take advantage of the asymptotic equivalence of functions x(t) and v(t). Let t − t

0

= ∆t. (Here t

0

6= ξ, where ξ is a singular point or symbol ∞.) Then on the interval [t

0

, ξ), where the magnitude |t

0

−ξ| 6= 0 is small enough, the coordinates of the function (13) are given by the formulae

v1(t) =

x1t+ (x1tx2tβ22+ x²_2tβ12)∆t

1+(β11x1t+β22x2t)∆t+(β11β22−β₁₂β21)x1tx2t(∆t)²,

(14)

v2(t) =

x2t+ (x1tx2tβ11+ x²_1tβ21)∆t

1+(β11x1t+β22x2t)∆t+(β11β22−β₁₂β21)x1tx2t(∆t)².

(15) Here x

1t

and x

2t

are coordinates of the solution x(t) at instant t 6= ξ.

It is obvious that from the point of view of stabil- ity, the most desirable situation is when the denomina- tors of the functions v

1

(t) and v

2

(t) are not equal to zero. In other words, on the interval [t

0

, ξ) the function f (λ) does not have real roots (or f (λ) has only negative roots on the real axis). According to the Routh-Hurwitz criterion, the last restriction is achieved in the case of (β

₁₁

x

_1t

+β

₂₂

x

_2t

) > 0 and (β

11

β

₂₂

−β

₁₂

β

₂₁

)x

_1t

x

_2t

> 0.

With no loss of generality it is possible to set β

₁₁

=

β

22

= 1 in (9). (This can always be achieved via a

suitable change of variables.) We introduce the notation

β

12

= p and β

21

= q. Then, using Corollary 1, we will

obtain the following result.

Theorem 5. Assume that in (9) we have n = 2 and β

11

β

22

− β

12

β

21

= 1 − pq > 0, p > 0, q > 0. Then any of the conditions:

(a) x

10

> 0, x

20

> 0;

(b) x

10

> 0, x

20

< 0, x

20

+ qx

10

> 0, det x

₁₀

x

₂₀

˙ x

10

x ˙

20

!

= −x

₂₀

(−x

²₁₀

+px

²₂₀

) + x

10

(−x

²₂₀

+ qx

²₁₀

) > 0;

(c) x

10

< 0, x

20

> 0, x

10

+ px

20

> 0, det x

10

x

20

˙ x

₁₀

x ˙

₂₀

!

= −x

₂₀

(−x

²₁₀

+px

²₂₀

) + x

10

(−x

²₂₀

+ qx

²₁₀

) > 0 is sufficient for the conditional stability of (9).

Proof. If (a) is true, the proof of the stability of the system (9) is reduced to the proof given by Belozyorov (2001).

Consider a solution to (9) in a small neighbourhood δ of any singular point ξ. Assume now that x

1t

> 0 and x

_2t

< 0 if |t − ξ| < δ. Then, according to Theorem 1 of (Belozyorov, 2001), a solution to (9) is conditionally stable if for some t

^∗

> 0 we have x

2

(t

^∗

) ≥ 0. Indeed, in this case x

₁

(t) > 0 and x

2

(t) > 0 for any t > t

^∗

and we have a situation described by condition (a). For this condition, it is obvious enough that in (15) the magnitude

∆t = t − t

^∗

= − x

2t

x

1t

x

2t

+ qx

²_1t

is positive. In turn, this inequality is equivalent to x

2t

+ qx

1t

> 0. In addition, it is necessary for the denominator of the function (15) to be positive on the analysed interval [t

^∗

, ξ):

1 + (β

11

x

1t

+ β

22

x

2t

)∆t

+ (β

11

β

22

− β

12

β

21

)x

1t

x

2t

(∆t)

²

= 1 − (x

1t

+ x

2t

)x

2t

x

_1t

x

_2t

+ qx

²_1t

+(1 − pq)x

1t

x

2t

x

²_2t

(x

1t

x

2t

+ qx

²_1t

)

²

> 0.

(This guarantees that the convergence conditions of the series (12) are satisfied.)

