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6Russian Journal of Nonlinear Dynamics, 2018, vol. 14, no. 3, pp. 387-407. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd180309
MSC 2010: 34K05, 34K20, 34K35, 37B25, 37B55, 45D05, 93D15, 93D20
On the Stability and Stabilization Problems of Volterra Integro-Differential Equations
In this paper, the stability and stabilization problems for nonlinear Volterra integro-differential equations with unlimited delay are considered. The development of the direct Lya-punov method in the study of the limiting properties of the solutions of these equations is carried out by using Lyapunov functionals with a semidefinite time derivative. The topological dynamics of these equations has been constructed revealing the limiting properties of their solutions. The assumption of the existence of a Lyapunov functional with a semidefinite time derivative gives a more complete solution to the positive limit set localization problem. On this basis new theorems on sufficient conditions for the asymptotic stability and instability of the zero solution of nonlinear Volterra integro-differential equations are proved. These theorems are applied to the problem of the equilibrium position stability of the hereditary mechanical systems as well as the regulation problem of the controlled mechanical systems using a proportional-integro-differential controller. As an example, the regulation problem of a mobile robot with three omnidirectional wheels and a displaced mass center is solved using the nonlinear integral controllers without velocity measurements.
Keywords: Volterra integro-differential equation, stability, Lyapunov functional, limiting equation, regulation problem
Received May 14, 2018 Accepted September 13, 2018
This work was supported by the grant of the Ministry of Education and Science of Russia within the framework of the State task [9.5994.2017/BP] and the Russian Foundation for Basic Research [18-01-00702, 18-41-730022].
Aleksandr S. Andreev asa5208@mail.ru Olga A. Peregudova peregudovaoa@gmail.com
Ulyanovsk State University
ul. L'va Tolstogo 42, Ulyanovsk, 432017, Russia
A. S. Andreev, O. A. Peregudova
Introduction
Volterra's fundamental research on the theory of functionals, integro-differential equations, and their application to the population dynamics and rheological mechanical systems [57] was intensively developed in the 1950s due to the need to take into account the phenomenon of aftereffect for the correct quantitative and qualitative description of the systems of automatic regulation as well as mechanical and biological processes. The corresponding studies have formed the theory of functional differential equations (FDE), which is characterized by the richness and nontriviality of mathematical and applied results. From a huge number of papers and a significant number of reviews and monographs on the theory of retarded functional differential equations (RFDE), we choose only the publications directly related to the topic of this paper.
The axiomatic construction of the phase spaces for RFDE with infinite delay is fully presented in the papers [18, 19, 26, 27, 49-52] (see also the survey [20]). This made it possible to find the conditions of the existence, uniqueness, continuation and continuous dependence on the initial values for the solution of RFDE with infinite delay. The realized construction of topological dynamics of RFDE made it possible to find the relationship between the stability property of the solution in both the norm of the phase space and Euclidean one as well as the stability of the solutions of original and limiting equations, and the stability property of the solution under perturbations [20, 26-29, 33, 41, 42].
The development of the Lyapunov functionals method in the stability study was investigated in the papers [11, 13-15, 18, 20, 27, 31-33, 37, 49]. This development consists in the modification and generalization of the corresponding theorems of N. N. Krasovskii, V. M. Matrosov, T. A. Burton, J. K. Hale, and others, previously proved for ordinary differential equations as well as RFDE with a bounded delay. The difficulty in obtaining such development consists in using the axiomatic constructions of functional spaces for RFDE with infinite delay.
The qualitative analysis of RFDE is used in [17, 18] to continue the studies of V. Volterra on the dynamics of rheological (hereditary) systems and processes.
In this paper, we consider nonlinear Volterra integro-differential equations (IDE) with unlimited but not infinite delay. These equations have some specific features as well as general properties inherent in RFDE. A wide study of IDE was initiated much earlier than that of RFDE (see [34]). In order to carry out the qualitative analysis of these equations, we will use the materials of the profound monographs [14, 16]. Some results on the investigation of the applied problems by using IDE are presented in [54-56].
This paper is organized as follows.
Section 1 considers the topological dynamics of nonlinear Volterra IDE, which makes it possible to find out the limiting properties of their solutions by constructing the limiting integro-differential equations with an infinite delay.
The assumption on the existence of a Lyapunov functional with a semidefinite time derivative gives a more complete localization of the positive limit set of the bounded solution. On this basis, new theorems on sufficient conditions for asymptotic stability and instability of the zero solution are proved. These results are presented in Section 2.
Section 3 presents a new result on sufficient conditions for the stability of the state of hereditary mechanical system.
The solutions to the motion stabilization problems of a controlled mechanical system by using PI and PID regulators lead to the study of Volterra IDE [1, 2, 7-10, 39, 43]. One of these problems was investigated in Section 4 on the basis of the theorems from Section 2. As an
example, the regulation problem of a mobile robot with three omniwheels and a displaced mass center has been solved using nonlinear PI regulators. Note that the dynamics of such a system was investigated in detail in [12, 38].
