JAN LIGĘZA*
THE DISTRIBUTIONAL SOLUTIONS OF SOME SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS
Abstract. In this paper we prove a theorem on the existence and uniqueness of the solution of the Cauchy problem for linear differential equations with distributional coefficients.
1. Introduction. One of the possibilities of generalization of the classical solution of a differential equation is the solution in the distributional sense.
Distributional solutions of ordinary differential equations have not been studied sufficiently, due to certain difficulties in defining operations on distributions:
some operations (e.g. multiplication, substitution, definite integral) cannot be defined for all distributions in a natural way. A particular large numbers of papers have been devoted to linear differential equations (see [6]— [ 11], [16]— [21], [24], [25], [27]— [31]. Another possibilities are considered in [12]— [15], [23]
and [26],
In this note we consider the system of equations
(1.0) y'(t) = A'(t)y(t) + B'(t),
where all elements A i } of the matrix A are functions of locally bounded variation on the interval (a, b) £ R 1 ( i j = 15 . n), the components Bt of the vector B are continuous functions on (a, b), the derivative is understood in the distributional sense and the product of two distributions is understood as generalized operation (see [3], [4]). The vector y = ( yl , . .. , y n) is an unknown distribution. We prove a theorem on the existence and uniqueness of the solution of the Cauchy problem for system (1.0) applying the sequential theory of distributions.
In papers [6]— [11], [16]—[21], [27]— [31] it has been examined linear differential equations with distributional coefficients but under another assump
tions than in our note.
2. Notation. We shall denote by T^loc(a, b) the space of real functions which are of bounded variation on every compact interval [c, d] cz (a, b).
We say that a distribution p is a measure on {a, b) if p is the first distributional derivative of a function of the class ^ "ioc(a, b). The symbol J t (a, b) denotes the set of all measures defined on (a, b).
By ^ ( a , b) we denote the space of all continuous real functions on (a, b) and by
^ ^ ( a , ^ ) we denote the set of all functions z such that z — v c , where v e ^ IOC(a, b) and c e ^ ( a , b).
A matrix A(t) = (ALj(t)) belongs to i / "ayc"(a, b) if and only if A , j e r loc(a,b) for i , j = 1, . .. , n.
R eceived N ovem ber 17, 1985.
A M S (M O S) Subject classification(1980), Primary 34A10, Secondary 46F10.
* Instytut M atem atyki U niw ersytetu Ślaskieeo. K atow ice, ul. B ankow a 14. Poland
We put
b) = y ioc(fl, b) x . „ x y [oc (a, b);
n times
b) = ?(a, b) x ... x ^ (o , ft}, ^ " ( a , 6) = / ^ ( a , 6) x ... x lT ^ (a, ft),
m times n times
/!'(£) = (A'ij(t)), A(t + ) = (Atj ( t + )), A(t —) = ( A i j ( t—)), AA(t) = ^ (t + ) - X ( r - ) , y0 = (y°, ..., y„°),
i4*(t) = y*(t) = (y*(t), . .. , y*(f)), where
^ e ft), y f eR 1, y, e m ? ( a , ft), (t) = ^ (t + ) + ^ ( t ~ ) > t e (fl> 6)> ^ + )>
(y,(^ —)) denotes the right (resp. left) hand side limit of the function y t at the point t for i — 1, ..., n.
A sequence of smooth, non-negative functions {5k} satisfying:
00
(2.1) J S k( r ) d t = l ,
— 00
(2.2) ók(t) = Sk( — t) for all ( e R 1, (2.3) Sk(t) = 0 for |f| ^ ak,
where { a j is a sequence of positive numbers with ak -> 0 as k -> oo, is called a 5-sequence.
The product of two distributions, the modulus of a distribution and inequalities between two distributions will be understood in this note as generalized operations (see [1], [2], [4], [5]).
Let P e V ^(a, b). Then we define
d
(2.4) $P' (t )dt = P * ( d ) - P * { c ) ,
C
where c , d e (a, b).
d
It is easy to observe that if P e "V)oc(a, ft) or P e ^ (a, ft), then J P'(t)dt exists for
' C
every c , d e (a, ft) (because P e i / '(<S(a, ft)).
