GENERALIZED GREEN ’ S FUNCTIONS FOR HIGHER ORDER BOUNDARY VALUE MATRIX DIFFERENTIAL SYSTEMS

In this paper, a Green's matrix function for higher order two point boundary value differential matrix problems is constructed. By using the concept of rectangular co-solution of certain algebraic matrix equation associated to the problem, an existence condition as well as an explicit closed form expression for the solution of possibly not well-posed boundary value problems is given avoiding the increase of the problem dimension.

systems of the type x(P)+ A X (p-l) p-1 (1) + + A X + A X f(t); 0 tb o Eli X (j-1)(O) + Fi =I s I q.
This classlcal approach has the inconvenience of the lack of expllcltness due to the relatlonship X(t) [I,O O] Y(t), as well as the computatlonal cost due to the increase of the problem dimension.In partlcular it needs the computation of the matrix exponential exp(tC) and it is well known that it is not an easy task [Ill.
These inconveniences motivates the study of some alternative approach that avoids the increase of the problem dimension.In [4], a solutlon for a very particular second order problem of the type (1.1]Is proposed avoiding the Increase of te problem dimension, however, the method is not applicable to more general problems.In a recent paper [7] a method for solving problems of the type (1.1) for the case p=2, without considering the extended system (1.2) have been proposed.
Results of [7] are based on the existence of an appropriate pair of solutions of the characteristic algebraic matrix equation Z-+ A Z + A O.
(1.4) o Unfortunately, equation (1.4) may be unsolvable [6] and in such case, the method given in [7] is not available.The aim of this paper is to study an existence condition for the solution of problem (1.1) as well as an explicit expression of a solution of the problem in terms of a generalized Green's matrix function G(t,s}, taking advantage, of the ideas developed in [7] but without the restriction of the existence of solutions of the associated algebraic matrix equation The paper is organlzed as follows.In sectlon 2, we Introduce the concept of rectanguIar co-solution for the equation (1.5} and we state some resuIts recentIy given [81, that will be used In the following sections.In sectlon 3, we construct a generallzed Green's matrix functlon of problem (I.I} by using an approprlate set of co-soIutions of equation {1.5) and a procedure analogous to the one develold In [5] for the scalar case.Finally, in section 4 an explicit closed form solution of problem (1.1) in terms of a generaIized Green's matrix function is given.S If S is a matrix in Cmxn, we denote by its Moore-Penrose pseudoinverse.We recall that an account of uses and properties of this concept may be found in [2] and that the computation of S is an easy matter using MATLAB [I0].
We begin by introducing the concept of rectangular co-solution of equation (1.5), recently given in [8].DEFINITION 2.1.We say that (X,T) is a (n,q) co-solution of equation (1.5)  Jk with Jj cRjxRj, m +...+ mk =np, such that then {(M J lsk} is a k-complete set of co-solutions of (I 5) 1S' J COROLLARY 1. ([8]) Let us suppose the notation of theorem and let {(M,s, 1-s-k} be a k-complete set of co-solutions of equation (1.5).Then, the general solution of the matrix differential equation (l.I) is given by where D ,, is an arbitrary matrix in Cm,xm.If W is the block partitioned matrix associated to the set {{M J Iss-<k} by definition 2 2 the only solution of (1.1) satisfying the Cauchy conditions X(J)(o) C OsJsp-I is given by (2.3) j' where the matrices D for li-k are uniquely determined by the expression (2.4)For the sake of clarity in the presentation, we recall a result about the solutions of rectangular systems of equations, that will be used in the following sections.
THEO 2. ([13,p.24])The matrix system SP=Q, where S, P, Q are matrices in Cmxn, C and C respectively, is compatible if and only if S S/Q Q and in this case, the solution of the system is given by P S+Q + (I S+S)Z, where Z is an arbitrary matrix in Cnxr.
Note that under the conditions of theorem 2, a particular solution of system SP=Q is given by P=S/Q.
Let us consider the homogeneous problem and let {(M J 1-<Isk} be the k-complete set of co-solutlons of equation {I 5  where the Cnl valued matrix functions Pi(s), Qi{s) have to be determined so that  On the other hand, by the continuity condition of the partlal derivatives of the Green's function unti] order p-2 at t=s, we obtain k k U (j), (s)P,(s} =U (j)l {s)Ql(s) J From (3.10) we have ql(s} P (s) R i( (3.17) Substituting (3.17 Let S be the block matrix Thus the following result has been established TIONE 3. Let {(Hzz J l$1k) be the k-Complete set of co-solutlons of equation {1.5) given by theorem and let {Ui(t} lik be defined by {3.RE.If the matrix S has full rank, then, from [2,p.12]S+S=I, only one solution and there exists a unique Green's matrix function. 4.-SOLUTION OF THE NON-HOMOGENEOUS BOUNDARY PROBLEM.x(P)+ A x(P-I) Let X(t) be defined by f:G(t,s) f(s) ds b oG(L,s) fCs} ds + Get,s} fCs} ds.
t Taking derivatives and using the LeIbniz' rule, we have " @ G(t,s) Jo at f(s)  Then, from the corollary 1, the form of the solutions of {4. .It is interesting to recall that the Jordan canonlcal form of a matrix may be efficiently computed with MACSYMA [9] and the matrix exponential of a Jordan block has a well known expression [12,p.66].
In the next example, we construct a generalized Green's matrix function for a not well-posed boundary value matrix problem.

EXANPLE.
Let us consider the second order differential equation, X''{t) + A X'(t} + A X{t) .
The standard approach to study such problems is based on the consideration of an extended first order problem Y'{t) C Y(t) + F(t); B Y(a) + B Y(b) B and B are b appropriate matrices in Cnqxnp, R is a matrix in Cnqxn, and C is the companion ) i' provided by corollary I.Then, the general solution of equation (3.I) is given by k X(t) U(t)D i=l where D is an arbitrary matrix in Cml and (3.3) Ui(t) M exp(tJ).

I
.-G(t,s) Is a contlnuous matrlx functlon in[O,b]x[O,b] and moreover, a(J)G/at (j) is a contlnuous function in (t,s}, for {t,s} In the trlangles Ost<sb and Oss<tsb for J=1 p-2. As a function of t, G(t,s} satisfies (3.1} and (3.2} in [O,b], if ts.From (3.5) the continuity condition at t=s of Green's function gives us that

U
-)[S)[P (s) ql(s)] I. matrix function U(s] defined by(3, 12} is Invertible for all s, because we may decompose U(s) in the form where is invertible since ((M .3 ), 1-<i<-k} is a k-complete set of co-solution of equation

From theorem 2 ,Pz
and let S be the Moore-Penrose pseudoinverse matrix the equation (3.21} is solvable if and only if z F U (J-z)(b)R (s) us suppose the algebraic equation {3.21) is compatible.Then, from theorem 2 and {3.20} a solution of {3.21} is given by and from (3.15),(3.17) it follows that Qi(s) Pl(s) R(s)( 1 T U:J-Z)(b) exp(-sJ Y } exp( 4).If condition {3.22) is given, then the boundary value matrix problem {3.1) {3.2) has a generalized Green's matrix function defined by {3.5), where P (s) and Qi{s} are given by (3.23) and (3.24).