1. On the positive limit set localization problem for the solution of a Volterra equation
1.1. Elements of topological dynamics
for a Volterra integro-differential equation
Consider a Volterra nonlinear integro-differential equation in the form
x(t) = f ^t,x(t), J g(t,r,x(r))dr^, x(to) = xo, (1.1)
where t0 £ R, x £ Rn is the phase vector, Rn is n-dimensional linear real space with some vector norm ||x||, the functions f and g are defined in the domains R x D x Rn and S x D, respectively, where D = {x £ Rn: ||x|| < H, 0 <H < and S = |(t,r) £ R x R: t < t}.
Assumption 1. Assume that the functions f and g are bounded and uniformly continuous in the domains R x D1 x D2 and S x D2, respectively, where D1 = {x £ Rn: ||x|| ^ H1 < H} and D2 = {y £ Rn: ||y|| ^ H2 < i.e., the following inequalities hold:
f (t, x, y)|| < m\(H\,H2) V(t,x,y) £ R x Dx x D2, f (t2,x(2),y(2)) - f (ti,x(1),y(1))|| < e V(tbx(1),y(1)), (t2,x(2),y(2)) £ R x Di x D2: (1.2) 112 - t11 < ¿1, ||x(2) - x(1)|| < 61, ||y(2) - y(1) || < ¿1, ¿1 = ¿1(e,H1,H2) > 0,
Hg(t,T,x)H < m2(H1) V(t,T,x) £ S x Dx, |g(t2,T2,x(2)) - g(t1 ,T1 ,x(1) )|| < e V(t1,T1 ,x(1)), (t2,T2,x(2)) £ S x D1: (1.3)
112 - t11 < 62, |T2 - T1I < 62, ||x(2) - x(1)|| < 62, 62 = 62(e, H1) > 0.
Assumption 2. Assume that for all (t,T,x) £ S x D1 the function g(t,T,x) satisfies the following inequality:
0
Hg(t,T,x)H < go (t - t,H1), J go(v,H1)dv < ma(H0 = const. (1.4)
—<x
Assumption 3. Assume that the functions f and g satisfy the Lipschitz condition in the domains R x D1 x D2 and S x D2, respectively, where D1 and D2 are defined in Assumption 1, i.e., the following inequalities hold:
f (t,x(2),y(2)) - f (t,x(1),y(1))|| < L1(H1,H2)(|x(2) - x(1)|| + ||y(2) - y(1)||)
V(t,x(2),y(2)), (t,x(1) ,y(1)) £ R x D1 x D2, (1.5)
0
Hg(t, t, x(2)) - g(t,T,x(1) )|| < g1(T - t,H0||x(2) - x(1)||, J g1 (v,H1)dv < m3 (H1) = const.
Statement 1 ([23]). Let Assumptions 1, 2 and 3 hold. Then for each initial point (t0, x0) E R x D there exists a unique solution x = x(t, t0,x0) (x(t0,t0,x0) = x0) of Eq. (1.1) defined on some interval [t0,a) and such that it depends continuously upon the initial data (t0,x0).
Statement 2. Let x = x(t,t0,x0) be some solution of Eq. (1.1) bounded by the compact domain D0 = {x E Rn: ||x|| ^ H0 < H} for all t ^ t0. Using conditions (1.2) and (1.4), one can find that for all ti,t2 E [t0, the following inequality holds:
||x(t2,t0,x0) - x(ti,t0,x0)|| ^ mi(H0,m3(H0))|t2 - ti|.
(1.6)
Proof of Statement 2. Indeed, using (1.4) for the function
t
y(t,t0) = y g(t,r,x(r,t0,x0))dr,
to
one can obtain the following estimate:
t
J Hg(t,r,x(r, t0,x0))Hdr
to
||y(t,t0)| <
<
t 0
Jg0(t - t,H0)dv ^ J g0(v,H0)dv ^ m3(H0).
to
Accordingly, using (1.2) for the solution x = x(t,t0,x0) of Eq. (1.1) for all t1,t2 E [t0, we have
||x(t2,t0,x0) - x(ti,t0,x0)| ^
t2
Hf (t, t, x(t, t0 ,x0),y(T, t0))|dt
ti
< mi(H0,ms(H0))|t2 - ti|.
Remark 1. It can be easily seen that for all H0 = const, 0 < H0 < H, the set M0 of the solutions of Eq. (1.1) defined as
Mo = {x = x(t,to,xo): ||x(t,to,xo)|| < H0 <H Vt > to} (1.7)
is a set of bounded and uniformly continuous functions with the uniform continuity estimate (1.6).
Let F and G be sets of all continuous functions defined in the domains R xD x Rn and S xD, respectively, with values in Rn. Define the convergence in the spaces F and G as a convergence in the compact open topology, i.e., a uniform convergence on any compact sets Ki C R x D x Rn and K2 C S x D, respectively. That convergence is metrizable and the spaces F and G are complete [53].
Using conditions (1.2) and (1.3), we can find that the families of translates
F(i) = {f(s)(t,x,y) = f(t + s,x,y), s E R}, G(i) = {g(s)(t,T,x) = g(t + s,T + s,x), s E R}
are precompact in the spaces F and G, respectively.
Define the closures [F(i)] and [G(i)]. The functions f(i) e [F(i)] and g(i) e [G(i)] satisfy conditions such as (1.2), (1.3), (1.4) and (1.5).
