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DIFFERENTIAL EQUATIONS AND
CONTROL PROCESSES N. 1, 2021 Electronic Journal, reg. N $C77-39410 at 15.04.2010 ISSN 1817-2172
http://diffjournal, spbu. ru/ e-mail: jodiff@mail.ru
Ordinary differential equations
Growth conditions for asymptotic behavior of solutions for certain time-varying differential equations
H.Damak, M.A.Hammami, A. Kicha
University of Sfax Tunisia, Faculty of Science of Sfax, Department of
Mathematics
Abstract. The question proposed in this paper is related to the study of the preservation of uniform h-stability and uniform boundedness of time-varying nonlinear differential equations with a perturbation using Gronwall's inequalities and Lyapunov's theory. Moreover, we show the linearization technique for the uniform h-stability of a nonlinear system and give necessary and sufficient conditions for the global boundedness of perturbed systems. The last part is devoted to the
h
Some examples and simulations are given to illustrate the main results.
h
h
1 Introduction
E-mail: MohamedAli.Hammami@fss.rnu.tn
The stability theory plays an important role in the area of the field of control systems and automation in engineering. There are different types of stability problems that arise in the study of dynamical systems, see [3, 9, 15] and has
produced a vast body of important results. In this work, we will investigate the concept of h-stability of time-varying nonlinear systems. This notion has been presented by M. Pinto in [20, 21]. He introduced it for differential systems under some perturbations and extended the study of exponential asymptotic stability to a variety of reasonable systems called h-systems. This notion is an important development of the exponential asymptotic stability within one common framework.
h
form stability, uniform Lipschitz stability (see [8]) and polynomial stability whose norm can increase not faster than exponentially. The most useful and general approach for studying the nonlinear control systems is the theory of Lyapunov. The relation between Lyapunov functions and various types of stability have been discussed by many authors, see [1, 4, 6, 7, 10, 11, 14, 19]. The general problem of motion stability includes two methods of stability analysis (the so-called linearization method and direct method) was first published in 1892. The linearization technique draws conclusions about a nonlinear system's local stability around an equilibrium point from the stability properties of its linear approximation, this result is proved in [14, 13] for the exponential stability. The direct method is not restricted to local motion, it determines the stability properties of a nonlinear system by constructing a scalar energy-like function for the system and examining the function's time variation, see [12]. Together, the linearization method and the direct method constitute the so-called Lyapunov stability theory. Nevertheless, there are some systems that may be unstable and yet these systems may oscillate sufficiently near this state that its performance is acceptable. To deal with this situation, we need a notion of stability that is more suitable than Lyapunov stability such a concept is called uniform boundedness. For the boundedness as well as the stability, the Lyapunov theory is very useful and the relation between Lyapunov functions and various types of boundedness are very similar to those between Lyapunov functions and various types of stability (see [6, 14, 16, 17, 18, 24]). It is concerned with quantitative analysis as opposed to Lyapunov analysis which is qualitative in nature.
The contribution of this paper is to construct a Lyapunov equation and use it to
h
h
point at the origin. In addition, we use the Lyapunov theory to establish the global uniform boundedness of nonlinear perturbed systems of differential equations. The topic of Laypunov stability of control systems described by a system of differential equations was an interesting research area in the past decades (see [2, 5]). Under appropriate growth conditions on the nonlinear perturbation, a
state control feedback is established based on the global uniform h-stabilizability of the nominal linear system to h-stabilize the perturbed control system using the Riccati differential equation. The remainder of this work is organized as follows. In Section 2, some preliminary results are summarized and the system description is given. The required assumptions and the statement of the main results are provided in Section 3. Section 4 is devoted to control applications. Finally, some numerical examples are given in Section 5 to demonstrate the effectiveness of the method put forward. Our conclusion is given in Section 6.
2 General definitions
We will use the following notations throughout this paper: R+ = [0, +œ) and Rn denotes the n-dimensional Euclidean space with appropriate norm ||.||. I and AT(t) denote the identity matrix and the transpose of the matrix A(t), respectively.
Consider the nonlinear system:
where t G R+ is the time, x G Rn is the state, f : R+ x Rn ^ Rn is continuous in (t, x) and locally Lipschitz in x.
Let x(t,to,Xo), or simply by x(t) the unique solution of (1) at time to starting from the point x0.
Firstly, let us introduce some basic definitions which we need in the sequel.
