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Periodic and almost periodic solutions for a class of systems described by coupled delay-differential and difference equations
NonlinearAnalysis, Theory, Methods& Applications,Vol. 1NO.3, pp. 197-206 PergamonPress,1977. Printedin GreatBritain.
PERIODIC AND ALMOST PERIODIC SOLUTIONS FOR A CLASS OF SYSTEMS DESCRIBED BY COUPLED DELAY-DIFFERENTIAL AND DIFFERENCE EQUATIONS
A . HALANAY Bucharest University, Faculty of Mathematics, 70109 Bucharest 1, Academiei 14, Romania and VL. R~.SVAN Institute for Power Equipment Research and Design, 74369 Bucharest IV, Vitan 236, Romania (Received 12 July 1976) Key words: Delay differential equations, difference equations periodic solutions, almost periodic solutions
1. I N T R O D U C T I O N
Trm COUPLEDdelay-differential and difference equations occur when studying transient processes for control systems containing water, steam or gas pipes, and also for electrical circuits with lossless LC transmission lines (see e.g.R.K. Brayton [1], C. Y. Ho and R. J. P. de Figueredo [2], A. N. Willson Jr. [3], VI. Rasvan [4], [5]). In the above references linear or nonlinear systems are considered, linear system stability and absolute stability of nonlinear systems are studied by frequency domain methods or by the method of Lyapunov function. The stability problem occurs for systems without forcing terms (e.g. electrical systems without sources or feedback control systems). If one considers, for instance, electrical systems with lossless LC transmission lines containing sources, the main problem of the electrical engineer is to find both the stationary and the transient currents and voltages of the electrical network. Mathematically speaking, one has to obtain the stationary and transient solutions (in most cases only the stationary one) of a dynamical system with forcing term which can be constant (for d.c. sources), periodic (e.g. for a.c. sources) or almost periodic (for modulated sources). From the mathematical point of view it is important to obtain existence conditions for such solutions. The existence of stable periodic and almost periodic solutions has also a practical value. A practician would however claim that these solutions always exist because he is speaking about their physical reality. One must in fact avoid confusion between the physical system and its mathematical model which can be correct or not. Therefore when a mathematical model is used existence theorems are needed to check its correctness. 2. A N E X A M P L E . T H E M A T H E M A T I C A L
PROBLEM
Consider for instance the electrical circuit from the paper ofR. K. Brayton and W. L. Miranker 197
198
A. HALANAYAND VL. RASVAN
[6], described by the following equations: av 0i ai --= L --= 02
at'
C
02
OV
at
v(O, t) = - E ( t ) - Roi(O, t),
(2.1)
v(l,O = vl( 0
CxOl(t) = - d/(vl(t) ) + i(l, t) v,(0) = v °,
i(~, 0) = io(~),
v(t, o) = Vo(~),
0 ~< ~ ~ t
These equations clearly define a mixed initial-boundary value problem for hyperbolic PDE. Using the transformation: v(2, t) = u1(2, t) + u2(2, t) i(2, t) = ~/C/L [u,(A, t) - u2(2, t)] the equations can be written in normal form and (2.1) become au 1 + 1 ~ul = O, ~u2 1 ~u2 = 0 ul(l, t) + u2(l, t) = Vl(t)
(1 + R o ~ - L ) Ux(0,t) + (1 - Ro C ~ ) C lbl(t) = - ~k(vl(t)) + ~ Vl(0) = %o
u2(0, t) = - E ( t )
[ul(/, t) - u2(l, t)]
(2.2)
.,(~, o) = ~[Vo(~) + ~J-V-CTo(~)], u2(~, o) = ~[Vo(~) - ~ ( , ~ ) ]
o .< ,~ ~ t
By integrating along the characteristics and denoting Ux(0, t) = rh(t), u2(l, t) = ~/2(t) one can find a one-to-one correspondence between the solutions of the mixed problem (2.2) and the solutions of the system
C16,(t) =
- v , ( t ) - O(Vl(t)) +
,/x(t)=
l1- a+-o ~ ~ c
2~/C/Ltl,(t -I~/LC)
r/2t"t - lx/L'-C)
(2.3)
,1~(0 = ~1(0 - 'h(t - &/L-~)
with theinitial conditiOnSo t/°(t) =~1[ v° ( ~ / ~ C )
+ x/L---~i° (
x/~CC)]
v,(0) = v 1, ~(t) = ~1[ Vo ( l +
x/~)--x/~/Ci°(
l+ x/~)]
-Ix/-~'t'O
Consequently a mixed initial-boundary value problem for hyperbolic PDE has been reduced to a Cauchy problem for a system of the type £(t) = Aox(t) + A l y ( t - h) - bo~(v(t)) + f ( t ) y(t) = A2x(t) + A3Y(t - h) - b~rk(v(t)) + g(t) v(O = c*x(O
(2.4)
Solutions for a class of systems
199
where f(t) and O(t) are the forcing terms. (Such reductions for more general mixed problems for hyperbolic P D E can be found in Abolinia and Myshkis [7], Cooke and Krumme [8] or Cooke [9]. They lead to various types of functional differential equations). It is now obvious that the problem of periodic and almost periodic solutions for the mixed problems can be reduced to the same problem for the corresponding FDE. Throughout the paper we shall study the problem of the periodic and almost periodic solutions for the system (2.4). 3. THE METHOD We shall follow in this paper the method used in a earlier paper of Halanay [ 10]. The formula of variation of constants (see Rhsvan [4]) enables us to write the system (2.4) as a couple of an integral and a difference equation. By usual techniques of the frequency method some evaluations of the solutions are obtained. Then use is made of a result which is essentially a special case of the general theorem of Kurzweil E11] on invariant manifolds. This result is the following: PROPOSITION 3.1. Suppose for a general flow in the Banach space ~ there exist real positive constants l, T, ~ < 1, kl, 2 such that (i)