Let Co be a set of the functions y = y(t) defined and continuous on t £ (-to,0]. For some sequence of numbers {rj} such that 0 < r1 < r2 < ... < rj < ..., rk — H for k — to, define the sets Kj C Co of the functions y = y(T) such that for all t, t1 and t2 the following inequalities hold:
|y(T )| < rj, ^y(T2) - y(T1)| < m1(rj ,m3(rj)) |t2 - T1|.
One can easily see that each set Kj is a compact.
oo
Note that the set r = |J Kj with the norm |||y||| = sup(y(s), -to < s ^ 0) is a complete j=1
separable Banach space.
For each pair (f,g) £ [F(1)] x [G(1)] define the functional-differential equation with infinite delay
-H-............) -
Note that the definition domain of Eq. (1.8) is R x r.
Denote by xt £ Co (t £ R) the part of the function x: R — D defined as xt(s) = x(t + s), -to < s ^ 0. For each initial point (a, y) £ R x r the solution x = x(t,a,y), xa(a,y) = y of Eq. (1.8) is unique and depends continuously on the initial data (a, y) £ R+ x r.
Let F* and G* be the shells of F(1) and G(1), respectively. Compose the pairs (f*,g*) of the functions f * £ F* and g* £ G* defined by the same sequence tm — +to.
For Eq. (1.1) define the family of limiting integro-differential equations as follows:
x(t) = f* ^t,x(t), j g*(t,T,x(T))dT), (f*,g*) £ F* x G*. (1.9)
Definition 1. Let x = x(t,t0,x0) be some solution of Eq. (1.1) defined for all t ^ t0. The point p £ D is said to be a positive limit point of that solution if there exists a sequence tk — +to such that x(tk,t0,x0) ^ p. The set of all positive limit points is said to be a positive limit set w+(t0,x0).
Definition 2. Let x = x(t,t0,x0) be some solution of Eq. (1.1) defined for all t ^ t0. The function ^ £ r is said to be a positive limit point of that solution in r if there exist two sequences tm — +to and Tm — +to such that x(tm + s,t0,x0) — ^(s) as m — to uniformly on s £ [-Tm; 0]. The set of all positive limit points ^ £ r is said to be a positive limit set Q+(t0,x0).
The following property of the sets w+(t0,x0) and (t0,x0) holds.
Theorem 1. Let x = x(t,t0,x0) be some solution of Eq. (1.1) bounded by some compact set K C D for all t ^ t0. Then for each positive limit points p £ w+(t0,x0) and ^ £ Q+(t0,x0) there exist both the limiting equation (1.9) and its solution x = x(t, 0,^) (t £ R) such that ■0(0) = p, {x(t, 0,0), t £ R}C u+(to,xo) and {xt(0,0), t £ R}C Q+(to,xo).
Proof of Theorem 1. Let p £ w+(t0,x0) and 0 £ Q+(t0,x0) be positive limit points defined by the sequence tk — +to, i.e., x(tk,t0,x0) — p and x(tk + s,t0,x0) — 0(s) as k — to. From the sequence {tk} choose the subsequence {tkj} such that the sequences {f(kj)(t, x,y) = = f(tk + t,x,y)} and {g(kj)(t,T,x) = g(tk. + t,T + tk.,x)} converge to the functions
f *: R x D x Rn and g*: S x D — Rn in the spaces F and G, respectively. For simplicity, denote by {tk} the sequence {tkj }.
From (1.6) using Remark 1, one can find that the sequence of the functions x (k) (t) = = x(t + tk, to, xo) is uniformly bounded and equicontinuous on t £ [—Ti; Ti], where Ti = ti — t0. Therefore, there exist both the subsequence {x(kl\t)} and the function x = x*(t), —Ti ^ t ^ Ti such that x(kl)(t) ^ x*(t) for t £ [—Ti;Ti]. From the sequence {x(kl\t)} choose the subsequence {x(km )(t)} which converges uniformly on t £ [—T2;T2], T2 = t2 — t0. Continuing that process further, one can find the subsequences t*m — Tm — and the func-
tion x = <(t) such that the sequence {x(m)(t) = x(t + t*m,t0,x0)} converges to x = <(t) uniformly on t £ [—Tm;Tm]. Since the function x = x(t,t0,x0) is a solution of Eq. (1.1), one can get the following equalities:
x(m)(t)
x(m)(t)
x(m)(t)
m+t
x(tm)+ J f ls,x(s),Jg(s, r, x(t))dr I ds,
to
t*m+s
x(tm )+ / f! t*m + s,x(t*m + s), [ g(t*m + s,r,x(r))dr j ds, 0 to
t I" \
x(tm) + J f(m) s,x(m)is) J g(m)(s,r,x(m)(r))dr | ds.