Definition 1 Assume that h : R+ ^ R+ is positive, continuous and bounded function. The system (1) is said to he:
1. Uniformly h-stable if there exist constants c > I and 5 > 0, independent of t0, such that for allt0 G R+ and for all x0 G Rn with ||x0|| < 5, the solution x(t)
2. Globally uniformly h-stable if there exists constant c > 1, such that for all t0 G R+ and a 11 x0 G Rn, the solution x(t) satisfies the estimation (2).
xx = f (t, x), x(t0) = x0, t > t0 > 0,
a)
x(t)|| < c|xo|h(t)h(to)-1, V t > t0.
(2)
Here,
Remark 1 For some special cases of h, the h-stability coincides with known types of stability:
i) If h(t) = a for a > 0, then the system, (1) is stable.
ii) If h(t) = e-Xt for X > 0, then the system (1) is uniformly exponentially stable.
iii) If h(t) = 1 ^^ for y > l, the system (l) is polynomially stable.
v) If h(t) is a strictly decreasing function, such that h(t) tends to 0 when t ^ then the origin is uniformly asymptotically stable. More precisely, the solutions of system (l) converge to the origin (i.e., limsup ||x(t)|| = 0).
Consider now the linear time-varying system:
x = A(t)x, x(t0) = x0, t > to > 0, (3)
where A(-) is n x n whose entries values are continuous functions oft £ R+.
The general solution of system (3) is given by:
x(t) = 0(t, t0)x0, x0 £ Rn, t > t > 0,
where 0(t, t0) is the state transition matrix associated with A(-). We define the
norm of matrices by: || AN = max ||Ax||.
i|x|<l
h
there exist c > l and a positive continuous bounded function h on R+, such that
||0(t,t0)|| < ch(t)h(t0)-1, V t > t0 > 0.
Remark 2 In linear systems there is the notion of an upper function which is related to upper Lyapunov exponent (see [3, 13, 16, 17, 18, 19, 22, 23]).
Definition 2 (See [23]) A bounded, function ^(t) is an upper function for system
X,
||0(t,s)||< X)dr^j, V t > s, where 0(t, s) is the fundamental matrix of the system.
Remark 3 In the case where ^ is an upper function and
h(t)h(t0)-1 = exp ( / ^(t)dr
vJ to
h
Remark 4 There is no relationship between the concept of polynomial stability and exponential stability as shown in the following example.
Example 1 Consider the scalar equation:
1
x =
1+ t
x e R, t > 0.
(4)
The state transition matrix 0(t, t0) is given by
0(t,t0) = tj+1, V t > t0.
Then, the system (4) is polynomially stable. On the contrary, if we suppose that (4) is exponentially stable, then there exists a > 0, such that
(t0 + 1) < e-a(t-to)(t + 1), V t > t0.
For t0 = 0 and t —> to, we obtain a contradiction and hence the system (4) is not exponentially stable.
We prove now the following lemma which will be used later.
Lemma 2 Consider the nonlinear system (1) with f (t, 0) = 0, for all t G R+. Let 0(t; t, x) be a solution of the system that starts at (t, x), and let 0x(t; t, x) =
—0(t; t,x) and dx
dx (t'x)
< L, where L is a positive constant. Then,
)lI V II 11/
II > ||x|| e
2 e-2L(T-t)
Proof. Let 0x(r; t,x) be the solution of
dT^T; t, x) = dX(t, ; t, x
0x(t; t,x) = I.
We have,
d
—4>T (t ; t,x)0(r; t, x)
= 20T(t; t,x)f (t,0(t; t,x))
< 2||0(r; t,x)||||f (t,0(t; t,x))
< 2L||0(r; t, x
x
Then,
d
—(r; t,x)0(r; t, x) > -2L||0(r; t,x)||2. (5)
dr
Setting ^(r) = —|0(r; t,x)|| and using (5), we conclude (as in [14], Example 3.9, pp. 103-104) that d+0(t) < 'L^(r), with
D+0(t) = lim sup - U(t + h) - ^(t)).
h V /
l
h
By the comparison lemma (see [14], pp. 102-103), we deduce that
(r;t,x)||2 > ||x||2e-2L(T-t).
□
In this work, it is worth to notice the origin is not necessarily an equilibrium point for system (1). This brings us to the notion of global uniform boundedness.
Definition 3 A solution of system (1) is said to be globally uniformly bounded if for every n > 0 there exists 0 = 0(n) > 0, such that
||x0|| < n ||x(t)|| < 0, V t > t0 > 0.
In order to solve the problem of such perturbed systems, we introduce the following technical lemma, that will be crucial in studying the global uniform boundedness of solutions.