Thus, after a transformation of the last inequality, we arrive at the system of inequalities

x

_2t

+ qx

_1t

> 0,

− x

2t

(−x

²_1t

+ px

²_2t

) + x

1t

(−x

²_2t

+ qx

²_1t

) > 0.

(16)

Assume that u = x

2t

/x

1t

. Then, from (16), we obtain the system

0 > u > −q, −pu

³

− u

²

+ u + q > 0.

At the beginning, consider equation g(u) = −pu

³

− u

²

+ u + q = 0. According to the Descartes Theo- rem (Demidovich and Maron, 1966), it has one positive root. Further, for a sufficiently small negative u we have g(u) > 0, and if u = −q, we have g(−q) = q

²

(pq −1) <

0 . Thus, because g(u) is a polynomial of the third de- gree, we come to the conclusion that there are two neg- ative roots of this polynomial. Let us denote by λ

_max

(resp. λ

min

) the greater (resp. the smaller) of these roots.

Thus, if for some t

^∗

∈ [t, ξ) inequalities (16) are fulfilled, x

₁

(t

^∗

) > 0 and x

2

(t

^∗

) > 0 and therefore the singular point t = ξ does not exist. Repeating similar reasoning for all singular points, including the first posi- tive a

1+

, we arrive at the conclusion that this point does not exist if a point t

^∗∗

such that x

1

(t

^∗∗

) > 0 is found and x

2

(t

^∗∗

) > 0. Therefore it is possible to set x

1t

= x

10

and x

2t

= x

20

in (14) and (15).

Note that taking advantage of the L’Hospital rule, the equation g(u) = 0 can be also obtained from the limit

t→a

lim

₁₊

˙ x

₂

(t)

˙

x

1

(t) = lim

t→a₁₊

x

₂

(t) x

1

(t) = lim

t→a₁₊

qx

²₁

(t) − x

²₂

(t)

−x

²₁

(t) + px

²₂

(t) , with

u = lim

t→a₁₊

x

2

(t) x

₁

(t) .

As the equation g(u) = 0 has one positive root, the segment [0, a

₁₊

) belongs to the domain of the conver- gence of the series (12), and for t ∈ [0, a

1+

) we have w = x

2

(t)/x

1

(t) ∈ (λ

max

, 0] and −pw

³

− w

²

+ w + q > 0. (In particular, this is also true for x

20

/x

10

∈ (λ

max

, 0].) From the previous analysis it is clear that λ

min

< −q < λ

max

. This completes the proof of Case (2a) and Theorem 5 if we take into account that the proof of Case (2b) (using (14)) reduces to the same result.

Denote by λ

_q

the maximum negative solution of

−pu

³

− u

²

+ u + q = 0 and by λ

p

the maximum negative solution of −qv

³

− v

²

+ v + p = 0. Then, from Theorem 5, it is possible to derive the following result.

Theorem 6. Let in (9) n = 2, β

11

β

22

−β

12

β

21

= 1−pq >

0, p > 0, q > 0. Then in the plane x

1

x

2

the domain of the conditional stability Ω of (9) represents a cone, which is the geometric place of the points described by

Ω = {x

1

− λ

p

x

2

≥ 0} ∩ {x

2

− λ

q

x

1

≥ 0}.

In addition, the apex angle of the cone Ω does not ex-

ceed π.

4. Domain of the Conditional Stability of Regular Homogeneous Quadratic Systems

Let

h

1

(v

1

, . . . , v

n

) = 0, . . . , h

n

(v

1

, . . . , v

n

) = 0 (17) be a regular system of n algebraic equations with re- spect to n unknowns v

1

, . . . , v

n

. (This system is called regular if its Jacobi determinant is not identically zero.) As is known from elimination theory (Fulton, 1984), using new variables z

1

= w

₁

(v

₁

, . . . , v

_n

), . . . , z

_n

= w

_n

(v

₁

, . . . , v

_n

) and equivalent transformations of the ini- tial system, it is possible to get one equation concerning one unknown (e.g., z

₁

):

ξ

₀

z

₁^k

+ ξ

₁

z

^k−1₁

+ · · · + ξ

_k

= 0,

where the coefficients ξ

i

, i = 1, . . . , k are complex num- bers. Thus all the remaining unknowns z

i

, i = 2, . . . , n are polynomials in z

1

. It is obvious that in general the number of solutions to (17) will equal k. The set Φ of all solutions to (17) is called an algebraic variety. The num- ber k of all elements of this set is called its degree. The degree of the algebraic variety is denoted by k =deg

C

Φ (Fulton, 1984).