0 V
(1.10)
to-tm
s
t
t
Hence, passing to the limit in (1.10) for m — ^ and using (1.2), (1.3) and (1.4), we obtain <(t) = P + J f * |s,<(s), j g*(s,r,<(r))dr j ds. (1.11)
Differentiating equality (1.11), we find that the function x = <(t), t £ R is a solution with the initial data (0,^) of Eq. (1.9), satisfying the initial condition ^(s) = <(s) for s £ Rand ^(0) = <(0) = p. Obviously, we obtain {x(t, 0,^) = <(t), t £ R} C w+(t0,x0) and {xt(0, ip) = ip(t + s), s £ R~, t £ R+} C n+(t0,x0). m
Remark 2. Using Theorem 1, one can easily see that the sets w+(t0,x0) and n+(t0,x0) are the unions of solutions of all Eqs. (1.9).
1.2. The theorem on a positive limit set localization
Let Vi £ Ci(R x D x R — R) and V2 £ Ci(S x D — R) be some functions defined along the solution x = x(t), t ^ t0 of Eq. (1.1). Define the Lyapunov functional candidate V(t,xt) as follows:
V(t,xt) = Vi It,x(t),jv2(t,r,x(r))dr j (1.12)
to
with the time derivative V(t, xt) such that
\ dt V dx J y={ g(t,T,x(T))dT
to
v= f V2(t,r,x(r))dr
where (•)' denotes the transpose operator.
Assume that the time derivative V(t, xt) satisfies the following inequality:
V(t,xt) < -W1 ^t,x(t), J' W2(t,T,x(T ))dT) < 0, (1.13)
where W1: R x D x R — R+ and W2: S x D — R are some functions satisfying conditions such as (1.2), (1.3) and (1.4).
Hence, in particular, for all (t,T,x) £ S x D2 the following inequalities hold:
t
W2(t, T, x) < Wo(t - t, H2), J Wo(v, H2)dv < ms(H2) = const. (1.14)
—o
Introduce the families of limiting functions {W*(t,x,y)} and {W*(t,T,x)}. Hence, in particular, for each continuous function x: R — D2 there exists an integral
W2(t,r,x(r ))dr.
Introduce also the limiting aggregate (f *,g*, Wf, Wf) defined by the same sequence tm —
Theorem 2. Assume that
1) One can find afunctional V (t,xt ) defined by the formula (1.12) with the function V\(t,x,v) bounded from below, i.e., V\(t, x, v) ^ m(K) for all (t,x,v) G R x K x R, where K C D is some compact set. The time derivative of the functional (1.12) along the solution of Eq. (1.1) satisfies inequalities such as (1.13) and (1.14).
2) The solution x = x(t,t0,x0) of Eq. (1.1) is bounded by the compact set K C D for all t ^ to.
Then the positive limit set (t0,x0) C D, as well as Q+(t0,x0) C r, is a union of the solutions x = y(t), t G R of all limiting equations (1.9) for all limiting aggregates (f *,g*, Wf, Wf) such that the following holds:
W* ft,p(t), j W2*(t,r,<p(r))dr j =0. (1.15)
o
t
Proof of Theorem 2. The function V(t) = V(t,xt(t0,x0)) is bounded from below and monotonically decreasing along the solution x(t) = x(t,t0,x0) of Eq. (1.1). Therefore, the following holds:
lim V(t) = V* = const. (1.16)
t—> + ^0
Let p e w+(t0,x0) be a positive limit point of the solution x = x(t,t0,x0) defined by the sequence tk — Using (1.13), for all t e [0;tk —10] one can find that
tk +t / s \
V (tk + t) — V (tk — t) < —J Wi I s,x(s), Jw2(s,r,x(r ))dT I ds. (1.17)
tk —t\to J
According to Theorem 1, one can find both Eq. (1.9) and its solution x = ^>(t) defined by the sequences t*m — and T, — +<x>. Using (1.17), for all t e [t*m — T,; t*m + T,] we have
tk+t
(
V(t*m + t) - V (C - t) < -J W(T) s,x(m (s), j w2m) (s,T,x(m) (t ))dT
tk—t\ to-tm
\
ds < 0. (1.18)
/
For a limiting pair (f *,g*) define the limiting aggregate (f *,g*,W1*,W2*). Hence, passing to the limit in the relation (1.18) for t, — and using (1.16), we obtain equality (1.15) for all t e R. U
s
2. On the solution to the stability problem
Assume that in Eq. (1.1) the following holds: D = {x e Rn: ||x|| < H, 0 < H < f (t, 0,0) = 0 and g(t,s, 0) = 0 for all (t,s) e S. Therefore, Eq. (1.1) has a zero solution x(t,t0,0) = 0. We will use the standard definitions on the stability and asymptotic stability as well as uniform stability and uniform asymptotic stability of the zero solution x = 0 of Eq. (1.1) [48].
Denote by a: R+ — R+ a function of Hahn type [48], i.e., a^(0) = 0 and a is continuous and strictly monotonically increasing (i = 1,2). Introduce the following domains: D1 = {x e D: ||x|| < Hi < H}, Di x Rn and S+ = {0 < t < t<
Using Theorem 2, one can obtain the following result.
Theorem 3. Assume that one can find a functional (1.12) with a function V1 = V1(t, x,^) such that
1) V1 (t, x, v) ^ a1(||x||) V(t, x, v) e R+ x D1 x R+;
2) the time derivative VV1(t,x,v) satisfies inequality (1.13).