Lemma 3 Let w, p : R+ ^ R be continuous functions and ^ : R+ ^ R+ is a function, such that
<£(t) < w(t)^(t) + p(t), V t > t0. (6)
t > t0 > 0,
^(t) < ^(t0)exp w(v)dv^ + J^ exp ^^ w(v)dv^p(s)ds.
3 Basic results
3.1 Sufficient conditions for uniform boundedness
Lyapunov's direct method allows us to determine the stability of a system without explicitly integrating the differential equation. This method is a generalization of
the idea that if there is an appropriate function of a system that satisfies certain conditions, then we can deduce the stability of this system. The following theorem discuss sufficient conditions on the global uniform boundedness of solutions of system (1) by using the Lyapunov's direct method.
Theorem 1 Suppose that h is a positive, hounded, continuous, decreasing onR+ with hh exists and continu,ous onR+. Moreover, suppose that there exist constants ai, a2, b > 0, k > 0 and a function V(t,x) satisfying the following properties:
(i) a^xll6 < V(t,x) < a2||x||b, (t,x) e R+ x Rn,
(ii) y(t,x) < h'(t)h(t)-iV(t,x) - kh/(t)h(t)-i, (t,x) e R+ x Rn.
Then, the system (1) is globally uniformly bounded.
Proof. Let x(t) = x(t, to, xo) be the solution of system (1) through (to,xo) e R+ x Rn. Then, it follows from condition (ii) that
y(t,x) < h/(t)h(t)-iV(t,x) - kh/(t)h(t)-i.
By using Lemma 3 and the decreasing of h, we get for all t > to and x0 e Rn the following estimation
V(t,x) < V(to,xo)h(t)h(to)-i - k£ exP h/(r)h(rh/(s)h(s)-ids < V(to,xo)h(t)h(to)-i + k.
We deduce for all t > to and all xo e Rn that
l*(t}|| < ("||xo||6h(i)h(io)-1 + -) b . (7)
a ai
1. If b > 1, % using the fact that (Ai + A2)e < Ai + A2, for all Ai, A2 > 0 and £ e]0,1[, one obtains from the decreasing of h,
"x(t)"< (a:)1 ixoi+(ai)1. («)
2. If b < 1. Since (Ai + A2)p < 2p-i(AP + A2), for all Ai, A2 > 0 and p > 1, one
h,
l|x(t)||< 2(02)1 ||x,| + 2(!) 1. (9)
This yields that the solutions of system (1) are globally uniformly bounded. Note that, if b > 1, the inequality (8) implies that
Ni)i|-( s i < © i mf k \ 1 1 Thus, for all t E R+, if we take ||x0|| > I — I , such that ||x(t)|| > I — I , we
\aij \aij
get that the solutions of system (1) approach to a compact set S, when t ^ to, given by
S = jx E Rn, ||x|| < ^^ 6
If b < 1 and by using a similar reasoning as above on (9), we can deduce that the solutions of system (1) approach to a compact set S', when t ^ to, given by
S; = jx E Rn, ||x|| < 2^ (k) 6
This completed the proof. □
3.2 Converse Theorem
Although Lemma 1 may not be very helpful as a stability test, we will see that it guarantees the existence of a Lyapunov function for the linear system (3). That is, if we can find a continuously differentiable, positive, bounded and symmetric matrix P(t), which is a solution of a differential equation for some continuous positive definite symmetric matrix Q(t), then V(t,x) is a Lyapunov function for the system. If the matrix Q(t) is chosen to be bounded in addition to being symmetric, continuous, positive definite and if A(t) is continuous and bounded, then it can be shown that when the origin is uniformly h-stable, there is a solution of system (3) that possesses the desired properties.
In this section, we state a converse theorem when the origin is a globally uniformly h-stable equilibrium point of the linear system (3), by defining a Lyapunov function that satisfies certain properties.
In what follows, we will denote by H the set of the functions h : R+ ^ [1, +to) with the property that:
/to
h(T)2dT < Mh(t), V t > 0.
Theorem 2 Let the origin he globally uniformly h-stable equilibrium point of system (3). Assume that h G H with hh exists and continuous onR+. Suppose that A(t) is continuous and bounded on Rn. Let Q(t) be a continuous, bounded, positive definite and symmetric matrix. Then, there is a continuously differentiable, bounded, positive and symmetric matrix P(t), which is the solution of the Riccati equation:
P(t) = h(t)h(t)-1P(t) - AT(t)P(t) - P(t)A(t) - Q(t). (10)
Proof. Assume that the system (3) is globally uniformly h-stable. Let 0(t; t,x) be the solution of system (3) that starts at (t, x). Due to linearity, 0(t; t,x) = 0(t, t)x. Let the matrix P(t) defined by
/to
f (T,t)Q(T )0(T,t)dT. (11)
Since Q(t) is positive definite and bounded matrix, then there exist positive constants k1 and k2, such that
kil < Q(t) < k2l, V t > 0. (12)
On the one hand, we have
/to
||0(t ; t,x)||2dT
/to
h(T )2dT ||x||2.