II ~
II~ II ~< l,
t e [~ + T,
3 + 2T] imply IIc(t; ~, ~
l
(ii) II~lll-
I1~11 ~
t~[[+ T,t+ 2z]implyllc(t;3,~l)-c(t;[,~2)ll..<~ll~a-~2ll
(iii) IIc(t; t, ~ ) - c(t; 3, ~2)II <~ kl e;'('-~) IIc, - c2 I1for t >/t Then there exists p:R --* ~ such that (1°) IIp(t)II ~< l (2 °) p(t) = c(t; t l, p_Ctl)) for t I> t 1 and t 1 e R
(3°) II c(t; 3, ~) - p(t)II ~ koe- ~" -~ [I~ - ~ ) II for II~ II -< t, t >/~ where k o > 0, v > 0 (4 °) If c(t + f~; ~ + f~, ~) =_ c(t; t, ~), then p(t + f~) =- p(t) (5 °) If every sequence {h.}., h. --, oo contains a subsequence h.k such that for every bounded solution c(t; 3, ~) the sequence c(t + h.~ ; ~ + h.~, ~) converges uniformly on each compact subset of the half-axis t >/t, uniformly with respect to 3, ~, for ~ ~ R, I]~ II ~ l, then p is almost periodic. Remark that the condition in (4 °) is satisfied if the flow is defined by a periodic system and the condition in (5 °) if the flow is defined by a system which is almost periodic with respect to t. One must also mention that the existence of periodic and almost periodic solutions for the linearized system (2.4) (i.e. when b o = b 1 --- 0) also follows from the above proposition. Indeed the solutions of a linear system are always globally Lipschitz and the condition (iii) follows at once. The conditions (i) and (ii) are straightforward consequences of the exponential stability of the linear homogeneous system. All the assumptions of the proposition being fulfilled, the existence of periodic and almost periodic solutions is proved (in the linear case). 4. THE RESULTS We shall consider the case of several nonlinear elements i.e. the system Yc(t) = aox(t ) + AlY(t - h) - Bodp(v(t)) + f ( t ) y(t) = A2x(t) + A3y(t - h) - BFk(v(t) ) + e(t) o(O = C*x(O
(4.1)
200
A. HALANAYAND VL. R~SVAN
where ~(v) = col
(~,(vi))
The initial conditions for system (4.1) are given by x(0) = x o, y(t) = Yo(t), - h <~ t <<.O, where Yo ~ L 2 ( - h ' 0 ; RP). Define the following matrices: aI
H(cr)
Ao
-- Ale -*h ]
- A2
I-A3
e-oh
.J
(4.2)
where e is a complex variable. We can state now the following: THEOREM4.1. Consider the system (4.1) and suppose the following conditions are fulfilled: 1°. (a) det H(a) # 0 for all tr such that Re a ~< -0t, where 0t > 0 (b) The eigenvalues of A 3 are inside the unit circle. 2 °. 0 ~< ~ j ~ l ) - q~j(~t2)~< # ; °~1
--
~j(0) = 0 (j = 1. . . . , m)
~2
(The nonlinearities are globally Lipschitz) 3 °. There exist zj > 0 and 6 > 0 such that Zdltd 1 + Re zaY(ia 0 >1 JI
(4.3)
for all real 09, where z d = diag (z 1. . . . . z=),/zd = diag (/q . . . . . #=) 4 °. ] f ( t ) I <~ M for all t, I#(t) i ~ M for all t, Then there exists a bounded on the real axis solution of the system, which is periodic, almost periodic respectively if f, g are periodic, almost periodic respectively. Moreover, this solution is exponentially stable. Here Re G(iog) means the matrix ~[G(ico) + G*(i~o)] and for two symmetric matrices G 1, G 2 the inequality G t /> G 2 m e a n s that G 1 - G 2 is a positive semidefinite matrix. To make it clear we mention also that the zero matrix in Y(a) has its dimensions such that the products of matrices have sense. F o r systems with a single nonlinearity the frequency condition (4.3) (which is, in fact, the usual Popov condition with the Popov parameter set to zero, but written for the multivariable case) becomes 1 -+ #
Re~(io~)/>6>0
As an example we shall check the possibilities to ensure the conditions of Theorem 4.1 for system (2.3). First, by introducing the following nonlinear function:
~(~) = ( ~
-
r)~ +
¢(~)
where 7 > 0 is an arbitrarily chosen number, the system (2.3) takes the following form
201
Solutionsfor a classof systems y vl(t ) _ ~1 qb(vl(t)) + ~2 r/,(t - lx/L-'~ t3,(t) = - C--~
rh(t) = --otxt/2(t - Ix/'-L-C ) - E(t) tl2(t) = vx(t) - rl,(t - Ix/-L-'C )
where 1 - Rox/~-/L oq -
1 + Ro
C~,/-~
The LHS of the characteristic equation has the following expression detH(~)=
a+
~
(1--ele-2~l'/Lc)+ ~
After some simple manipulation one can find that condition (a) of Theorem 4.1 can be satisfied for ~<%
1 1 <2~ L ~ l n ] ~ l - -~
where % is the positive root of the equation 7
2
I ,l e2 'vLc
= 0
and ~ is chosen such that
The condition (b) of Theorem 4.1 is automatically satisfied because ]g,[ < 1. It remains to check the fulfilment of the frequency condition. The transfer function 3,(a) has the following expression: ] -- ~ l e - 2 a l ~ / L C
,(0")=( a + ~T) (1 _ otle_2#,~/L-~)+ ~20~. x/-C~Le-2~,,/g-d After some manipulation we get Re v(ico) = (1-o,
cos 2~/x/L-C)[~--~ + ~-~1 ot' (2C ~ -
ot2 y) cos 2og/,fL--c] + ~-~1(2x/-~~)sin 2 2colx/%-C ] det H(ioJ) l 2
It can be easily seen that if y is chosen in order that 24-d
I or, [ 2x/C/L 1 + I•t , I < r <
then Re 7(it0) t> 0 and the frequency condition (4.3) is fulfilled for/~ > 0 arbitrarily large. Therefore the existence of a forced regime is ensured for
A. HALANAYANDVL. RASVJ~N
202
_ I --l~tIICx/~-~ + I "~ 1~I ]
~ ~ ~(VI ) _ ~(V2 ) ~-~I, l~l -- V2
where e > 0 and e' > 0 are arbitrarily small. This is the same sector as the Hurwitzian one. For any nonlinearity belonging to this sector the free system (i.e. the system without sources) is absolutely stable (see Infante [12], where this fact was proved by Lyapunov methods, or R~svan [5], where the proof is done by frequency methods). In fact the system (2.3) has an "almost linear" behaviour (under usual assumptions, of course).
