DSpace at VNU: ON INITIAL AND BOUNDARY VALUE PROBLEMS FOR IMPLICIT DYNAMIC EQUATIONS ON TIME SCALES

DOI 10.1007/s40306-013-0026-zON INITIAL AND BOUNDARY VALUE PROBLEMS FOR IMPLICIT DYNAMIC EQUATIONS ON TIME SCALES Nguyen Chi Liem Received: 29 December 2011 / Revised: 10 September 2012

DOI 10.1007/s40306-013-0026-z

ON INITIAL AND BOUNDARY VALUE PROBLEMS

FOR IMPLICIT DYNAMIC EQUATIONS ON TIME

SCALES

Nguyen Chi Liem

Received: 29 December 2011 / Revised: 10 September 2012 / Accepted: 24 September 2012

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer

Science+Business Media Singapore 2013

Abstract This paper is concerned with the index concept, the unique solvability of the

initial value problem, the two point boundary-value problem and Green’s function for a class of linear implicit dynamic equations with index-1 on time scales The results are a generalization of the previous ones for differential and difference-algebraic equations

Keywords Time scales· Implicit dynamic equation · Linear dynamic equation · Index ·

Boundary-value problem· Green’s function

Mathematics Subject Classification (2000) 39A11· 34A09 · 65L80 · 65F20 · 34D20

1 Introduction

In the last decades, the theory of differential-algebraic equations (DAEs for short) has been

an intensively discussed field in both theory and practice The general form of DAEs is

f

t, x(t ), x(t )

and its linearization has the form

where A . and B .are given matrix functions Equations (1) and (2) can be seen in many real problems, such as in electric circuits, chemical reactions, vehicle systems,

If the matrices At are invertible for all t∈ R, we can multiply both sides of (2) by A−1t

to obtain an ordinary differential equation However, in the case where there is at least one

t0 ∈ R such that A t0is singular, some further assumptions need to be posed One of the ways

N.C Liem (B)

Department of Mathematics, Mechanics and Informatics, Vietnam National University,

334 Nguyen Trai, Hanoi, Vietnam

e-mail: liemlkqn2005@gmail.com

to investigate (2) is to introduce the index concept of the equation Based on this concept,

we can study (2) by decomposing it into an ordinary differential equation and algebraic relations Results on the solvability of the Cauchy problem for (2) can be found in [10]; for the boundary value problems, we can refer to [17]

Together with the theory of differential-algebraic equations, there has been a great in-terest in singular difference equations (SDEs) (also referred to as descriptor systems, im-plicit difference equations) because of their appearance in many practical areas, such as in the Leontiev dynamic model of multisector economy, the Leslie population growth model, singular discrete optimal control problems and so forth (see [6,7]) On the other hand, SDEs occur in a natural way of using discretization techniques for solving DAEs and par-tial differenpar-tial-algebraic equations , which have already attracted much attention of re-searchers (cf [6,10,13])

Bondarenko and his colleagues in [5] considered a special class of implicit

nonau-tonomous difference equation T (n)x(n + 1) + x(n) = f (n), where T (n) are

degener-ate matrices, and established the solvability of initial value problems (IVPs) and periodic boundary-value problems (BVPs) for this special class of SDEs

The index notion for linear SDEs with time-varying coefficients

was introduced in [8,16] and the solvability of IVPs as well as multipoint BVPs are studied

in [1,3] Later on, the index notion has been extended to nonlinear cases f (n, x(n + 1),

x(n))= 0 [2]

There is a close relation between linear SDEs and linear DAEs, namely, the explicit Euler method applied to a linear index-1 DAE leads to a linear index-1 SDE (see [1,3]) and the unique solutions of the discretized IVP/BVP converge to the solutions of the corresponding continuous problems

Further, in recent years, to unify the continuous and the discrete analyses or to describe the process of numerical calculation with non-constant steps, a new theory was born and is more and more extensively concerned, that is, the theory of the analysis on time scales The most popular examples of time scales areT = R and T = Z Using the “language” of time

scales, we rewrite (2) and (3) in the form

or in the general form

f

t, x Δ (t ), x(t )

= 0,

with t in time scale T and Δ being the derivative operator on T.