Thus,
xTP(t)x < c2k2M = cjx||2. A(t)
L, such that
||A(t)|| < L, V t G R+.
From Lemma 2, we have
t;t,x)||2 > ||x||2e-2i(T-t).
/TO
e-2L(T-tW ||x||2
= ^||x|2 > 0, v t > 0. 2L N N > ' >
Therefore, P(t) is positive and bounded. In addition, the definition ofP(t) shows that it is symmetric and continuously differentiable. To calculate the differentiable of P(t), we use the following property
d
-0(r,t) = -0(r,t)A(t).
Hence,
1
p(t) = h'(t}h(t)-1P (t) + h(t)
it
Q(t )0(r,t)dr
■>oo
+ h(t) / / (r,t)Q(r)
't
d , / x
dr - Q(t)
= h/(t)h(t)-iP(t) - AT(t)P(t) - P(t)A(t) - Q(t).
□
h
h e % and h/ exists continuous on R+. Suppose that A(t) ¿5 continuous and bounded, on Rn. Thus, there exists a function V(t,x) satisfying the following properties:
(i) ||x||2 < V(t,x) < (ci + 1)||x||2, (t,x) e R+ x Rn, (ii) V/(t,x) < h/(t)h(t)-iV(t, x), (t, x) e R+ x Rn,
ci
Proof. We choose a matrix Q(t) continuous, bounded, positive definite, symmetric on R+, and there exists c2 > 0, such that
xT(Q(t) + h/(t)h(t)-iI - A(t) - AT(t))x > C2|x|2. (13)
h
P(t)
Lyapunov function V : R+ x Rn ^ R+ by
V(t, x) = xTP(t)x + ||x||2.
It is easy to verify that,
V(t,x) < (ci + 1)||x||2.
Another,
V(t,x) > kih(iW e-2L(T-i)dr||x||2 + ||x||2
= ^iM|x|2 + |x|2 >||x||2. 2L
Hence, the first inequality of the theorem is hold.
Now, we shall show (ii). By taking the derivative of V(t, x) along the trajectories of the linear system (3), we get
V"(t,x) = xT P (t)x + xT P(t)x + xT P (t)x + xT AT (t)x + xT A(t)x = h'(t)h(t)-1V(t,x) - Q(t)||x||2 + A(t)||x||2 + xTAT(t)x + xTA(t)x - h'(t)h(t)-1xTx.
It follows from (13) that
y(t,x) < h'(t)h(t)-1V(t, x), which prove (ii). This ends the proof. □
Remark 5 The above theorem is an extension of the global uniform exponential stability in [14], for h(t) = with ft > 0 and the polynomial stability, for
h(t) = 7-— with Y > 1.
W (1+ t)Y
h
system (3). Assume that h G H and hh exists continuous on R+. Suppose that A(t) is continuous and bounded on Rn. If Q(t) = CT(t)C(t), where C(t) ¿5 a continuous matrix in t G R+, then the Riccati equation is given by
P(t) = h'(t)h(t)-1P(t) - AT(t)P(t) - P(t)A(t) - C(t)TC(t).
3.3 h-Linearized stability of nonlinear systems
The existence of Lyapunov functions for linear systems per Theorem 2 will now be used to prove a linearization result and to determine the uniform h-stability of the nonlinear system. In this section, the result of the preceding section is combined to obtain one of the most useful results in Lyapunov stability theory namely: linearization method. The advantage of this method lies in the fact that, under certain conditions, it enables one to draw conclusions about a nonlinear
system by studying the behavior of a linear system.
Consider the nonlinear non-autonomous system (1), where f : R+ x D ^ Rn is continuously differentiate and D is a domain that contains the origin. Suppose that the origin is an equilibrium point of the system, with f (t, 0) = 0 and assume
" df ■
that the Jacobian matrix
dx
D,
is, there exist positive constants Li and L2, such that
< Li, V x e D, V t > 0,
dx (t,x)
- (t,xi) - - (t,x2)
< L20x1 - x2||, V x1, x2 e D, V t > 0.