5. PROOF OF THETHEOREM A. Basic estimates
Let (x, y) be a solution of (4.1) defined for t >i t; by the variation of constants formula we can write x(t) = Z 1l(t -- t) X(t) +
-h
f:
Zl2(t - s) y(s) ds -
[Z 1l(t - s) B 0 + Z12(t
-
s)
B1] ~(v(s)) ds
[ z ~ l ( t - s ) f ( s ) + Zx2(t - s)e(s)] ds
+
hence v(t) = C * Z 1 l(t -
l)x(l)
+
C*Zx2(t - s) y(s) ds -h
-
f:
[ C * Z 1 l(t - s) B 0 + C*Z12(t - s) B1] qb(v(s)) ds
[ C * Z 1 l(t - s) f ( s ) + C*Z12(t - s) if(s)] ds
+
Denote ?(t) = v(t + /) and write down the nonlinear integral equation
?(t) = ~(t) - fl K(t - s) #~(~(s)) ds the same as in [10]. Here
fo ~(t) = C*Z11(t)x(t ) +
K(t) = C*Zll(t )B 0 + C*ZI2(t) BI
C*Z12(t - s)y(t + s)ds +
[ C * Z l l ( t - s ) f ( t + s)
-h
+ C*Z12(t - s)g(t + s)]ds
We need now to check the Laplace transform of K(t); we have [4]
g(0") = C*Zll(a)Bo +
C*Z12(a)BI=[C
=[C*
*
0IV ~11(0")
LaZ21(a)
2 1 2 ( o " ) ] [ B0] aZ22(a) j
O]H-l(a)[BB:]=
y(a)
B,
Solutions for a class of systems
203
The condition (4.3) in the statement allows us to write ~d#~ I + R e z ~ Y ( i o 9 -- p)>~-fi2 1
Y~o
for p > 0 sufficiently small. This follows from the uniform conitinuity of Y(a) for - ~t < Re tr ~< O, where 0t is the one in the statement. The above frequency condition allows us to obtain the same estimates as in [10]
, .o
< '
J
I;(01 ~< k,e-~"(Ix(~)l + IlY~II)+ ka.
0
< 2 <.
where t'o y~(s) = y(t + s ) a n d Ily~ll2 = | d-
[y~(s)12ds h
The inequality can now be written as
<~ [k,(lx(~)l
fr[•tq)j(Yj(t))]2dt
Ilyill)+
+
kseUr] 2
(5.1)
1 dO
F r o m here, as in [7], we get the main estimate
Ix(t)l <
k9 +
k,oe-"('-~'(Ix(~)l + Ily~ll)
(5.2)
In the same way, for two different solutions
[Xl(t) -
x2(t)l ~ K2e-"~'-~'([xl(D - x2(/)l + IIYI.~ - Y2.ill)
(5.3)
We need now the corresponding estimates for y. We have y(t) = A2x(t) + A a y ( t - h) - B~(p(v(t)) + g(t)
Denote for a while A 2 x ( t ) - Bltp(v(t)) + O(t) = F(t)
and write k-1
y(t) = A~y,(t - ~ - kh) +
~, A ~ F ( t - j h )
0 for~+ (k-
1)h~
kh, h e n c e k-2
y(t + s) = A k - l y ~ t + s - l - (k - 1)h)+
~ AJaF(t + s - j h )
0 for-h~s<~+
(k-
1 ) h - t , and k-1
y(t + s) = Akayt-(t + s - Z - kh) + ~, AJ3F(t + s - j h )
0 for~ + (k - 1 ) h - t ~< s < 0.Therefore
(5.4)
204
A. HALANAYANDVL. RASV~,N k-1
[y(t + s)[ ~< k,,6 k-1 ]yr(s)l + k,~ Z pJlF(t + s - jh) I
0
0
and k-1
[}Y,I[ ~< k~2P'-tllYz[I + k,2 ~, PJIIFt-jhll 0
where liFt[J2 = f ~ h IF(t+ s)[2ds
We have further:
IF(t) I <~ M + Mdx(t) I <<.kla + k,,e-""--°(lx(~) I + IIY~II) hence:
[IF,_jh[I <~kls + k16e-~(t-Jh-b(lx(3)[-t- [lY~[[) Consequently one can write k-I
k-1
0
0
[JY,[[ ~< k12pk-ll[Y~[] + k12k15 ~ PJ+ k,2k,6(lx(~)l + [[Y~[[e-~''-° ~ (Peuh)i and, by choosing # such that pe ~k < 1
Ily,II ~ k,~-~llY~ll
+ k,7 +
k~se-""-~'(Ix(3)l + IIY,II)
But, ifp e"h < 1, then, for 3 + (k - 1)h ~< t < 3 + kh, pk < e-~,-~). Therefore
[lYtl[ <~ k,7 + k,9e-U"-i'(lx(3)[ + IlY~II) In the same way the corresponding estimate is obtained for the difference of two solutions.