Consider the following system of real quadratic equations with respect to the unknown vector v = (v

1

, . . . , v

n

)

^T

:

−v

₁

= v

^T

B

₁

v, . . . , −v

_n

= v

^T

B

_n

v. (18) Let W ⊂ C

ⁿ

be the algebraic variety of all solutions to (18). Its degree equals deg

C

W.

Definition 6. The system of equations (18) is called com- plete if deg

C

W =deg

C

V, where V ⊂ C

ⁿ

is the variety of all solutions to (18), for which it is supposed that all n

²

(n + 1)/2 of the elements of matrices B

1

, . . . , B

n

are not numbers but independent parameters.

Theorem 7. Every complete system (18) has at least one nontrivial real solution.

Proof. Assume that one from among variables v

1

, . . . , v

_n

(e.g., v

n

) is not equal to zero. Then the system (18) can be represented as

w

1

w

_n

= w

^T

B

1

w

w

^T

B

_n

w , . . . , w

n−1

w

_n

= w

^T

B

n−1

w w

^T

B

_n

w or, equivalently, as

f

1

(w

1

, . . . , w

n

) = 0, . . . , f

n−1

(w

1

, . . . , w

n

) = 0, (19)

where, by virtue of the completeness of the system (18), all forms f

1

, . . . , f

n−1

are cubic with respect to n vari- ables. Again, by virtue of completeness, numbers β

i

∈ R can be always found such that

n−1

X

i=1

β

_i

f

_i

(w

₁

, . . . , w

_n

)

= γ

₁

w

₁³

+ · · · + γ

_n

w

³_n

+ Q(w

₁

, . . . , w

_n

). (20) Here the degree of any variable included in the form Q does not exceed 2. (Note that in (20) all γ

i

6= 0.) Then from (Fulton, 1984) it follows that the system (19) (and, consequently, (18)) has at least one nontrivial real solu- tion.

Let, e.g., the system (6) be regular and complete.

Then it is easy to check that for n = 2 we have deg

_C

W = 3. On the other hand, if for n = 2v

^T

B

1

v = (δ

1

v

1

+ δ

2

v

2

)(ν

1

v

1

+ ν

2

v

2

) and v

^T

B

2

v = (δ

1

v

1

+ δ

2

v

2

)(ξ

1

v

1

+ ξ

2

v

2

), where both forms have a common linear factor and δ

1

, δ

2

, ξ

1

, ξ

2

, ν

1

, ν

2

∈ R, then the sys- tem (6) is incomplete; for this case deg

_C

W = 2 and a nontrivial real solution cannot exist.

In what follows, we will need the following trivial corollary of Theorem 2 taken from (Belozyorov, 2001).

Theorem 8. Assume that for a regular system (6) the fol- lowing conditions are fulfilled:

(a) initial values x

i0

≥ 0;

(b) forms (x

₁

, . . . , x

_i−1

, 0, x

_i+1

, . . . , x

_n

)B

_i

(x

₁

, . . . , x

_i−1

, 0, x

_i+1

, . . . , x

_n

)

^T

are positive definite;

(c) positive numbers r

_i

can be found such that the form x

^T

P

n

i=1

r

_i

B

_i

)x is negative definite, i, j = 1, . . . , n.

Then any solution to (6) is conditionally asymptotically stable.

Assume that (6) is reduced to the form (9), where the matrix B is real.

Corollary 2. Assume that for a regular system (9) the following conditions are fulfilled:

(a) initial values x

_i0

≥ 0;

(b) b

ij

≥ 0 (i 6= j) and b

ii

> 0;

(c) the elements of the matrix (−B)

⁻¹

are non- negative, i, j = 1, . . . , n.

Then any solution to (9) is conditionally asymptotically

stable.