We can write /(t, x) in the form
f (t,x) = f (t, 0) + dx (t,z)x, where z e]0,x[. Since f (t, 0) = 0, then we have
/ (t,x) = dx (t,z)x
= ix0)x+
f ^) - f(t0)
x
= A(t)x + x(t,x),
where A(t) = —(t, 0) and x(t,x) =
dx ^ - dx(t,0)
x.
3y its linearization in a
Therefore, we may approximate the nonlinear system (1) small neighborhood of the origin.
h
stability of the origin in the non-autonomous case. We will see that, the uniform hh system.
Theorem 4 Let the origin be an equilibrium point for the nonlinear system
x = f (t, x), x(t0) = x0, t > t0 > 0, (14)
where f : R+ x D ^ Rn is continuously differentiate with D = {x e Rn/||x|| <
Suppose that the Jacobian matrix t.
dx
is bounded, and Lipschitz on D, uni-
A(t) = dx (t,x)
x=0
Then, the origin is uniformly h2 -stable for system (14) if it is uniformly h-stable for the linear system x = A(t)x, with h GH.
Proof. We can write the nonlinear system (14) as:
x = A(t)x + x(t,x), (15)
such that x defined on R+ x D is continuous in (t, x), locally Lipschitz in x and verifies the following condition:
||x(t,x)||< Л^Ц2, V x G D, V t > 0, (16)
with Л is a positive constant. In addition, we assume that the linear system (3) has a uniform h-stable equilibrium point at the origin, h G H with h' exists and continuous on R+, and A(t) is continuous and bounded on Rn. Then, Theorem 2 ensures the existence of a continuously differentiable, bounded, positive and symmetric matrix P(t) that satisfies (10), where Q(t) is continuous, bounded, positive definite and symmetric matrix that verifies (13). By Theorem 3, there exists a Lyapunov function V(t,x) having the properties (г) and (гг). The derivative of V(t, x) along the trajectories of system (15) is given by:
V"(t,x) = xT P (t)x + xT P(t)x + xT P (t)x + xT x + xT x
< h'(t)h(t)-1V(t,x) - xTQ(t)x + 2(||P(t)|| + l)x(t,x)||x|| + AT(t)||x||2 + A(t)||x||2 - h/(t)h(t)-1|x|2.
By using the inequality (13), we get
V"(t,x) < h/(t)h(t)-1V(t,x) - c2||x||2 + 2(||P(t)|| + l)x(t,x)||x||.
From condition (16) and the property on P(t), we obtain
V"(t, x) < h/(t)h(t)-1V(t, x) - (c2 - 2рЛ(с1 + l)) ||x||2, V ||x|| < p. By choosing p < min ^ , we °btain
V(t,x) < V(t0,x0)h(t)2h(t)-2.
Therefore, for all t > t0 and all x0 G D the solution x(t) of system (15) is as follows:
||x(t)|| < V(C1 + 1)||x0|h(t)2h(t0)-1,
which ensures that V"(t, x) is negative definite in ||x|| < p. Hence, we conclude that the origin of the nonlinear system (14) is uniformly h2-stable. □
Remark 6
- The previous theorem is a generalization of uniform exponential stability, that is, the nonlinear system is uniformly exponentially stable if the linearized system is uniformly exponentially stable (see [14], Theorem 4-13).
- The linearization result is hold for polynomial stability: h(t) = ^ ^ with
Y > 1.
Corollary 2 If the nonlinear system is autonomous, that is, x = f (x), we can draw conditions about the stability of the origin as an equilibrium point for the
system by investigating the stability for the linearization of the system, where A = f
dx
, such that h(t) = e ßt, for all t £ R+ and ß > 0.
x=0
3.4 Boundedness of solutions of perturbed systems
We can use Lyapunov's indirect method to show the global uniform boundedness of the solutions. We consider a nonlinear perturbed system and we give sufficient conditions on the perturbed term. Our conditions are expressed as relations between the Lyapunov function and the interconnection term.
Theorem 5 Consider the perturbed system:
x = A(t)x + x(t,x), x(t0) = x0, t > t0 > 0, (17)
where A(t) is continuous and bounded on Rn, x : R+ x Rn ^ Rn continuous in (t,x), locally Lipschitz in x and satisfies the following assumption:
Ilx(t,x)|| < <p(t)||x|| + ^(t), V x £ Rn, V t > 0,
with (p and ^ are non-negative continuous integrable functions on R+. Assume that the system (3) is globally uniformly h-stable with h £ H is decreasing and exists continuous onR+, then the solutions of system (17) are globally uniformly bounded.