A natural question is whether the existing results for (2) and (3) can be extended and unified for the implicit dynamic equations of the form (4) The purpose of this paper is to answer a part of that question We will study the solvability of the Cauchy problem and some matters concerning the boundary-value problem of (4)

The organization of this paper is as follows In Sect.2we summarize some results about the analysis on time scales In Sect.3, we introduce the index-1 concept and deal with the Cauchy problem of linear implicit dynamic equations (4) (LIDEs) The technique we use

in this section is somewhat similar to the one in [8,10] However, we need some improve-ments because of the complicated structure of a time scale Section4deals with the solution uniqueness for the two point boundary-value problem and constructs the Green function

2 Preliminaries

This section surveys some notions on the theory of the analysis on time scales which was introduced by Stefan Hilger in 1988 [11] A time scale is a nonempty closed subset of the real numbers R, and we usually denote it by the symbol T We assume throughout

that a time scale T is endowed with the topology inherited from the real numbers with

the standard topology We define the forward jump operator and the backward jump

op-erator σ, ρ : T → T by σ(t) = inf{s ∈ T : s > t} (supplemented by inf ∅ = sup T) and

ρ(t ) = sup{s ∈ T : s < t} (supplemented by sup ∅ = inf T) The graininess μ : T → R+∪{0}

is given by μ(t) = σ(t) − t A point t ∈ T is said to be right-dense if σ(t) = t and t < sup T, right-scattered if σ (t) > t , left-dense if ρ(t) = t and t > inf T, left-scattered if ρ(t) < t, and isolated if t is right-scattered and left-scattered For every a, b ∈ T, by [a, b] we mean the

set{t ∈ T : a  t  b} The set T kis defined to beT if T does not have a left-scattered

max-imum; otherwise it isT without this left-scattered maximum Let f be a function defined

onT, valued in Rm We say that f is delta differentiable (or simply: differentiable) at t∈ Tk provided there exists a vector f Δ (t )∈ Rm , called the derivative of f , such that for all  > 0 there is a neighborhood V around t with f (σ(t)) − f (s) − f Δ (t )(σ (t ) − s)  |σ(t) − s|

for all s ∈ V If f is differentiable for every t ∈ T k , then f is said to be differentiable onT

IfT = R then delta derivative is f(t )from continuous calculus; ifT = Z, the delta

deriva-tive is the forward difference, Δf , from discrete calculus A function f defined on T is

rd-continuous if it is continuous at every right-dense point and if the left-sided limit exists

at every left-dense point The set of all rd-continuous functions fromT to a Banach space

X is denoted by Crd( T, X) A matrix function f from T to R m ×m is said to be regressive if

det(I + μ(t)f (t)) = 0 for every t ∈ T k

Theorem 1 (See [4]) Let A( ·) be an rd-continuous m × m-matrix function Then, for any

t0∈ Tk , the IVP

has a unique solution x( ·) defined on t  t0 Further, if A(·) is regressive, this solution exists

on t∈ Tk

The solution of the corresponding matrix-valued IVP X Δ (t ) = A(t)X(t), X(t0)= I ,

which is called the Cauchy operator of the dynamic equation (5) and denoted by ΦA(t, t0), always exists for t  t0, even A(·) is not regressive (see [12,19]) If we suppose further that

A( ·) is regressive, the Cauchy operator Φ A(t, t0) is defined for all t, t0∈ Tk

It is seen that any solution x( ·) of the dynamic equation (5) can be written as x( ·) =

ΦA( ·, t0)x0and the cocycle property

ΦA(t, τ ) = Φ A(t, s)ΦA(s, τ )

is valid for all τ  s  t.

If A(t) commutes with its integral t

t0A(s)Δs, (in particular, A(t)≡ constant matrix

satisfies this straightforwardly) then we denote eA(t, t0) instead of ΦA(t, t0).

Theorem 2 (Constant variation formula, see [12]) Let A: Tk→ Rm ×m and f : Tk× Rm→

Rm be rd-continuous and there exists a solution x(t), t  t0for the dynamic equation x Δ=

A(t )x + f (t, x), x(t0)= x0 Then

x(t ) = Φ A(t, t0)x0+

 t

t

ΦA

t, σ (s)

f

s, x(s)

We refer to [12,18] for more information on analysis on time scales.