B. Application of Proposition 3.1 The system (4.1) will define a flow in the Hilbert space R n × L2( - h, 0; R p) with the usual norm; we shall write c(t; 3, ~) for (x(t; 3, xff), Yt(" )), Yfl, x(/), y~( • ))( • )), where y~e L2(-h, 0; RP). The first two definition properties of the flow c(~; t, ~) =
c(t2;tl, c(tl;3,~) ) = c(t 2 + t~;3, P),t 2 >/tl /> 3 follow at once from the definition of a solution of (4.1), and from the fact that this system satisfies the uniqueness conditions. The (uniform) continuity of c( •, ~, ~) follows from the fact that x(., 3, x(t), t,(" )) is a Caratheodory solution, hence it is absolutely continuous, and from the uniform L2-continuity ofyfl, x(/), L(" ))(" ) with respect to t
[[~,,~_ y,~[lZ = fo d- h
where ~m ~t(p) = 0.
lyre(s)- ytt(s)12ds <<.ot(t2 - t x )
Solutionsfor a class of systems
205
(This inequality follows by careful computation, using the representation formula (5.4)). By performing estimates in the same way as above it is easy to see that condition (iii) of Proposition 3.1 is satisfied. The basic estimates give
])c(t, 7:, c)]l <~ k2o + k2,e-"°-btlc-H lift, ~, ~.,) - c(t, ~, e=)ll -< k22 e-"-"lle,
and
- e=ll
From here conditions (i) and (ii) are immediately checked and we may use Proposition 3.1 to get the result. 6. FINAL REMARKS It can be seen that the proof has been performed in two steps: the first one for x(t) which is an element of the Euclidean space R" and the second one for Yt(" ) which is an element of the space /_,2(- h, 0; RP). I f f and 0 are periodic (almost periodic) then there exists a stable bounded solution (~(t), J)(t)) of (4.1) which is periodic (almost periodic); however, all these properties are true in the usual sense only for ~(t): (a) I (t) I < M 2 (b) ~(t + ~) -= ~(t) if f and g are periodic with the period (c) ~(t) is almost periodic in the sense of Bohr if f and g are almost periodic in the sense of Bohr. When about 9(t), all these properties are true in Lz( - h , 0; R p)
(a)
IIj),l[ = {|-a
IP'(s)12ds}
(b) IIJ)t+n - Y, II2
=f_:,
=
,-h IP(s)12ds
~< M2
- ~,(s)[Zds = O, or
:p(t + n) = ~(t)a.e. (c) For any e > 0 there exists a relatively dense set of real numbers {~} such that ]].gt+¢ - j),ll2 =
fo -h
Lpt+¢(s) - yt(s)12ds =
]j)(x + s) -
~s)12ds ~<
t-h
It follows that j)(t) is almost periodic in the sense of Stepanov (more precisely it is a S2-function) even i f f and g are almost periodic in the sense of Bohr. The fact that the properties of the forcing term are reproduced by y(t) only in a weakened sense is a consequence of the fact that y(t) is given by a difference equation whose solutions are discontinuous even if the initial conditions are continuous. This is another illustration of the propagation of singularities, a phenomenon which is typical for hyperbolic PDE. Returning again to the language of the electrical engineer, the vector x is the state vector of the lumped circuits: in the lumped part of the network the properties of the forcing term are fully reproduced. The vector y corresponds to the distributed variables of the network: in the distributed part of the network the properties of the forcing term are reproduced only in a generalized (weakened) sense. REFERENCES 1. BRAYTONR. K., Small-signal stability criterion for electrical networks containing lossless transmission lines. IBMJ. Res. Dev. 12, 431-40 (1968).
206
A. HALANAY
AND
VL. R~SVAN
2. FIGUEREDOR. J. P. de & Ho C. Y., Absolute stability of a system of nonlinear networks interconnected by lossless transmission lines. LE.E.E. Trans. Circuit Theory CT-17, 575--84 (1970). 3. WILLSONJR. A. N., Stability and instability criteria for nonlinear distributed networks. LE.E.E. Trans. Circuit Theory CT-19, 615-22 (1972). 4. RAsw,N VL., Absolute stability of a class of control systems described by coupled delay-differential and difference equations. Rev. Roum. Sci. Teehn.-Electrotechnique Energ. 18, 329-46 (1973). 5. RXSVANVL., Some results concerning the theory of electrical networks containing lossless transmission lines. Rev. Roum. Sci. Teehn.-Eleetrotechnique Energ. 19, 595--602 (1974). 6. BRAYTONR. K. & MIRANKERW. L., Stability theory for mixed initial boundary value problems. Arch. Ration Mech. Analysis 17, 358-76 (1964). 7. ABOLINIAV. E. & MYSHKISA. D., A mixed problem for an almost linear two-dimensional hyperbolic system. Mat. Sb. $0:92, 423-42 (1960). 8. COOKEK. L. & K R ~ D W., Differential-difference equations and nonlinear initial boundary value problems for linear hyperbolic partial differential equations. J. math. Analysis Applic. 24, 372-87 (1968). 9. COOKEK. L., A linear mixed problem with derivative boundary conditions (unpublished). Pomona College, (1971). 10. HALANAYA., Almost periodic solutions for a class of nonlinear systems with time-lag. Rev. Roum. Math. pures, appl. 14, 1269-76 (1969). 11. K~RzwE~L J.~ ~nvariant manif~ds f~r ~ws~ in Differential Equati~ns and Dynamical Systems (Eds Hale & La Sa~e)~ pp. 431-38, Academic Press, NY (1967). 12. INrArCr~E. F., Some results and applications of generalized dynamical systems, in dapan--U.S. Seminar on Ordinary Differential and Functional Equations. Springer Veflag, Berlin (1972).