Proof. We have the system (3) is globally uniformly h-stable, A(t) is continuous and bounded on Rn. Let Q(t) be a continuous, bounded, positive definite and symmetric matrix, such that (13) is hold. Then, Theorem 3 ensures that there exists a Lyapunov function candidate
V (t,x) = xT P (t)x + ||x||2
that satisfies the certain properties. The derivative of V(t,x) along the trajectories of system (17) is as follows:
y (t, x) = h'(t)h(t)-1 V(t, x) - xTQ(t)x + h'(t)h(t)-1xTx + 2xT (||P(t) || + l)x(t, x)
+ xT A(t)x + xT AT (t)x.
Using the inequality (13), we get
V(t,x) < h'(t)h(t)-1V(t,x) + 2(c1 + l)^(t)||x||2 + 2(c1 + l)^(t)||x||
-1
x)
< (h'(t)h(t)-1 + 2(c1 + l) <p(t)J V (t, x) + 2(c1 + l) ^(t)V V (t,x). Put, ro(t) = ^V(t, x), then
w(t) =-, •
v 7 2v/VM)
This yields,
w(t) < (ih'(t)h(t)-1 + 2(ci + l)^(t))w(t) + 2(ci + l)^(t).
,2
By Lemma 3 and the decreasing of h, we have for all t > to
ro(t) < ro(t0)exp (2(c1 + l)M2Jh(t)2h(t0)-2 + 2(c1 + l) exp (2(c1 + l)M2JM1; where M1 = ^(s)ds and M2 = ^(s)ds. Therefore,
0
x
(t)|| < /(ci + 1) exp(2(ci + 1)M2) ||xoHh(t)2h(to)-1 + 2(ci + 1) exp(2(ci + 1)M2)Mi.
Consequently, the solutions of system (17) are globally uniformly bounded. From the decreasing of h, we obtain
||x(t)|| - 2(ci + 1) exp (2(ci + 1)M2)Mi < /(ci + 1) exp ^2(ci + 1)M2) ||xo|
Hence, for all t G R+, if we take ||x0|| > 2(c]_ + l) exp ^(c1 + l)M2JM1, such
that ||x(t)|| > 2(c1 + l) exp ^2(c1 + l)M2^ M1, then the solutions of system (17) approach, when t goes to infinity, to the compact set S defined by
S = {x G Rn, ||x|| < 2(c1 + l) exp ^2(d + l)M^M^ .
0
□
As a particular case of the forgoing theorem, when ^(t) = 0 we obtain the following corollary.
Corollary 3 Consider the perturbed, system, (17) where A(t) is continuous and bounded, on Rn, x ^ defined on R+ x Rn continuous in (t, x) and locally Lipschitz in x. Suppose that x satisfies the following assumption:
||x(t,x)||< p(t), V t > 0, (18)
where p is a non-negative continuous integrable function onR+. If the system (3) is globally uniformly h-stable with h G H is decreasing and hh exists continuous on R+, then the solutions of the perturbed, system (17) are globally uniformly bounded.
h
h
the form:
X = A(t)x + B (t)u(t) + F (t,x,u), (+ \ ^ J
x(to) = xo,
where x G Rn is the state ve ctor, u(t) G Rm is the control in put, A(t) G Rnxn, B(t) G Rnxm are matrices whose elements are continuous bounded functions on R+. The function F : R+ x Rn x Rm ^ Rn is continuous in (t, x, u) and satisfying the following inequality:
||F(t,x,u)|| < A(t)||x|| + Y(t)|u|, V x G Rn, V u G Rm, V t G R+, (20)
with A and y are non-negative continuous integrable functions on R+. The corresponding system without perturbation called the nominal system is described by
x = A(t)x + B (t)u(t),
( ) ( ) ( ^ 21
x(to) = xo,
Definition 4 we say that the system (19) is stabilizable, if there exists at least a
u( t) ,
u( t) u( t)
Definition 5 Let h : R+ ^ R+ be a continuous bounded function. The linear
h
control system u(t) € Rm and a const ant c > 1, such that for all t > t0 and all x0 € Rn the solution x(t) of the closed-loop system satisfies the estimation:
||x(t)|| < c||x0||h(t)h(t0)-1.
The goal of this section is to found a state feedback controller u(t), such that the system (19) is globally uniformly h-stabilizable.