3 Linear implicit dynamic equations on time scales

LetT be a time scale We consider the linear implicit dynamic equation of the form

The homogeneous equation associated to (7) is

where A., B ∈ Crd(Tk ,Rm ×m ), q. ∈ Crd(Tk ,Rm ) In the case where the matrices At are

in-vertible for every t∈ T, we can multiply both sides of (7) by A−1t to obtain an ordinary dynamic equation

x Δ = A−1

t Bt x + A−1

t qt, t ∈ T,

which has been well studied If there is at least a t such that A tis singular, we cannot solve

explicitly the leading term x Δ In fact, we are concerned with a so-called ill-posed problem where the solutions of the Cauchy problem may exist only on a submanifold or even they

do not exist One of the ways to solve this equation is to impose some further assumptions stated under the form of indices of equation

We introduce the so-called index-1 of (7) Suppose that rank At = r for all t ∈ T and

let T t ∈ GL(R m ) be such that T t|ker A t is an isomorphism between ker A t and ker A ρ(t );

T ∈ Crd(Tk ,Rm ×m ) Let Q

t be a projector onto ker A t satisfying Q . ∈ Crd(Tk ,Rm ×m ) We

can find such operators Tt and Qt in the following way: let At possess a singular value decomposition

At = U t ΣtV t , where Ut, Vt are orthogonal matrices and Σt is a diagonal matrix with singular values

σ1

t  σ2

t  · · ·  σ r

t > 0 on its main diagonal Since A . ∈ Crd(Tk ,Rm ×m ), in the above

decomposition of A t we can choose V t ∈ Crd(Tk ,Rm ×m ) (see [7]) Hence, by putting

Qt = V t diag(0, Im −r )V t and Tt = V ρ(t )V t , we obtain Qt and Vt as the requirement

Let Qt and Tt be such matrices and put Pt := I − Q t We suppose further that Qρ(t )is rd-continuously differentiable onTk

It is known that ρ(σ (t)) = t if and only if t is not right-dense and left-scattered at the

same time We consider the case where t = t0is right-dense and left-scattered at the same

time (ρ(t0) < t0 = σ(t0)) Then, from the continuity of Qρ(·) and Q·at t0we get the equali-ties limt →t0Qρ(t ) = Q ρ(t0)and limt →t0Qρ(t ) = Q t0 Therefore, Q ρ(t0) = Q t0

Thus, by the above assumptions of the projector Qt we always have Qρ(σ (t )) = Q tfor all

t∈ Tk

From the relation



P ρ(t ) x(t )Δ

= P ρ(σ (t )) x Δ (t ) + (P ρ(t ) ) Δ x(t ) = P t x Δ (t ) + (P ρ(t ) ) Δ x(t ), for all t∈ Tk, we get

At x Δ (t ) = A tPt x Δ (t ) = APρ(t )x(t )Δ

− (P ρ(t )) Δ x(t )

.

Therefore, the implicit dynamic equation (7) can be rewritten as

At(Pρ(t )x) Δ=At(Pρ(t )) Δ + B t



x + q t, t∈ Tk

Thus, we should look for solutions of (7) from the space C1

N:

C N1

Tk ,Rm

=x( ·) ∈ Crd



Tk ,Rm

:P ρ(t )x(t ) is differentiable at every t∈ Tk

Note that C1

N does not depend on the choice of the projector function since the relations

Pt Pt= P tand PtPt= Pt are true for each two projectors Ptand Pt along the space ker At.

Let

S t=x∈ Rm : B t x ∈ imA t



.

Under these notations, we have:

Lemma 1 [9] The following assertions are equivalent

1 kerA ρ(t ) ∩ S t= {0},

2 The matrix Gt = A t − B t Tt Qt is nonsingular,

3 Rm = ker A ρ(t ) ⊕ S t for all t∈ Tk

Lemma 2 [9] Suppose that the matrix G t is nonsingular Then we have the following asser-tions:

Pt = G−1

Qt = −G−1

− T tQtG−1t Bt is the projector onto ker A ρ(t ) along S t, (11)

Qt G−1t Bt = Q t G−1t BtPρ(t ) − T t−1Qρ(t ), (13)

TtQt G−1t does not depend on the choice of T t and Q t (14)

Definition 1 The LIDE (7) is said to be of index-1 if for all t∈ Tk, the following conditions hold:

(i) rankAt = r= constant (1  r  m − 1),

(ii) ker A ρ(t ) ∩ S t= {0}

Assume that (7) is of index-1

We now describe briefly the decomposition technique for (7)

By Lemma1, since (7) has index-1, Gt is nonsingular for all t∈ Tk Multiplying (7) by

Pt G−1t and Q tG−1t , respectively, yields

PtG−1t At x Δ = P tG−1t Btx + P t G−1t qt ,

Q t G−1t A t x Δ = Q t G−1t B t x + Q t G−1t q t

Applying (9), we have

Ptx Δ = P t G−1t Btx + P tG−1t qt,

Due to (12) and (13), (15) becomes

⎧

⎪

⎪

(Pρ(t )x) Δ = (P ρ(t )) Δ (I + T tQtG−1t Bt)(Pρ(t )x) + P t G−1t Bt(Pρ(t )x)

+ (P t + (P ρ(t )) Δ Tt Qt)G−1t qt, Qρ(t )x = T tQt G−1t BtPρ(t )x + T tQtG−1t qt.

(16)

Therefore, with u(t) := P ρ(t )x(t ), v(t ) = Q ρ(t )x(t ), (16) becomes an ordinary dynamic equation onT

u Δ = (P ρ(t )) Δ

I + T tQt G−1t Bt

u + P t G−1t Btu+Pt + (P ρ(t )) Δ TtQt

G−1t qt, (17) and an algebraic relation

v = T tQtG−1t Btu + T tQt G−1t qt (18)

Let t0∈ Tk Solving u(t) from (17) with the initial condition u(t0) = P ρ(t0)x0 and using the relation (18), we get an expression of the solutions of the index-1 LIDE (7) x(t)=

u(t ) + v(t) for t  t0.

Inspired by the above decoupling procedure, we state initial conditions for the index-1 LIDE (7) as

P ρ(t0)

x(t0) − x0



It follows that

u(t0) = P ρ(t0)x(t0) = P ρ(t0)x0,

but we do not expect

x(t0) = x0

as in the case of ordinary dynamic equations on time scales

Denote Qtcan := −T tQtG−1t Bt, Ptcan := I − Q tcan By (11), Qtcan projects onto ker Aρ(t ) along St and is called the canonical projector for the index-1 case Note that Qtcan is

rd-continuous and independent from the choice of Q t and T t The solutions of (7) with the initial condition (19) are represented by

x(t ) = P ρ(t ) x(t ) + Q ρ(t ) x(t ) = (I + T t Q t G−1t B t )u + T t Q t G−1t q t

where u ∈ C1

rd solves from the inherent ordinary dynamic equation (17) with the initial

condition u(t0)= P ρ(t0)x0

By multiplying both sides of the homogeneous equation associated to (17) with Qtand using the fact that

0= (Q ρ(t )Pρ(t )) Δ = Q t (Pρ(t )) Δ + (Q ρ(t )) Δ Pρ(t ) =⇒

Q t (P ρ(t ) ) Δ = −(Q ρ(t ) ) Δ P ρ(t ) ,

one has

Qtu Δ = Q t(Pρ(t )) Δ Ptcanu = −(Q ρ(t )) Δ Pρ(t )Ptcanu = −(Q ρ(t )) Δ Pρ(t )u.

Further, since Q t u Δ = (Q ρ(t )u) Δ − (Q ρ(t )) Δ uwe get

(Qρ(t )u) Δ = (Q ρ(t )) Δ (Qρ(t )u).

Hence, if Qρ(t0)u(t0) = 0 then Q ρ(t )u(t ) = 0 for all t  t0 Therefore, (17) has the invariant property:

if x(t0) ∈ im P ρ(t0) then x(t) ∈ im P ρ(t ) for all t∈ Tk (21) Consider the homogeneous equation (8), i.e., q t ≡ 0 Let Φ0(t, t0)be the matrix solution

of the dynamic equation

(Φ0(t, t0)) Δ = (P ρ(t )) Δ (I + T tQtG−1t Bt)Φ0(t, t0) + P tG−1t BtΦ0(t, t0), t  t0,

Φ0(t0, t0) = I.