PERIODIC AND ALMOST PERIODIC SOLUTIONS FOR A CLASS OF SYSTEMS DESCRIBED BY COUPLED DELAY-DIFFERENTIAL AND DIFFERENCE EQUATIONS
A . HALANAY Bucharest University, Faculty of Mathematics, 70109 Bucharest 1, Academiei 14, Romania and VL. R~.SVAN Institute for Power Equipment Research and Design, 74369 Bucharest IV, Vitan 236, Romania (Received 12 July 1976) Key words: Delay differential equations, difference equations periodic solutions, almost periodic solutions
1. I N T R O D U C T I O N
Trm COUPLEDdelay-differential and difference equations occur when studying transient processes for control systems containing water, steam or gas pipes, and also for electrical circuits with lossless LC transmission lines (see e.g.R.K. Brayton [1], C. Y. Ho and R. J. P. de Figueredo [2], A. N. Willson Jr. [3], VI. Rasvan [4], [5]). In the above references linear or nonlinear systems are considered, linear system stability and absolute stability of nonlinear systems are studied by frequency domain methods or by the method of Lyapunov function. The stability problem occurs for systems without forcing terms (e.g. electrical systems without sources or feedback control systems). If one considers, for instance, electrical systems with lossless LC transmission lines containing sources, the main problem of the electrical engineer is to find both the stationary and the transient currents and voltages of the electrical network. Mathematically speaking, one has to obtain the stationary and transient solutions (in most cases only the stationary one) of a dynamical system with forcing term which can be constant (for d.c. sources), periodic (e.g. for a.c. sources) or almost periodic (for modulated sources). From the mathematical point of view it is important to obtain existence conditions for such solutions. The existence of stable periodic and almost periodic solutions has also a practical value. A practician would however claim that these solutions always exist because he is speaking about their physical reality. One must in fact avoid confusion between the physical system and its mathematical model which can be correct or not. Therefore when a mathematical model is used existence theorems are needed to check its correctness. 2. A N E X A M P L E . T H E M A T H E M A T I C A L
PROBLEM
Consider for instance the electrical circuit from the paper ofR. K. Brayton and W. L. Miranker 197
198
A. HALANAYAND VL. RASVAN
[6], described by the following equations: av 0i ai --= L --= 02
at'
C
02
OV
at
v(O, t) = - E ( t ) - Roi(O, t),
(2.1)
v(l,O = vl( 0
CxOl(t) = - d/(vl(t) ) + i(l, t) v,(0) = v °,
i(~, 0) = io(~),
v(t, o) = Vo(~),
0 ~< ~ ~ t
These equations clearly define a mixed initial-boundary value problem for hyperbolic PDE. Using the transformation: v(2, t) = u1(2, t) + u2(2, t) i(2, t) = ~/C/L [u,(A, t) - u2(2, t)] the equations can be written in normal form and (2.1) become au 1 + 1 ~ul = O, ~u2 1 ~u2 = 0 ul(l, t) + u2(l, t) = Vl(t)
(1 + R o ~ - L ) Ux(0,t) + (1 - Ro C ~ ) C lbl(t) = - ~k(vl(t)) + ~ Vl(0) = %o
u2(0, t) = - E ( t )
[ul(/, t) - u2(l, t)]
(2.2)
.,(~, o) = ~[Vo(~) + ~J-V-CTo(~)], u2(~, o) = ~[Vo(~) - ~ ( , ~ ) ]
o .< ,~ ~ t
By integrating along the characteristics and denoting Ux(0, t) = rh(t), u2(l, t) = ~/2(t) one can find a one-to-one correspondence between the solutions of the mixed problem (2.2) and the solutions of the system
C16,(t) =
- v , ( t ) - O(Vl(t)) +
,/x(t)=
l1- a+-o ~ ~ c
2~/C/Ltl,(t -I~/LC)
r/2t"t - lx/L'-C)
(2.3)
,1~(0 = ~1(0 - 'h(t - &/L-~)
with theinitial conditiOnSo t/°(t) =~1[ v° ( ~ / ~ C )
+ x/L---~i° (
x/~CC)]
v,(0) = v 1, ~(t) = ~1[ Vo ( l +
x/~)--x/~/Ci°(
l+ x/~)]
-Ix/-~'t'O
Consequently a mixed initial-boundary value problem for hyperbolic PDE has been reduced to a Cauchy problem for a system of the type £(t) = Aox(t) + A l y ( t - h) - bo~(v(t)) + f ( t ) y(t) = A2x(t) + A3Y(t - h) - b~rk(v(t)) + g(t) v(O = c*x(O
(2.4)
Solutions for a class of systems
199
where f(t) and O(t) are the forcing terms. (Such reductions for more general mixed problems for hyperbolic P D E can be found in Abolinia and Myshkis [7], Cooke and Krumme [8] or Cooke [9]. They lead to various types of functional differential equations). It is now obvious that the problem of periodic and almost periodic solutions for the mixed problems can be reduced to the same problem for the corresponding FDE. Throughout the paper we shall study the problem of the periodic and almost periodic solutions for the system (2.4). 3. THE METHOD We shall follow in this paper the method used in a earlier paper of Halanay [ 10]. The formula of variation of constants (see Rhsvan [4]) enables us to write the system (2.4) as a couple of an integral and a difference equation. By usual techniques of the frequency method some evaluations of the solutions are obtained. Then use is made of a result which is essentially a special case of the general theorem of Kurzweil E11] on invariant manifolds. This result is the following: PROPOSITION 3.1. Suppose for a general flow in the Banach space ~ there exist real positive constants l, T, ~ < 1, kl, 2 such that (i)
II ~
II~ II ~< l,
t e [~ + T,
3 + 2T] imply IIc(t; ~, ~
l
(ii) II~lll-
I1~11 ~
t~[[+ T,t+ 2z]implyllc(t;3,~l)-c(t;[,~2)ll..<~ll~a-~2ll