Theorem 6 Assume that the linear system x(t) = A(t)x(t) is globally uniformly h-stable with h € % and h' exists continuous on R+. We choose Q(t) continuous, bounded, positive definite and symmetric matrix that verifies (13), then the Riccati differential equation (10) has a solution P(t) bounded, positive, symmetric continuously differentiate matrix, and the nonlinear control system (19) is h2-stabilizable by the feedback control
u(t) = -BT(t)P(t)x(t), t > to. (22)
Proof. Suppose that the linear system x(t) = A(t)x(t) is globally uniformly h-stable where A(t) is continuous and bounded on Rn. Then, Theorem 2 ensures the existence of a continuously differentiable, bounded, positive and symmetric matrix P(t)
V(t, x) along the solutions x(t) of system (19) using the chosen feedback control (22) is given by
V (t,x) = xT P (t)x + xT P (t)x + xT P (t)x + xT x + xT x
= h'(t)h(t)-1 V (t, x) + 2F (t, x, u)xT P (t) + uT (t)BT (t)P (t)x + xTP(t)B(t)u(t) - xTQ(t)x - h'(t)h(t)-1xTx + xTAT(t)x + xTA(t)x + UT (t)BT (t)x + xT B (t)u(t) + 2F (t, x, u)xT
< (h'(t)h(t)-1 + 2(ci + 1) (A(t) + ciY(t)||B||^)) V(t, x), where ||B||TO = sup ||B(t)||. Hence,
i to
t> 0
V(t,x) < V(to,xo)h(t)h(to)-1 exM2(cx + 1W (A(s) + ci7(s)||B|U)ds
to
< V(to,xo)h(t)h(to)-1 exp (2(ci + 1) (M + ci||Bh(X,
/>TO fTO
with M1 = A(s)ds and M2 = y(s)ds. Therefore, Jo Jo
||x(t)|| < c||x0||h(t)1 h(t0)-2,
where c = \/(ci + 1) exp ^(ci +1) ^Mi + c^B . This yields that the system (19) is h2-stabilizable. The proof is completed. □
Next, we state another sufficient condition for the h-stabilizability of system (19) in the case when the linear system x = A(t)x is not globally uniformly h-stable,
h
h
h G H and h' exists continuous on R+. Then the nonlinear system, (19) is h2-stabilizable.
Proof. Assume that the system (21) is h-stabilizable, then there exists K(t) G Rmxn, such that
x = (A(t) + B(t)K(t))x = A(t)x, Vt > to (23)
is globally uniformly h-stable. We choose Q(t) a positive, continuous, bounded and symmetric matrix that satisfies (13), then we consider the Lyapunov function V(t,x) = xTP(t)x + ||x||2 from Theorem 3. Hence, by taking the derivative of V(t, x) along the solutions x(t) of system (19) using the chosen feedback u(t) = K(t)x(t), we obtain
V"(t,x) = xT P (t)x + xT P(t)x + xT P (t)x + xT x + xT x
< h'(t)h(t)-1V(t,x) + 2F(t,x,u)||P(t)||||x|| + 2A(t)||x||2 + 2F(t,x,u)||x|
< (h'(t)h(t)-1 + 2(ci + 1) (A(t) + 7 (t)||K ||<)) V (t,x),
with ||K||< = sup ||K(t)||. Thus, to
V(t,x) < V(to,xo)h(t)h(to)-1 ex^2(cx + 1) ^ (A(s) + Y(s)||K|U)ds < V(to, xo)h(t)h(to)-1 exp + 1) (Mi + ||c
^00 />00
with M1 = / A(s)ds and M2 = / y(s)ds. Therefore, for all t > to and all
oo xo G Rn the solution of system (19) is given by:
||x(t)|| < c||xo||h1 (t)h(to)-2, where c = //(c1 + 1) exp ^(c1 + 1) ^M1 + M2||K. This ends the proof. □
Example 2 Consider the second order problem:
xi = X2 =
-1 ^ e
- 1)xi +
1+ t -1 TTt
-t
- 1) X2 +
1+ t 1
1 + t2
u(t) +
X2 +
x1e
t
(1 + t2^1 + xi
=u(t)
x1 e
t
\/1 +
x
rU(t),
(24)
t > 0
The above system is exactly the system (19), where
A(t) =
and
/
V
1 +1 0
1
0
\
1 +1
1
t
b (t) =
/
1 +1 0
x(t) =
xi(t) x2(t)
F(t, x, u) =
1 + t2
xi x2
+
xi
e
t
x
1 + t2
t
u(t).
e
The nominal system x(t) = A(t)x(t) ¿5 globally uniformly h-stable with h(t) =
--Then, for a matrix
1 + t 7 J
Q(t) =
/
V
1
2(1+ t) 0
+1
0
\
+1
)
2(1+ t)
P(t) ,
p (t) =
V0 2
/
satisfies the Riccati equation (10). Moreover, the function F satisfies the assumption (20) with A(t) ¿2 an^ Y(t) = V/2e-t.