Then, due to (20), the solution of the matrix equation

A t (Φ(t, t0)) Δ = B t Φ(t, t0), t  t0,

Pρ(t0)(Φ(t0, t0) − I) = 0

can be expressed by the formula

Φ(t, t0)=I + T t Q t G−1t B t

Φ0(t, t0)P ρ(t0) = P tcanΦ0(t, t0)P ρ(t0) , t  t0. (22)

It is easily verified that ker Φ(t, t0) = ker A ρ(t0) and im Φ(t, t0) = im P tcan = S t hold The

matrix solution Φ(t, t0) t t0is called the Cauchy operator of (8)

Further, due to the invariant property (21) of the solutions of (17), we have

Pρ(t )Φ(t, s) = P ρ(t )Ptcan Φ0(t, s)Pρ(s) = Φ0(t, s)Pρ(s), t  s. (23) Therefore

Φ(t, s)Φ(s, τ ) = Φ(t, τ) for all τ  s  t,

and the unique solution of (7) with the initial condition (19) can be given by the constant variation formula

x(t ) = P tcanΦ0(t, t0)Pρ(t0)x0

+

 t

t0 PtcanΦ0

t, σ (s)

Pρ(σ (s))

Ps + (P ρ(s)) Δ TsQs

G−1s qsΔs

+ T tQt G−1t qt , t  t0,

or equivalently

x(t ) = Φ(t, t0)Pρ(t0)x0+

 t

t0

Φ

t, σ (s)

Ps + (P ρ(s)) Δ TsQs

G−1s qsΔs

+ T t QtG−1t qt, t  t0.

(24)

Now suppose that u = P ρ(t )x satisfies the homogeneous equation associated (17)

corre-sponding to some operators Tt and Qt, i.e.,

(Pρ(t )x) Δ = (P ρ(t )) Δ PtcanPρ(t )x + P t G−1BtPρ(t )x.

Let Tt be another linear transformation fromRmontoRmsuch that Tt|ker A t is an

iso-morphism between ker At and ker Aρ(t ) and Qt be a projector onto ker At such that Qt is rd-continuous and Qρ(t )is rd-continuously differentiable By denoting Pt = I −  Qt,  Gt=

At − B t TtQt, we have

(  Pρ(t )x) Δ = ( Pρ(t )Pρ(t )x) Δ

= Pt(Pρ(t )x) Δ + ( Pρ(t )) Δ Pρ(t )x

= Pt(Pρ(t )) Δ PtcanPρ(t )x+ PtG−1t BtPρ(t )x + ( Pρ(t )) Δ Pρ(t )x

= ( Pρ(t )Pρ(t )) Δ PtcanPρ(t ) Pρ(t )x − ( Pρ(t )) Δ Pρ(t )Ptcan Pρ(t )x

+ PtG−1t BtPρ(t )x + ( Pρ(t )) Δ Pρ(t )x

= ( Pρ(t )) Δ Ptcan Pρ(t )x − ( Pρ(t )) Δ Pρ(t )x

+ PtG−1t BtPρ(t )x + ( Pρ(t )) Δ Pρ(t )x

= ( Pρ(t )) Δ Ptcan Pρ(t )x + Pt G−1

t BtPρ(t )x.

Moreover, it is easy to prove that Pt G−1t = Pt G−1

t which implies



PtG−1t Bt Pρ(t )x= Pt G−1

t BtPρ(t )x= Pt G−1

t Bt Pρ(t )Pρ(t )x = Pt G−1

t Bt Pρ(t )x.

Therefore we obtain

(  Pρ(t )x) Δ = ( Pρ(t )) Δ Ptcan Pρ(t )x + Pt G−1

t Bt Pρ(t )x.

Furthermore, we also have the relation

Qρ(t )x = −Q tcanPρ(t )x ⇒ Qρ(t )x = −Q tcan Pρ(t )x.