Assume also that for any limiting aggregate (f *,g*, W*, W2*) there are no solutions x = ^>(t) (t e R) of Eq. (1.9) satisfying the equality
W
*
t,v(t), J W2(t,T,<p(T))dT I = 0 (2.1)
except for the zero solution x = 0. Then the zero solution x = 0 of Eq. (1.1) is asymptotically stable.
Proof of Theorem 3. Using conditions 1 and 2 of the theorem one can easily obtain the stability property of the zero solution x = 0 of Eq. (1.1). From (2.1) using Theorem 2, one can conclude that for each bounded solution x = x(t,t0,x0) of Eq. (1.1) the following holds:
lim x(t, t0,x0) = 0. t—>+^0
In other words, the zero solution x = 0 of Eq. (1.1) is attractive. Thus, we obtain the asymptotic stability property of the zero solution x = 0 of Eq. (1.1). ■
Assumption 4. Assume that the function V2(t,T,x) satisfies conditions such as (1.4), i.e., V2(t, t, x) is uniformly continuous on (t,T,x) £ S x K (K c D) and the following inequality holds:
0
\V2(t,r,x)\ ^ v(t - t,K) V(t,r,x) £ S x K, J v(v)dv <
—oo
Remark 3. Using Assumption 4, one can easily see that the family of translates {V^ (t,r,x) = = V2(t + v,r + v,x), v £ R} is precompact in some functional space {Vj^} and for each function <p : R- ^ K the integral
t
J V2(t,r,V>(r))dr
is convergent.
Theorem 4. Assume that one can find a functional (1.12) with the functions V\(t,x,v) and V2(t,r,x) such that
1. Assumption 4 holds and the following inequalities hold:
ai(||x||) < Vi(t,x,v) ^ a2(||x|| + \v\) V(t,x,v) £ R+ x Di x R.
2. The time derivative V(t,xt) satisfies inequality (1.13).
3. One can find at least one limiting aggregate (f *,g*,V2*,Wr1*,Wr2*) such that for any c = = c0 = const > 0 there are no solutions x = ^>(t) of Eq. (1.9) satisfying the relationships
+
V2*(t,r,V(r ))dr
> Co
and
W it,p(t), j WZ(t,r,<p(r))dr j = 0.
Then the zero solution x = 0 of Eq. (1.1) is uniformly stable and asymptotically stable uniformly on x0.
Proof of Theorem 4. For any e > 0 choose 5 = ¿(e) > 0 such that 5 < a-1(a1(e)). Let t0 ^ 0 be some initial time instant. Using (1.13), for each solution x = x(t,t0,x0) of Eq. (1.1) with initial point ||x0|| < 5, we obtain the following inequalities:
ai(||x(t,t0,x0)||) ^ V(t,xt(t0,x0)) ^ V(t0,x0) ^ a2(|x01|) < ai(e)
t
for all t ^ t0. Thus, it follows
||x(t,t0,x0)|| < e t ^ t0.
Hence, we obtain the uniform stability property for the zero solution x = 0 of Eq. (1.1).
Let x = x(t, t0,x0) be a solution of (1.1) bounded by the domain D1 = {||x|| ^ H1}. Defined by the sequence tk — the aggregate (f *,g*, V2*, Wf, W2*) is limiting due to condition 3 of the theorem. According to Theorem 2, the sequence of translates {x(k)(t) = x(tk + t,t0,x0)} converges to the solution x = ^>(t) of Eq. (1.9) as tk — such that the following equality holds:
Wf lt,^(t), y Wf(t,r,V(r))dr j = 0.
Notice that the functional V(t,xt(t0,x0)) is a time function V(t) along the solution x = = x(t, t0,x0) of Eq. (1.1). Let us prove that V(t) — 0 as t — Assume the contrary, namely, that the following holds: V(t) ^ Vo = const > 0 for all t ^ t0. Then, according to the inequality V\(t,x,v) ^ a2(||x|| + |v|), we obtain
|x(t)|| +
V2(t, r, x(r))dr
to
^ C0 = a-1(V0) > 0.
Thereafter, we obtain
|x(tfc +t)|| +
tk+t
V2 (tk + t,r,x(r ))dr
to
> C0
for the sequence tk — and for all t > 0. In other words, the following inequality holds:
|x(k)(t)|| +
t
J V2(k)(t,r,x(k) (r))dr
o-tk
> C0.
In particular, for the solution x = ^>(t) of Eq. (1.9) this yields
t
+
Vf(t,r,V(r ))dr
^ C0 > 0.
This contradicts condition 3 of Theorem 4.
We therefore conclude that for the solution x = x(t,t0,x0) of Eq. (1.1) the following holds: V(t) = V(t,xt(t0,x0)) \ 0 as t — Using the property of the continuous dependence on initial data .To for the solutions of Eq. (1.1), we complete the proof. ■
Theorem 5. Let conditions 1 and 2 of Theorem 4 hold. Assume that condition 3 of Theorem 4 holds for each limiting aggregate (f *,g*, Vf, W-f, Wf). Then the zero solution x = 0 of Eq. (1.1) is uniformly asymptotically stable.