(iii) IIc(t; t, ~ ) - c(t; 3, ~2)II <~ kl e;'('-~) IIc, - c2 I1for t >/t Then there exists p:R --* ~ such that (1°) IIp(t)II ~< l (2 °) p(t) = c(t; t l, p_Ctl)) for t I> t 1 and t 1 e R
(3°) II c(t; 3, ~) - p(t)II ~ koe- ~" -~ [I~ - ~ ) II for II~ II -< t, t >/~ where k o > 0, v > 0 (4 °) If c(t + f~; ~ + f~, ~) =_ c(t; t, ~), then p(t + f~) =- p(t) (5 °) If every sequence {h.}., h. --, oo contains a subsequence h.k such that for every bounded solution c(t; 3, ~) the sequence c(t + h.~ ; ~ + h.~, ~) converges uniformly on each compact subset of the half-axis t >/t, uniformly with respect to 3, ~, for ~ ~ R, I]~ II ~ l, then p is almost periodic. Remark that the condition in (4 °) is satisfied if the flow is defined by a periodic system and the condition in (5 °) if the flow is defined by a system which is almost periodic with respect to t. One must also mention that the existence of periodic and almost periodic solutions for the linearized system (2.4) (i.e. when b o = b 1 --- 0) also follows from the above proposition. Indeed the solutions of a linear system are always globally Lipschitz and the condition (iii) follows at once. The conditions (i) and (ii) are straightforward consequences of the exponential stability of the linear homogeneous system. All the assumptions of the proposition being fulfilled, the existence of periodic and almost periodic solutions is proved (in the linear case). 4. THE RESULTS We shall consider the case of several nonlinear elements i.e. the system Yc(t) = aox(t ) + AlY(t - h) - Bodp(v(t)) + f ( t ) y(t) = A2x(t) + A3y(t - h) - BFk(v(t) ) + e(t) o(O = C*x(O
(4.1)
200
A. HALANAYAND VL. R~SVAN
where ~(v) = col
(~,(vi))
The initial conditions for system (4.1) are given by x(0) = x o, y(t) = Yo(t), - h <~ t <<.O, where Yo ~ L 2 ( - h ' 0 ; RP). Define the following matrices: aI
H(cr)
Ao
-- Ale -*h ]
- A2
I-A3
e-oh
.J
(4.2)
where e is a complex variable. We can state now the following: THEOREM4.1. Consider the system (4.1) and suppose the following conditions are fulfilled: 1°. (a) det H(a) # 0 for all tr such that Re a ~< -0t, where 0t > 0 (b) The eigenvalues of A 3 are inside the unit circle. 2 °. 0 ~< ~ j ~ l ) - q~j(~t2)~< # ; °~1
--
~j(0) = 0 (j = 1. . . . , m)
~2
(The nonlinearities are globally Lipschitz) 3 °. There exist zj > 0 and 6 > 0 such that Zdltd 1 + Re zaY(ia 0 >1 JI
(4.3)
for all real 09, where z d = diag (z 1. . . . . z=),/zd = diag (/q . . . . . #=) 4 °. ] f ( t ) I <~ M for all t, I#(t) i ~ M for all t, Then there exists a bounded on the real axis solution of the system, which is periodic, almost periodic respectively if f, g are periodic, almost periodic respectively. Moreover, this solution is exponentially stable. Here Re G(iog) means the matrix ~[G(ico) + G*(i~o)] and for two symmetric matrices G 1, G 2 the inequality G t /> G 2 m e a n s that G 1 - G 2 is a positive semidefinite matrix. To make it clear we mention also that the zero matrix in Y(a) has its dimensions such that the products of matrices have sense. F o r systems with a single nonlinearity the frequency condition (4.3) (which is, in fact, the usual Popov condition with the Popov parameter set to zero, but written for the multivariable case) becomes 1 -+ #
Re~(io~)/>6>0
As an example we shall check the possibilities to ensure the conditions of Theorem 4.1 for system (2.3). First, by introducing the following nonlinear function:
~(~) = ( ~
-
r)~ +
¢(~)
where 7 > 0 is an arbitrarily chosen number, the system (2.3) takes the following form
201
Solutionsfor a classof systems y vl(t ) _ ~1 qb(vl(t)) + ~2 r/,(t - lx/L-'~ t3,(t) = - C--~
rh(t) = --otxt/2(t - Ix/'-L-C ) - E(t) tl2(t) = vx(t) - rl,(t - Ix/-L-'C )
where 1 - Rox/~-/L oq -
1 + Ro
C~,/-~
The LHS of the characteristic equation has the following expression detH(~)=
a+
~
(1--ele-2~l'/Lc)+ ~
After some simple manipulation one can find that condition (a) of Theorem 4.1 can be satisfied for ~<%
1 1 <2~ L ~ l n ] ~ l - -~
where % is the positive root of the equation 7
2
I ,l e2 'vLc
= 0
and ~ is chosen such that
The condition (b) of Theorem 4.1 is automatically satisfied because ]g,[ < 1. It remains to check the fulfilment of the frequency condition. The transfer function 3,(a) has the following expression: ] -- ~ l e - 2 a l ~ / L C
,(0")=( a + ~T) (1 _ otle_2#,~/L-~)+ ~20~. x/-C~Le-2~,,/g-d After some manipulation we get Re v(ico) = (1-o,
cos 2~/x/L-C)[~--~ + ~-~1 ot' (2C ~ -
ot2 y) cos 2og/,fL--c] + ~-~1(2x/-~~)sin 2 2colx/%-C ] det H(ioJ) l 2
It can be easily seen that if y is chosen in order that 24-d
I or, [ 2x/C/L 1 + I•t , I < r <
then Re 7(it0) t> 0 and the frequency condition (4.3) is fulfilled for/~ > 0 arbitrarily large. Therefore the existence of a forced regime is ensured for
A. HALANAYANDVL. RASVJ~N
202
_ I --l~tIICx/~-~ + I "~ 1~I ]
~ ~ ~(VI ) _ ~(V2 ) ~-~I, l~l -- V2
where e > 0 and e' > 0 are arbitrarily small. This is the same sector as the Hurwitzian one. For any nonlinearity belonging to this sector the free system (i.e. the system without sources) is absolutely stable (see Infante [12], where this fact was proved by Lyapunov methods, or R~svan [5], where the proof is done by frequency methods). In fact the system (2.3) has an "almost linear" behaviour (under usual assumptions, of course).