In the case when P , Q e t ~ioc(a, ft) and u e %>(a, ft), then it has been proved in [1], [3], [4] that
(2.5) P - Q ' e J t ( a , b ) , u-Q' e J / ( a , b ) , (2.6) P ' ( Q u ) = ( P ' Q ) u = ( P ' u ) Q ,
(2.7) |JP(t)Q'(t)dt\ < |J|P |(t)ie'l(t)dt| ^ sup |F|*W |}ieiW dt|
c c te[c ,d ] c
and
otk
(2.8) lim J P( t — s)5k{s)ds = P*(t) k^ x - a k
for every t e (a, b).
It is easy to check that if P' ^ 0 on (a, b) and a < c < d < b, then i
(2.9) j- P' (t) dt = P * ( d ) ~ P* (c) ^ 0
(see [2], Theorem 6). Hence
] p ' ( t ) d t ^ J(2'(t)df where P',Q' e J / ( a , b ) , P' < Q' and a~< c < d < b.
3. Cauchy problem for system (1.0). Let A e b), B e (a, b) and y € ‘V (ś n (a, b). Moreover, let y satisfy system (1.0) in the distributional sense. Then we call y the solution o f equation (1.0).
TH EOREM 3.1. Assume that
(3.1) A e r i * n(a,b),
f or every t e (a, b)
(3.2) d e t ( 2 I - A A ( t ) ) # 0 and det(2/ + AX(0)-# 0, where I denotes the identity matrix,
(3.3) Be<#n(a,b).
Then the problem
fy ' ( t) = A'(t)y(t) + B'(t) 1 y * ( t 0) = y 0
(3.4)
has exactly one solution in the class t ' (6 n{a, b).
REMARK 3.1. Assumption (3.2) is essential. This can be observed from the following examples:
(3.5)
and
(3.6)
y'(t) = 2S(t)y(t) y * ( - l ) = 0
j.z '(t) = - 2 5 { t ) y( t )
z*( 1) = 0
where 5 denotes the Dirac’s delta distribution. It is not difficult to show that the distributions
y = cH and z = c(H — 1)
are solutions of problem (3.5) and (3.6) respectively (H denotes the Heaviside’s function).
We shall prove some lemmas before giving the proof of Theorem 3.1.
LEMMA 3.1. I f F e%>(a,b) and g e t " loc(a, b), then the product F ' ■ g exists.
P r o o f of Lemma 3.1. L etF t = F*5ka n d g k = g*Sk, where {(5J is an arbitrary (5-sequence (the asterisk denotes the convolution). Then
(3-7) (Fkgk)' = F’kgk + Fkg'k.
Hence, by (2.5), we have
(3.8) F'g = ( Fg ) ' -F g ' ,
which completes the proof of Lemma 3.1.
LEMMA 3.2. I f G' e M ( a , b), G' ^ 0 (on (a , b)) and Fe%>(a,b), then there exists a number , such that
d d
(3.9) J F(t)G'{t)dt = li 1 J G'(t)dt = /i1(G*(Ą-G *(c)),
where a < c < d < b, fi1 e [m, M ], m = inf F(t) and M = sup F(t).
te [c , d ] te[c ,d ]
P r o o f of Lemma 3.2. If G*(d)—G*(c) = 0, then by (2.9), Lemma is obvious.
Let G*{d) — G*{c) > 0 and let Fk = F*dk, G'k = G'*5k, where {<5t } denotes the (5-sequence. For a fixed e > 0 there exists a number k0 such that
d d d
(3.10) ( m - e ) $ G ' k{t)dt ^ J F k(t)G’k{t)dt < (M + e) J G'k(t)dt
c c c
if k ^ k0 . We shall prove that
d d
lim $ Fk(t)G’k(t)dt = f F ( t ) G \ t ) d t .
k^cc c c
In this purpose we consider the sequence {gk} defined as follows
(3.11) gk = uk + vk,
where
uk = \ ( F k(t )- F( t) )G' k(t)dt and vk = ] F(t)(G'k( t ) - G ' ( t ) ) d t .
C C
d
Since |F* —F| => 0 on {a, b) (almost uniformly) and {J |Gfc(t)|dt} is a bounded
C
sequence (see [1]), we see by (2.7) that.
(3.12) lim uk = 0.
k - > co
We claim that
(3.13) lim vk = 0.
k~* oo
In fact, for e > 0 there is a positive number y such that
|F(f) — F(f)| c for \t — f| < y and t, f e [ c , d].