The proof of Theorem 5 is similar to that of Theorem 4 with the particularity that V(t) = = V(t,xt(t0,x0)) \ 0 as t — uniformly on (t0,x0) £ R+ x {x0 £ D : ||x0|| < 5}.
t
3. On the equilibrium position stability of a hereditary mechanical system
The motion of a holonomic mechanical system between the points of which the hereditary elastic interaction takes place is described by n generalized coordinates and can be represented by Lagrange equations in the form [57]
t
d (d)T\ dT dn . f. . . . , . .
M (w) - -q = --q - - J *<«• T>"{T)dT + {iA>
where q = (51,52,...,qn)' is the vector of generalized coordinates, T = q'A(q)q/2 is the kinetic energy of the system, n = n(t, q) is the potential energy of the external forces, n £ C2, Q = Q(t, q, q) is the vector of nonpotential generalized external forces, the force Q consists of dissipative and gyroscopic components, i.e., Q(t,q, 0) = 0 and q'Q(t,q,q) ^ 0, the matrices B £ Rnxn and $ £ Rnxn characterize the relaxation (viscoelastic, hereditary elastic) properties of the system. According to the physical meaning of these properties, we assume that the matrix $(t, t) satisfies the following inequalities:
0
||$(t,T)|| < $o(t - t), J $o(v)dv <
T)X < 0, X > /3(T - i)|M|2 V.r e Rn,
0
where the scalar function /: R- — R+ is such that 3(v) > 0 Vv £ R- and f /(v)dv <
Assume that dn/dq = 0 for q = 0. Thus, Eq. (3.1) has a zero equilibrium position q = q = 0.
Assume also that all the conditions imposed on Eq. (1.1) hold for Eq. (3.1). Thereby, for Eq. (3.1) one can find limiting equations such as
t
d fdT\ dT _ <911* dt \dq J dq dq
(dT\ dT dn* {
{-dj)-di = -W~ m)q" J **{t'T)q[T)dT + g*(i'q'(3'2)
Introduce the following matrix and functional:
t
$i(t) = | $(t,T)dT, Bi = B + $1, V = T(q,q)+n(t,q)+ni(t,qt),
0
t
ni(i, qt) = q'(t)Bi(t)q(t)/'2 - ± J(q(t) - q(T))'$(t, r)(q(t) - q{r))dr.
Remark 4. The functional ni(t, qt) was defined in [57] as a potential energy of hereditary forces. One can find the time derivative of the functional V along the trajectories of Eq. (3.1) as
follows:
V(t, qt, №) = №Q{t, q(t),q(t)) + ^ + \ ?(t) -
\ f (Q(t) - q(r)y^f±(q(t) - q(T))dr < | (ll(i, 5) + \
Statement 3. Assume that the potential function n0(t, q) = n(t, q)+q'B1(t)q/2 is definite positive, non-increasing on t and has a nondegenerate isolated minimum at the point q = 0, i.e., no(t,q) ^ ai(||q||), dn(t,q)/dt < 0 and ||dno/dq|| ^ a2(||q||) for all small ||q||. Then the equilibrium position q = q = 0 of the system (3.1) is uniformly asymptotically stable.
Proof of Statement 3. One can easily see that for the functional V all conditions of Theorem 5 are satisfied. For the time derivative of the functional V one can obtain the following estimate:
t
V(t, qt, q(t)) < ~W(t, qt) = j P(r - t) ||g(i) - q(r) ||2dr < 0.
o
The functional limiting to W(t, qt) is such as
t
1
W*(t,qt) = --2 J (3(t - t.)\\q(t.) - q{t)\\2d,t.
Hence, one can conclude that no solution of the limiting system (3.2) can stay forever in the set {W*(t,qt) = 0} = |q(r) = q(t) Vt ^ t}, other than the trivial solution q = q = 0. Finally, using Theorem 5, we complete the proof. ■
Remark 5. In the classical case, when the matrix $(t, t) is such as $(t, t) = $(r —t), the function n does not depend on time, n(0) = 0 and B = const, Statement 3 has the following form. Assume that the effect of the hereditary forces is such that
x'$(r - t)x < 0, x'(B + $0)x > 0, (=0 ^ x = 0)
0
ds
$?= / $(s)ds, x'^^x ^-/3{s)\\x\\2, /3{s)> 0 Vs > 0. ds
Then the equilibrium position q = q = 0 of the system (3.1) is uniformly asymptotically stable.
4. On the position stabilization problem of a controlled mechanical system using a nonlinear integral regulator
4.1. Problem solution
Consider the controlled holonomic mechanical system given by the Lagrange equation
d (dT\ dT ^ TT
dt\lM)-^ = Q + U> (41)
where q £ Rn denotes the generalized positions, T = q'A(q)q/2 is the kinetic energy of the system, A = {ajk(q)}, A £ Rnxn is the inertia matrix, Q = Q(t, q, q) is the vector of generalized uncontrolled forces, and U is the vector of generalized control forces. Represent the force Q as
Q(t,q,q)= Q1(t,q)+Q2(t,q,q), (4.2)
where Q1(t, q) = Q(t, q, 0) and Q2(t, q, q) = Q(t, q, q) - Q1(t, q).