5. PROOF OF THETHEOREM A. Basic estimates
Let (x, y) be a solution of (4.1) defined for t >i t; by the variation of constants formula we can write x(t) = Z 1l(t -- t) X(t) +
-h
f:
Zl2(t - s) y(s) ds -
[Z 1l(t - s) B 0 + Z12(t
-
s)
B1] ~(v(s)) ds
[ z ~ l ( t - s ) f ( s ) + Zx2(t - s)e(s)] ds
+
hence v(t) = C * Z 1 l(t -
l)x(l)
+
C*Zx2(t - s) y(s) ds -h
-
f:
[ C * Z 1 l(t - s) B 0 + C*Z12(t - s) B1] qb(v(s)) ds
[ C * Z 1 l(t - s) f ( s ) + C*Z12(t - s) if(s)] ds
+
Denote ?(t) = v(t + /) and write down the nonlinear integral equation
?(t) = ~(t) - fl K(t - s) #~(~(s)) ds the same as in [10]. Here
fo ~(t) = C*Z11(t)x(t ) +
K(t) = C*Zll(t )B 0 + C*ZI2(t) BI
C*Z12(t - s)y(t + s)ds +
[ C * Z l l ( t - s ) f ( t + s)
-h
+ C*Z12(t - s)g(t + s)]ds
We need now to check the Laplace transform of K(t); we have [4]
g(0") = C*Zll(a)Bo +
C*Z12(a)BI=[C
=[C*
*
0IV ~11(0")
LaZ21(a)
2 1 2 ( o " ) ] [ B0] aZ22(a) j
O]H-l(a)[BB:]=
y(a)
B,
Solutions for a class of systems
203
The condition (4.3) in the statement allows us to write ~d#~ I + R e z ~ Y ( i o 9 -- p)>~-fi2 1
Y~o
for p > 0 sufficiently small. This follows from the uniform conitinuity of Y(a) for - ~t < Re tr ~< O, where 0t is the one in the statement. The above frequency condition allows us to obtain the same estimates as in [10]
, .o
< '
J
I;(01 ~< k,e-~"(Ix(~)l + IlY~II)+ ka.
0
< 2 <.
where t'o y~(s) = y(t + s ) a n d Ily~ll2 = | d-
[y~(s)12ds h
The inequality can now be written as
<~ [k,(lx(~)l
fr[•tq)j(Yj(t))]2dt
Ilyill)+
+
kseUr] 2
(5.1)
1 dO
F r o m here, as in [7], we get the main estimate
Ix(t)l <
k9 +
k,oe-"('-~'(Ix(~)l + Ily~ll)
(5.2)
In the same way, for two different solutions
[Xl(t) -
x2(t)l ~ K2e-"~'-~'([xl(D - x2(/)l + IIYI.~ - Y2.ill)
(5.3)
We need now the corresponding estimates for y. We have y(t) = A2x(t) + A a y ( t - h) - B~(p(v(t)) + g(t)
Denote for a while A 2 x ( t ) - Bltp(v(t)) + O(t) = F(t)
and write k-1
y(t) = A~y,(t - ~ - kh) +
~, A ~ F ( t - j h )
0 for~+ (k-
1)h~
kh, h e n c e k-2
y(t + s) = A k - l y ~ t + s - l - (k - 1)h)+
~ AJaF(t + s - j h )
0 for-h~s<~+
(k-
1 ) h - t , and k-1
y(t + s) = Akayt-(t + s - Z - kh) + ~, AJ3F(t + s - j h )
0 for~ + (k - 1 ) h - t ~< s < 0.Therefore
(5.4)
204
A. HALANAYANDVL. RASV~,N k-1
[y(t + s)[ ~< k,,6 k-1 ]yr(s)l + k,~ Z pJlF(t + s - jh) I
0
0
and k-1
[}Y,I[ ~< k~2P'-tllYz[I + k,2 ~, PJIIFt-jhll 0
where liFt[J2 = f ~ h IF(t+ s)[2ds
We have further:
IF(t) I <~ M + Mdx(t) I <<.kla + k,,e-""--°(lx(~) I + IIY~II) hence:
[IF,_jh[I <~kls + k16e-~(t-Jh-b(lx(3)[-t- [lY~[[) Consequently one can write k-I
k-1
0
0
[JY,[[ ~< k12pk-ll[Y~[] + k12k15 ~ PJ+ k,2k,6(lx(~)l + [[Y~[[e-~''-° ~ (Peuh)i and, by choosing # such that pe ~k < 1
Ily,II ~ k,~-~llY~ll
+ k,7 +
k~se-""-~'(Ix(3)l + IIY,II)
But, ifp e"h < 1, then, for 3 + (k - 1)h ~< t < 3 + kh, pk < e-~,-~). Therefore
[lYtl[ <~ k,7 + k,9e-U"-i'(lx(3)[ + IlY~II) In the same way the corresponding estimate is obtained for the difference of two solutions.