VFe conclude that the conditions of Theorem 6 are hold. Therefore, the system (24) ¿5 globally uniformly h2-stabilizable under the closed-loop linear feedback
u(t) = -BT (t)P (t)x(t) =
t
2(1 + t)
(xi(t) + x2(t)).
For simulation of system (24) we select ¿he initial state ^x1(0),x2(0)^ = (1,1).
1.
1
1
1
1
0 5 10 15 2 0 2 5 30 35 40 45 50 Time (s)
Figure 1: The trajectory of the state x(t) = (#i(t), x2(t)) of system (24). 3.6 Examples
In this section, we give some numerical examples and simulations to prove the applicability of the theoretical results.
Example 3 Consider the scalar equation:
2
x
x =
+
Setting, V(t, x) = x2 and h(t) =
t + sin x 1 + t' 1
x G R, t G R+. (25)
,
(i+1)2
and decreasing on R+ with h' exists and is continuous on R+. Then, Theorem 1 holds with ai = a2 = 1 and b = k = 2. This yields the global uniform boundedness of system (25), that is, the solutions of system (25) approach to a compact set S', when t ^ +<, given by:
= {x G R, |x| < -^2} .
For simulation of system (25) we select the initial state x(0) = 1. The result is
2
Example 4 We consider the second order problem:
_x3 + x2x2 _
x 1 +
a
x 1
x2 = —x3 + x1 sin x2 —
1 +1
x1 a
1 +1
x2,
a 1,
(26)
mo
0
Figure 2: The trajectory of the state x(t) of system (25).
where x = (x1, x2)T € D C R2 and t € R+. The Jacobian matrix of the nonlinear system (26) is given by:
df ( )
dx (x1,x2)
a 0
(0,0)
10+t I = A(t).
1 +1
The linear system x(t) = A(t)x(t) is uniformly h-stable with c = 1 and h(t) = € %. By applying Theorem 4, the system (26) is uniformly 1 a -stable.
(1+ t)a • * rr » » » v y J * (1+t)2 For simulation of system (26) we select the initial state ^x1(0),x2(0^ = (1, 2) and a = 3. The result of simulation is depicted in Figure 3
Figure 3: The trajectory of the state x(t) = (x^x2) of system (26).
Example 5 We consider the first order problem:
x = -(ZT^ + A x + + , x G R, t G R+. (27)
V i +1 ) i +13 i +t2 + v;
The previous system can be written as:
x(t) = A(t)x(t) + x(t,x), t > 0. The linear system x(t) = A(t)x(t) ¿5 globally uniformly h-stable and h(t) = ^ ^ 2 G % is positive, bounded and decreasing where hh exists and is continuous
on R+. Then, for a function Q(t) = ^^f +1 ver''fies (13), ¿here exists P(t) = i satisfies the Riccati equation (10). Furthermore,
\ 1
|x(t,x)| < |x| + , V x G R, V t G R+.
\ 1
Pm¿7 ^(t) = ^ and ^(t) = ^ which are non-negative continuous and integrable functions on R+. From Theorem 5, we deduce the global uniform bound-edness of system (27).
t
compact set S given by:
S = {x G R, |x| < — en}.
2
For simulation of system (27) we select the initial state x(0) = 1. The result of simulation is depicted in Figure 4
4 Conclusion
We have introduced some new conditions for global uniform boundedness of nonlinear systems of differential equations. A converse theorem has been established to guarantee the global uniform h-stability of a nonlinear system when its linearization has a global uniform h-stability equilibrium point. One of the main interests of this paper is that it serves to establish that property for nonlinear
h
been showed with Lyapunov theory. We have illustrated this use in the global h
h
vided and sufficient conditions has been given. The effectiveness of the conditions obtained in this paper has been verified in some numerical examples.
0123456789 10 Time (s)
x(t)
Annex
Proof of Lemma 3. We write (6) as
<p(t) - ro(t)^(t) < p(t), V t > t0.
On the other hand,
^V(s) exp ^- J^ )dr^ ^ < p(s) exp ^- J^ )dr^ , s > t0. t0 €
-d ( ^(s) exp ( - [ )dr) ) < [ p(s) exp ( - [ )dr ) ds, V t > t0,
'to
ds
'to
'to
'to
which implies
^(t) exp I - / )dr I - ^(t0) < / p(s) exp I - / )dr I ds.
to to to
Then,
^(t) < ^(t0) exp ( / )dr I + / p(s) exm / )dr I ds, V t > t0.
to to s
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