Now, we divide the interval [c, d~] into a finite numbers of intervals [tr_ j , fr] such that c = t 0 < t t . .. < tp — d, | tr_ Ł — tr\ < y and r = 1 ,... , p. Thus
vk = t ] [(F (0 — F(tr)) (G'k(t) — G'{t)) + F{tr) (G'k{t) — G’(f))] d t .
r= 1 lr-1
This implies (3.13) (by (2.7)—(2.8)). By relations (2.8), (3.10)—(3.13), we obtain
d d d
m j G'{t)dt s; J- F(t)G'(t)dt ^ M J G'(t)df.
c c c
Putting in the last inequalities
i
\ F ( t ) G' ( t ) d t
Mi G*(d) — G*(c)
we have (3.9) which completes the proof of Lemma 3.2.
LEMMA 3.3. Let F e ^ ( a , b) and let g e ^ joc(a, b). Then the function T defined as follows
(3.14) T(t) = } F’(s)g(s)ds = F ( t ) g * ( t ) ~ F ( c ) g * ( c ) ~ i F(s)g'(s)ds
C C
is continuous on the interval ( a, b).
P r o o f of Lemma 3.3. Let g = g \ —g 2-< where g\ ^ 0 anc g'2 ^ 0 on (a,b).
Then, by (3.14), we have
T ( t ) - T ( t ) = \ F'( s)g(s)ds = ( F ( t ) - F ( t ) ) g * ( t ) + F ( t ) ( g * ( t ) - g * ( t ) ) -
t
F(s)g'i(s)ds + j F(s)g'2{s)ds.
t t
Hence, by (3.9), we infer that
T(t) - T(t) = (F(t) — F(i))g* (t) + F(t) ( g f ( t) - g f (rj) - Ml(g *(t)-g * (t)) + + F(F)(gf(t)-g$(t)) + Ju2(gUt)-g$(Fl) = ( F ( t )~ F(t))g*(t) + + ( F ( t ) - V i ) ( g * (0 - g*(t))+ (FCt) - v . } ) { g W ) - g2 (0),
where
Hi ,H2 e [ inf F(s), sup F(s)].
s e [ f ,f ] s e [ r ,f ]
The last equality finishes the proof of Lemma 3.3.
LEMMA 3.4. I f y l , y 2 e i / 'Xoc(a,b) and c e ^ ( a , b ) , then
(3-15) (y 1y 2)cl = y 1(y2c').
P r o o f of Lemma 3.4. The existence of the products ( y1y 2)c' and y 1{y2c') follows from Lemmas 3.1 and 3.3. We get, by (3.8) and (2.6)
(3.16) [ y i y 2) ć = ( y , y 2c ) ' ~ ( y l y 2)'c = ( y l y 2c y - ( y ' l y 2) c - y , ( y ' 2c) and
(3.17) y! {y 2 c ) = ^ [(y2c ) ' - / 2c] = y ^ y 2c)'- y ^ c ) =
= ( y i y 2c ) ' - y ' i ( y 2c ) - y i ( y '2c) = (y1y 2c ) ' - ( y ' l y ^ c - y ^ c ) , which completes the proof of Lemma 3.4.
LEMMA 3.5. Let A = (Aitj) satisfies assumptions (3.1)—(3.2) and let Z = (ztj ) be a matrix such that Z e i /70xcn(a, b), Z*(t) = (zf j (t)) and
f Z'(t) = A'(t)Z(t) 1 Z * ( t 0) = I.
(3.18)
Moreover, let
(3.19) u(t) = det(zfj(t)).
Then
(3.20) u e t \ oc(a,b),
f or every compact interval [ c , d~] a (a, b)
(3.21) inf |w(t)| > 0
t e [ c , d ]
(3.22) - e ^ oc(a,b).
u
P r o o f of Lemma 3.5. The existence and uniqueness of matrix Z follow from [21]. Since Z e i ^ * cn{a,b), we obtain that « e f j oc(a,6). It is enough to prove (3.21). Let [c,d ] be an arbitrary compact interval such that [c,d ] c: (a , b ), r0 e [c, d] and inf |u(t)| = 0. Then there exists a point a t e [c, d] possessing the
t e [ c ,d ]
following properties:
(i) «(at) = 0
(ii) there exists a sequence {t k} convergent to a 1 and lim u(tk) = u(a1 —) = 0 or lira u(tk) = u(a1 + ) = 0.