Note that the component Q2(t, q, q) in (4.2) represents the dissipative and gyroscopic forces, i.e., q'Q2(t,q,q) ^ 0.
Consider the dynamic feedback controller
t
U = Ui(t, q) - J R(t, v )q(v )dv, (4.3)
0
where the function U1(t,q) is such that the following equalities hold:
Ul(t,q)+Ql(t,q) = -^^, II G C1, n(i, 0) = 0, ^(i,0) = 0,
dq dq
where n G C 1([0, +œ) x Rn — R) is the potential energy.
Assume that the function n = n(t, q) is non-increasing in time, i.e., dn(t,q)/dt ^ 0. Assume also that the matrix R(t, v) (R G C 1(R+ x R- — Rnxn)) and its derivative dR(t, v)/dt satisfy the following conditions:
0
a1(v - t)||x||2 ^ x'R(t,v)x ^ a2(v - t)||x||2, J ak(s)ds < k = 1,2, (4.4)
dR(t, v) = w(/)/,(/ ^ ^qII^H2 <; x'R0(t)x < [h\\x\\2, /30 = const, >0, pi = const > 0,
dt
x' (//,,!/).\/i/) + M(t)Ro(t) - x < -ßo\\x\\2, R0(t) = R(t,t).
In view of (4.4), for the time derivative of the functional
t
I
V = i q'A(q)q + II(i, q) + ^ I J R(t, v)q{v)dv R^{t) J R(t, v)q{v)dv
y0
one can find that
t -i, ^
V = q'Q2(t,q,q) + ^^ + -2 j R(t,u)q(u)du\ ^^ ( [ R(t,u)q(u)du\ +
t
+ IJ R(t,u)q(u)du j RöHt) \^Ro(t)q(t) + J -q'(t) J R(t,u)q(u)du <
< R(t,u)q(u)d^ (^j R(t,u)q(u)d^ +
+ ^ j' R(t, v)q(v)dvj R-i(t)M(t) ^ j R(t,v)q(v)dvj =
t \'
RöHt) J R(t, u)q(u)du + //.. • / ) A / i / • + M (t) Ro(t)^j x
0 J
t \ t
x l RöHt) j R(t, v)q{v)dv j <-A j R(t,u)q(u)di
0
t
t
The functional limiting to W(t, qt) is
W*(t, qt) =
R*(t,v )q(v )dv
On can easily find that {W*(t, qt) = 0} = {q(v): q(v) = 0, —to < v < t}. In accordance with Theorem 5, one can obtain the following result.
Statement 4. Assume that the potential function n = n(t, q) is such that the following inequalities hold:
a1(||q||) < n(t, q) < a2(HqH),
dn(t, q)
dq
^ 0(e) Vq £{0 < e < M < A}
and conditions (4.4) are valid.
Then the controller (4.3) solves the stabilization problem of the zero position q = q = 0 of the system (4.1) providing its uniform asymptotic stability.
Remark 6. Let the equality R(t,v) = R(v -1) hold. Then we get R0(t) = R(0) = R0. In such a case, if we choose the matrix R(s) according to the condition dR(s)/ds = MR(s), then conditions (4.4) can be written as
x'(R0M + MR0)x > ^0||x||2, f30 = const > 0.
2
t
4.2. On the Position Stabilization Problem of a Mobile Robot with Three Omniwheels and a Displaced Mass Center
Let us consider the model of a mobile robot moving without slipping on a horizontal plane [38]. Following [38], we assume that Oxyz is a fixed coordinate system associated with a horizontal reference plane Oxy, the axis Oz is directed vertically upwards, C is the the center of an equilateral triangle at the vertices of which the centers of the robot wheels are located, Cx1y1z1 is a moving coordinate system rigidly attached to the robot platform, the axis Cx1 is parallel to the axis of rotation of the first wheel, the axis Cz1 is vertical, the axes Cy1, Cz1 and Cx1 form the right-handed orthogonal triad, and the planes Oxy and Cx1y1 are parallel.
The robot is considered as a system of four absolutely rigid bodies which are a platform and three wheels. The mass and dimensions of the rollers are neglected.
The distances from the center of the platform to the center of each identical wheel are equal
to a, r is the radius of the wheels. The mass center of the system C0 is displaced relative to the
-►
center C by a distance d. The angle between the axis Cx1 and the vector CC0 is denoted by a.
Assuming that the rollers of all wheels move without slipping, the motion of the entire system is determined by the time variation of the three coordinates which are two Cartesian coordinates x and y of the center C in a fixed system Oxyz and the angle of rotation of the platform around the vertical measured from the axis Ox. In accordance with [38], let m0 and m1 be the mass of the platform and the mass of robot wheel, respectively, m = m0 + 3m1. Let also Is be a reduced inertia moment of the system.
Y
O
X
Fig. 1. Omnimobile robot scheme.
The equations of motion of the robot in the variables x, y, y presented in [38] can be written as
mx — m0d sin(a + y)y — 3m1yy — m0dy2 cos(a + y) = = i ^sin 0Mi + sin(tp + ^-)M2 + sin(0 + ^L)M3 my + m0d cos(a + y)y + 3m1xy — m0dy2 sin(a + y) = (4.5)
= ^ cos *l>Mi ~ cos(^ + ^f)M2 ~ cos (if) +
a
— rnods'm(a + ip)x + rnodcos(a + ip)y + Isy = —(Mi + M2 + Ms),
where Mj is the torque applied to the jth wheel.