B. Application of Proposition 3.1 The system (4.1) will define a flow in the Hilbert space R n × L2( - h, 0; R p) with the usual norm; we shall write c(t; 3, ~) for (x(t; 3, xff), Yt(" )), Yfl, x(/), y~( • ))( • )), where y~e L2(-h, 0; RP). The first two definition properties of the flow c(~; t, ~) =
c(t2;tl, c(tl;3,~) ) = c(t 2 + t~;3, P),t 2 >/tl /> 3 follow at once from the definition of a solution of (4.1), and from the fact that this system satisfies the uniqueness conditions. The (uniform) continuity of c( •, ~, ~) follows from the fact that x(., 3, x(t), t,(" )) is a Caratheodory solution, hence it is absolutely continuous, and from the uniform L2-continuity ofyfl, x(/), L(" ))(" ) with respect to t
[[~,,~_ y,~[lZ = fo d- h
where ~m ~t(p) = 0.
lyre(s)- ytt(s)12ds <<.ot(t2 - t x )
Solutionsfor a class of systems
205
(This inequality follows by careful computation, using the representation formula (5.4)). By performing estimates in the same way as above it is easy to see that condition (iii) of Proposition 3.1 is satisfied. The basic estimates give
])c(t, 7:, c)]l <~ k2o + k2,e-"°-btlc-H lift, ~, ~.,) - c(t, ~, e=)ll -< k22 e-"-"lle,
and
- e=ll
From here conditions (i) and (ii) are immediately checked and we may use Proposition 3.1 to get the result. 6. FINAL REMARKS It can be seen that the proof has been performed in two steps: the first one for x(t) which is an element of the Euclidean space R" and the second one for Yt(" ) which is an element of the space /_,2(- h, 0; RP). I f f and 0 are periodic (almost periodic) then there exists a stable bounded solution (~(t), J)(t)) of (4.1) which is periodic (almost periodic); however, all these properties are true in the usual sense only for ~(t): (a) I (t) I < M 2 (b) ~(t + ~) -= ~(t) if f and g are periodic with the period (c) ~(t) is almost periodic in the sense of Bohr if f and g are almost periodic in the sense of Bohr. When about 9(t), all these properties are true in Lz( - h , 0; R p)
(a)
IIj),l[ = {|-a
IP'(s)12ds}
(b) IIJ)t+n - Y, II2
=f_:,
=
,-h IP(s)12ds
~< M2
- ~,(s)[Zds = O, or
:p(t + n) = ~(t)a.e. (c) For any e > 0 there exists a relatively dense set of real numbers {~} such that ]].gt+¢ - j),ll2 =
fo -h
Lpt+¢(s) - yt(s)12ds =
]j)(x + s) -
~s)12ds ~<
t-h
It follows that j)(t) is almost periodic in the sense of Stepanov (more precisely it is a S2-function) even i f f and g are almost periodic in the sense of Bohr. The fact that the properties of the forcing term are reproduced by y(t) only in a weakened sense is a consequence of the fact that y(t) is given by a difference equation whose solutions are discontinuous even if the initial conditions are continuous. This is another illustration of the propagation of singularities, a phenomenon which is typical for hyperbolic PDE. Returning again to the language of the electrical engineer, the vector x is the state vector of the lumped circuits: in the lumped part of the network the properties of the forcing term are fully reproduced. The vector y corresponds to the distributed variables of the network: in the distributed part of the network the properties of the forcing term are reproduced only in a generalized (weakened) sense. REFERENCES 1. BRAYTONR. K., Small-signal stability criterion for electrical networks containing lossless transmission lines. IBMJ. Res. Dev. 12, 431-40 (1968).
206
A. HALANAY
AND
VL. R~SVAN
2. FIGUEREDOR. J. P. de & Ho C. Y., Absolute stability of a system of nonlinear networks interconnected by lossless transmission lines. LE.E.E. Trans. Circuit Theory CT-17, 575--84 (1970). 3. WILLSONJR. A. N., Stability and instability criteria for nonlinear distributed networks. LE.E.E. Trans. Circuit Theory CT-19, 615-22 (1972). 4. RAsw,N VL., Absolute stability of a class of control systems described by coupled delay-differential and difference equations. Rev. Roum. Sci. Teehn.-Electrotechnique Energ. 18, 329-46 (1973). 5. RXSVANVL., Some results concerning the theory of electrical networks containing lossless transmission lines. Rev. Roum. Sci. Teehn.-Eleetrotechnique Energ. 19, 595--602 (1974). 6. BRAYTONR. K. & MIRANKERW. L., Stability theory for mixed initial boundary value problems. Arch. Ration Mech. Analysis 17, 358-76 (1964). 7. ABOLINIAV. E. & MYSHKISA. D., A mixed problem for an almost linear two-dimensional hyperbolic system. Mat. Sb. $0:92, 423-42 (1960). 8. COOKEK. L. & K R ~ D W., Differential-difference equations and nonlinear initial boundary value problems for linear hyperbolic partial differential equations. J. math. Analysis Applic. 24, 372-87 (1968). 9. COOKEK. L., A linear mixed problem with derivative boundary conditions (unpublished). Pomona College, (1971). 10. HALANAYA., Almost periodic solutions for a class of nonlinear systems with time-lag. Rev. Roum. Math. pures, appl. 14, 1269-76 (1969). 11. K~RzwE~L J.~ ~nvariant manif~ds f~r ~ws~ in Differential Equati~ns and Dynamical Systems (Eds Hale & La Sa~e)~ pp. 431-38, Academic Press, NY (1967). 12. INrArCr~E. F., Some results and applications of generalized dynamical systems, in dapan--U.S. Seminar on Ordinary Differential and Functional Equations. Springer Veflag, Berlin (1972).