fc-+ oo k~* oo
Hence, we have
(3.23) £ C j z f i a J = 0 or £ CjZj{at - ) = 0 or ^ CjZj{a! + ) = 0,
; = i } = i i= i
where cj + . .. + c2 > 0, z,. = (zu , ..., znJ), 0 = (o, . . . , o ) and j = l , . . . , n . We put
n
(3.24) v(t) = £ Cjzf(t) for fe (a ,6 ).
i= i Then v is a solution of the problem
r u'(t) = A'(t)v(t) (3.25)
L(y) = 0,
where L(v) = v*(a1) or L(v) = v( a1—) or L(i>) = 17(0 !+ ). Taking into account [21] we infer that
n
Z C j Z f { t ) = 0 for t e ( a , b ) ,
;= 1
which is impossible (because u((0) = 1). Thus our assertion follows.
LEMMA 3.6. Let A = (AiyJ) satisfies assumptions (3.1)—(3.2). Then the problem
(3.26) y'(t) = A'(t)y(t)
y* { t 0) = 0, t0 e ( a , b) /ias on/y t/ie zero solution in the class y fś n(a,b).
P r o o f of Lemma 3.6. If y e i / 'cś n(a, b) and if y satisfies (3.26), then y ’i(t) = Z Ai ji t i vj i ^Cji t j ),
j= 1
where A'ij e J i ( a, ft), Vj e i^oe(a, b), c,- e r^(a, ft) and i , j = 1, ..., n. By (2.5)—(2.6) we obtain that j/; e J i ( a , b)(i = 1, ..., n). Hence y e y",”c (a, ft). An application of [21] (Theorem 2.1) completes the proof of Lemma 3.6.
The matrix Z = (zfj) defined in Lemma 3.5 will be called the fundamental matrix o f solutions.
P r o o f of Theorem 3.1. Let Z(t) = (z*j{t)) be the fundamental matrix of solutions. Then there exists the inverse matrix Z 1 (?) for t e (a, ft) (by Lemma 3.5) and Z 1 g i \ l * n(a, ft). We claim that
(3.27) y = Z g ,
where
(3.28) g'(t) = Z~Ht )B' (t ), g*(t 0) = y 0 is a solution of problem (3.4). In fact, by the equality
(Z “ 1B') = ( Z ~ l B)'—{ Z ~ l )'B,
Lemmas 3.1, 3.3, and 3.5 and relation (2.5) we obtain that g ^ e^"(a,b). Using Lemma 3.4 we infer that Z g satisfies (1.0). Moreover, by (3.27)—(3.28), we get 9* (to) = y 0- Uniqueness assertion follows from Lemma 3.6.
REMARK 3.2. Let all assumptions of Theorem 3.1 be satisfied. Then it is not difficult to show (by Lemmas 3.1,3.3— 3.6) that the system (1.0) with the condition y (£0 + ) = y 0 or y(f0 —) = y0 has exactly one solution in the class i r(ś n(a, b).
REFERENCES
[1] P. ANTOSIK [IT AHTOCHK ], IJopndoK orrmocumejibHo Mepbi u eeo npuMeitenue k uccAedo- eahwo npou3eedeHux o6o6ufeunbix (fiytiKąuu, Studia Math. 26 (1966), 247—262.
[2 ] , On the modulus o f a distribution, Bull. Acad. Polon. Sci. Ser. Sci. Math.
Astronom. Phys. 15 (1967), 717— 722.
[3 ] , The symmetric value o f a distribution at a point, In : Proceedings of the Scientific Research Centre in Zabrze, Polish Academy of Sciences, Institute o f Mathematics, Warsaw, Polish Academy o f Sciences, 1973.
[4 ] '--- , J. LIGĘZA, Products o f measures and functions o f finite variation. In:
Proceedings of the conference on generalized functions and operational calculus, Varna 1975.
Sofia 1979, 20— 26.
[5 ] , J. M IK USIŃSK I, R. SIKORSKI, Theory o f distributions. The sequential approach, Amsterdam — Warsaw 1973.
[6] G. BUTTAZZO, D. PERCIVALE, On the approxim ation o f the elastic bounce problem on Riem annian manifolds, J. Differential Equations 47 (1983), 227— 245.