Consider the stabilization problem of the robot's position
x = x(0) = const, y = y(0) = const, y = = const.
In the case of absence of control torques, the existence of the energy integral makes it possible to apply Statements 3 and 4 to the stabilization problem. In accordance with those statements, one can obtain the following stabilizing control torques:
(
M = P(tp)Mo, P(4>) = -
sin y — cos y
sin (y + ^y j — cos (y +
sin (y + ^ j — cos (y +
l N
2 a
2TT\ 1
t) 2 a
4n \ 1
-) 2a J
where M0 = (M01 ,M02 ,M03)' is such that
M0k = —fkyk + l»hj eak(T t]yudT I, fkak > fxk, k = 1,2,3,
(4.6)
or
Mok = -9k Vk - Ilk J (T "t) yd I, k = 1, 2, 3, (4.7)
gk, ak, f3k, ¡ik and Yk being some positive constants (k = 1,2,3) and y = (y1,y2,y3)T =
T
= (x — x(0), y — y(0) ,y — y(0))
Note that the regulator (4.7) may be applicable without velocity measurement. Indeed, applying the integration by parts formula, from (4.7) we obtain
M0k=—(fe+ik)yk—'^"yk(0))+hi^ykdT) <4-8)
The robot parameters are given as
m0 = 20 kg, m1 = 1 kg, Is = 0.48 kg • m2, a = n/6 rad, a = 0.25 m, r = 0.1 m, d = 0.05 m.
15 10 5 0
-5 -10
0 10 20 30 40 50 60
Fig. 2. The time response of the robot's positions and references for the controller (4.6).
The robot's desired position is chosen as
x(0) = 1 m, y(0) = 2 m, ^(0) = n/4 rad. (4.9)
The initial conditions are taken as
xo = x(0) + 50 m, yo = y(0) - 70 m, ^o = ^(0) - 2 rad, x0 = 25 m/s, y0 = -15 m/s, xp0 = 10 rad/s.
We fixed the control gains to
(4.10)
fi = 35, f2 = 35, f3 = 0.13, ¡¡i = 25, ¡¡2 = 25, ¡3 = 0.1,
ai = 2, a2 = 2, a3 = 1, gi = 5, g2 = 5, g3 = 0.13, (4.11)
7i = 20, 72 = 13, 73 = 0.1, A =3, fa = 3, & = 0.5.
In Figures 2 and 3 we show the robot's coordinates x, y and y as well as references using the controllers (4.6) and (4.7). From these results, it can be seen that controllers (4.6) and (4.7) provide smooth fast convergence of the robot's posture to the reference.
60 40 20 0
-20 -40
0 10 20 30 40 50 60
40 20 0 -20 -40 -60 -80 -100
0 10 20 30 40 50 60
8 6 4 2 0 -2 -4 -6
0 10 20 30 40 50 60
Fig. 3. The time response of the robot's positions and references for the controller (4.7).
5. Conclusion
In this paper, we construct the topological dynamics of a nonlinear Volterra integro-differential equation. It allows one to find the important properties of their solutions including the property of quasi-invariance of the positive limit set of each bounded solution. The dynamics constructed in the paper has some specific features in comparison with well-known constructions of RFDE from the works [18, 19, 26, 27, 49, 50, 52]. These features are based on the specific conditions (1.2)-(1.5) with respect to the right-hand side of IDE as well as on a definition of the functional space of limiting equations with infinite delay and also on a special deduction of the quasi-invariance property. It is well known that the main method of stability analysis of nonlinear differential equations is the direct Lyapunov method. The absence of a universal technique for finding the Lyapunov functions and functionals stimulates numerous studies on the development of this method in the direction of the extension of the traditional Lyapunov theorems on various types of equations including RFDE [11, 13-15, 20, 27, 31-33, 37, 49] and their modifications, and the development of algorithms for constructing them for specific classes of various types of equations. An effective approach to the study of various theoretical and applied problems on limiting behavior and asymptotic stability is the development of the direct Lyapunov method in the direction of using Lyapunov functions and functionals with a semi-definite time derivative [3-6, 18, 21, 22, 24, 25, 30, 35, 36, 40, 44, 46, 47, 58]. The theorems obtained in this paper are the development of these results for nonlinear integro-differential equations. As already noted in the Introduction, one of the factors of an intensive study of Volterra integro-differential equations consists in their application in modeling the hereditary problems of mechanics [17, 45, 57]. Sufficient conditions for uniform asymptotic stability of the equilibrium position of the hereditary mechanical system are found in the paper taking into account the action of external forces.
The development of the Lyapunov functionals method in the study of the solution stability of nonlinear IDEs allows us to justify new types of nonlinear regulators with integral components for solving automatic control problems. The problem of equilibrium position stabilization for a holonomic mechanical system is solved by constructing the regulator of a new type. As an example, the problem of position stabilization of a three-wheeled mobile robot is solved by means of the new regulators constructed in the paper.
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