LARGE SCALE SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS

In this paper an alternative approach to the method of asymptotic expansions for the study of a singularly perturbed linear system with multiparameters and multiple time scales is developed. The method consists of developing a linear non-singular transformation that transforms an arbitrary n-time scale system into a diagonal form. Furthermore, a dichotomy transformation is employed to decompose the faster subsystems into stable and unstable modes. Fast, slow, stable and unstable modes decomposition processes provide a modern technique to find an approximate solution of the original system in terms of the solution of an auxiliary system. This method yields a constructive and computationally attractive way to investigate the system.

1 Introduction Singular perturbations of two point boundary value problem is an active subject of research with long history.By employing the asymptotic expansion of such systems under strong conditions on the coefficient matrices, existence, uniqueness and approximations of solutions of such systems are studied in [12,13,14].In [2,5,15,16], under less demanding conditions on the coefficient matrices, boundary value problems for two-time scale linear systems are analyzed.Furthermore, the fast and slow mode decomposition approach [2,9,16] provides a modern alternative technique to study the singularly perturbed systems.Moreover using a dichotomy transformation to decompose the faster subsytems into stable and unstble modes yields a constructive and computationally atractive procedure to investigate such problems [10,15,17,18].
In this paper, by following a hierarchial order reduction scheme of a joint multiparameters and multi-time scale singular perturbation of linear system [6,8,11], a linear non-singular transformation which totally decouples an arbitrary n-time scale and multiparameter linear singularly perturbed system, is developed.Furthermore, a dichotomy transformation is employed to decompose the faster subsytems into stable and unstble modes.Sufficient conditions are given for existence and uniqueness of solution of an auxiliary boundary value problem which guarantees the existence, uniqueness and approximation of solution of given boundary value problem.The obtained conditions are given in terms of known coefficient matrices.This paper is organized as follows: In section 2, by following the argument [6,8,11], a joint multi- time scale multiparameter singularly perturbed two point boundary value problem is formulated.For the sake of convenience and simplicity a few notations are defined in section 3.These notations will be used throughout this paper.In section 4, by following the decoupling procedure in [6], a totally decoupling process is briefly discussed.Moreover, the asymptotic behaviour and the representation of the transformation in terms of the given coefficient matrices is given.In section 5, by developing a dichotomy transformation which decomposes the faster subsystems of totally decoupled system into stable and unstable subsystems, the existence, uniqueness and approximation of the original problems are investigated.An example illustrating the decomposing procedures and finding an approximate solution is presented in section 6.This example exhibits the generalization and extension of the work of Wilde and Kokotovic [17,18].
n N-En i.
i=l In (2.1) all the coefficient matrices are continuous on IR + and have appropriate dimensions.For jeJ(1, ri) and e J( 1, n), parameters 'i's are positive real numbers.For fixed i J(1,n), ali's have the same order with respect to j.
following inequality is valid: This means that for fixed E J(1,n), the _.e _< < T j,ke J(1,ri).

S , S
where_e , are some positive numbers.Furthermore, for any j J(1,ri), have different order with respect to i.
For simplicity we omit the arqument of the coefficient matrix functions.
are block matrices being defined by: The matrices in (2.5) that are formed in an obvious way from the coefficient matrices in (2.1) with Di's where Iir" are the identity matrices of appropriate dimensions.
of the D matrices are bounded, that is,
For the sake of simplicity and convenience to our future presentation, let us introduce the following notations.

Lm
In-m + en-mLmMmJ is the inverse of Tn-m, for mJ(0,n-2), and hence 'n-m is invertible.Denoting n-m as the inverse of 'n-m, $2= nO, n_lo...o2 is an inverse of T2.Therefore the solution of the original system can be given interms of the solution of the totally decoupled multi time scale system (4.10)--(4.11)by X(t) S2(t)U(t) where U(t) is the slution of (4.10)-(4.11).
The validity of the transformation is established, by establishing the existence, uniqueness, boundedness and other fundamental properties of (4.7) for rn E J(0,n-2) in [7].For the sake of completeness, we present the desired assumptions.
For this purpose let us introduce a few notations.
From the choice of Pl' (5.12), we get, n e k IlANm(t)ll _< P- O(i k "i).

k--n--rn
This completes the boundedness of solution of (5.7).It is obvious that Nm defined by Nm(t) m(t) + ANm(t) is a solution of (5.2).This together with the boundedness of N m and ANm implies the existence of bounded solution N m of (5.2).
Proof: Proof is analogous to the proof of Theorem 5.1.Therefore, we omit the details.Now we will apply a transformation to the totally decoupled system (4.10)so that it decomposes every faster sub-systems into stable and unstable subsystems and leaves the slowest subsystem as it is.
We apply the transformation E defined by T T T (U1T U?I u2T2...Unl Un2 (Wl T w2T1 w2T2 ...Wnl Wn2 ) = Z T T )T to totally decoupled system (4.10) with boundary condition (4.11).(5.15) Now we establish the existence and uniqueness of a bounded solution of the totally decoupled system (4.10) with boundary condition (4.11) and the original boundary value problem (2.5)-(2.6).
Our objective in the rest of this section is to find an approximate solution of the totally decomposed system (5.14)-(5.15)by a solution of a boundary value problem with the given coefficient matrices and hence to find an approximate solution of the original boundary value problem (2.5)-(2.6).
For this purpose let us consider an suxiliary system corresponding to the totally decomposed system (5.14) as follows; where E = diag'Inl.

Inn_m _km
We observe that the fundamental matrix solution of (5.20) approximates the fundamental matrix solution of (5.14).Now we assume the following assumptions.
Remark 5.1" We note that the dichotomy transformation is determined by the system of differential equation in (5.2) and (5.3).The results of this section generalize the results of Wilde and Kokotovic  [17,18] in a systematic way.This fact can be justified in the succeeding sections.

Application
In this section we consider an optimal control problem.The preceeding results are applied to find an approximate solution of the boundary value problem associated with the necessary optimality condition.
Let us consider an optimal control problem.

observed that
From this, it can be -1
Since P(0) and -II22 +N(1) are invertible, it is enough to verify the invertibility of +21 ( 1) +22 (1) To verify the above statement, we present the following lemma.
Now by the application of Theorem 5.4, the expression in (5.23) and (5.24) with respect to (6.2) and (6.3) are given by n