[7 ] M. CARIE, E. PASCALI, II problem m a del rim balzo unidem ensionale e sul approssim azioni con penalizzazioni non convesse, Rend. Mat. 13 (1980), 541— 553.
[8 ] T. DŁO TKO, J. KLAMKA, J. LIGĘZA, C o n trollability and m inim um energy control o f linear system s, Podstawy sterowania 6 (1976), 211— 217.
[9 ] V. DOLEŻAL, D ynam ics o f linear system s, Praha 1964.
[10] H. M . rE JIbO A H fl, T. UIHJIOB [I. M . G ELFA N D , G. J. SHILOV], 0 6 o 6 u fe m b ie t fy m q u u u deucmeuH nad hum u, T. I., MocKBa 1958.
[11] A. XAJIfeAHAfl, BEKCJIEP [A. HALANAY, D. WEXLER], K anecm eem aH m eopun uMnyAbCHbix cucmeM, MocKBa 1971.
[12] T. H. HILDEBRANDT, On system s o f linear differential S tieltjes - inteqral equations, Illinois J.
Math. 3 (1959), 352— 373.
[13] H .T . K H ry P A fl3 E [I. T. K IG U RA DZE], H eK om opue cunzyAHpHbie K paeeue 3adanu óah o6biKHoeeHHbix ducfiffiepehifuaAbHbix ypaeHenuu, H3fl. TÓHjiHCCKoro YHHBepcHTeTa 1975.
[14] J. KURZWEIL, G eneralized ordinary differential equations and continuous dependence on a param eter, Cechoslovak Math. J. 7 (1957), 418—449.
[15 ] , G eneralized ordinary differential equations, Cechoslovak Math. J. 8 (1958), 360— 389.
[16 ] , L inear differential equations w ith distributions as coefficients, Bull.
Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 7 (1959), 557— 560.
[17] A. LASOTA, A pplications o f generalized fu n ctio n s to contingent equations and control theory, In:
Seminar Lectures of the University o f Maryland 1971— 1972, Maryland, 41— 53.
[18 ] -- F. H. SZAFRANIEC, A pplication o f differential equations with d ist
ributional coefficients to optim al control theory, In: Prace Matematyczne 12. Kraków UJ 1968, 31— 37.
[19 ] --- , J. TRAPLE, N ico le tti boundary value problem fo r system s o f linear differential equations with distributional perturbations, In: Prace Matematyczne 15. Kraków UJ 1971, 103— 108.
[20] J. LIGĘZA, On generalized periodic solutions o f som e linear differential equations o f order n, Ann.
Polon. Math. 33 (1977), 210—218.
[21 ] , On distributional solutions o f som e system s o f linear differential equations, Casopis Pest. Mat. 102 (1977), 30— 36
[22 ] ---,O n a n integral inequality, Prace Nauk. Uniw. Śląsk., Prace Matematyczne 7 (1977), 22— 25.
[23] A .f l. MŁIIHKMC, A . M. CAMOHJIEHKO [A .D . MYSHKIS, A. M. SAM OLENKO ], Cucm eM a c moAHKaMU e sadanbie MOMenmbi epeMeHuu, M aT. c6opHHK 6:2 1967, 202—208.
[24] R. PFA FF, Gewóhnliche lineare D ifferentialgleichungen n-ter O rdnung m it D istributions- koefizienten, Proc. Edingburgh Math. Soc. 85 (1980), 291— 298.
[25 ] --- , G eneralized system s o f linear differential equations. Proc. Edingburgh Math. Soc. 89 (1981), 1— 14.
[26] S. SCHWABIK, M. TVRDY, O. VEJVODA, D ifferential and integral equations, Praha, 1979.
[27] J. TRAPLE, B oundary value problem fo r linear differential equations with distributional perturbations, In: Prace Matematyczne 17. Kraków UJ 1975, 119— 126.
[28] Z. W YDERKA, On the linear concave convex differential open - loop gam e with m easures as coefficients, Prace Nauk. Uniw. Śląsk., Prace Matematyczne 8 (1978), 88— 92.
[29 ] , Som e optim al control problem f o r linear system s, Systems Sci. 4 (1978), 53— 59.
[30 ] , Som e problem s o f optim al control fo r linear system s with measures as coefficients, Systems Sci. 5 (1979), 425— 431.
[31 ] , On the attainable set fo r linear system s w ith m easures as coefficients.
Systems Sci. 6 (1980), 385— 390.