This shows that the Cauchy operator of (8) does not depend on the choice of T t and Q t, and

hence neither the expression of x by (24)

Consider the case where the right-hand side of the homogeneous equation associated to (17), i.e., the matrix



At = (P ρ(t )) Δ Ptcan + P tG−1t Bt,

is regressive (obviously, At is rd-continuous) With this assumption, the inherent dynamic equation (17) has a unique solution defined onTk Thus,

Theorem 3 Given an index-1 LIDE (7), then

1 for each t0∈ Tk , x0∈ Rm , q ∈ Crd(Tk ,Rm ), the LIDE (7) with the initial condition (19)

is uniquely solvable for t  t0,

2 further, with the assumption  At to be regressive, exactly one solution of the homogeneous equation x(t) of (8) passes through each x0∈ S t0 at t0

Remark 1

1 WhenT = R (ρ(t) = t for all t ∈ R) we choose T t = −I to see the result mentioned in

[10] For the caseT = Z, the result can be seen in [3]

2 On the solvability of IVP for the quasi-linear implicit dynamic equations A t x Δ = B tx+

f (t, x), with small perturbation f (t, x) and for the equations At x Δ = f (t, x), with the

assumption of differentiability for f (t, x), we can refer to [9]

To finish this section, we give an example

Example 1 Consider the time scaleT =∞k=0[2k, 2k + 1] and the equation A t x Δ = B t x+

qt, with

At=



1 1

t t









1 2



.

We have ker A t = span{(1, −1) T }, rank A t = 1 for all t ∈ T It is easy to verify that Q t=

1

2

1 −1

−1 1



is a projector onto ker A t Let us choose T t = I and observe that

Gt = A t − B tTtQt=



1 1

t t



−





1 2





=



t−1

2 t+1 2



Since det Gt= 1 = 0, (7) has index-1 We get

G−1t =



t+1

2 −1

2 1



, Tt QtG−1t Bt=



t2− t t2− t + 1

−t2+ t −t2+ t − 1



,

TtQt G−1t qt=



t− 2

−t + 2



, Ptcan=



t2− t + 1 t2− t + 1



.

Note that At=1

2

t −1 t−1

t −1 t−1



is regressive (and rd-continuous) Indeed,

det

I + μ(t) At

=

1 if t∈∞k=0[2k, 2k + 1)

2t if t∈∞

Therefore the equation A t x Δ = B t x + q t with the initial condition P ρ(t0) (x(t0) − x0)= 0 is

uniquely solvable onTkand its solution is represented by

x = P tcanu + T t QtG−1t qt

=



t2− t + 1 t2− t + 1



u+



t− 2

−t + 2



,

where u satisfies u Δ=1

2

t −1 t−1

t −1 t−1+



2

t −1 t−1

t −1 t−1



, also the initial condition u(t0)=

P ρ(t0) x0.

Put h(t):=1

2(t − 1), we can find

Φ0(t, t0)=



e2h(t, t0)−t

t0e2h(t, σ (s))h(s)Δs t

t0e2h(t, σ (s))h(s)Δs e2h(t, t0)− 1 −t

t0e2h(t, σ (s))h(s)Δs 1+t

t0e2h(t, σ (s))h(s)Δs



∀t, t0∈ Tk

Therefore the Cauchy operator of the above equation is

Φ(t, t0) = P tcanΦ0(t, t0)Pρ(t ) = 2e 2h(t, t0)Ptcan.

4 Boundary value problem and Green’s function

By using the results mentioned in Sect.3, we study the existence and uniqueness of solutions

of two point BVPs of LIDEs Let t0, t1∈ Tk, we consider the dynamic equation

with t ∈ [t0, t1], and the boundary condition for this equation is

where D1, D2 are constant square matrices of order m, and b ∈ M := im((D1, D2)). Throughout this section we suppose that (25) is of index-1 By (16), if x(t) is a solution

of (25), it must be defined on t ∈ [t0, σ (t1)], instead of on [t0, t1], because we have to define

the derivative of u(t) = P ρ(t ) x(t ) at t = t1.

We recall the notion of Moore–Penrose pseudoinverse of a matrix

Let X be an m × m-matrix Then there exists a unique matrix Y satisfying the conditions

(i) Y XY = Y ,

(ii) XY X = X,

(iii) XY = R⊥,

(iv) Y X = P⊥,

where Q⊥= I − P⊥ (resp R⊥) is the orthogonal projector onto ker X (resp onto im X).

We call the matrix Y - the Moore–Penrose pseudoinverse of X and denote it by X+ If X is nonsingular then X+= X−1.

We keep all notations and hypotheses as in Sect.3on the matrices At, Bt and on the

operators Tt, Qt Note that here the assumption that these matrices and operators are defined

only on[t0, σ (t1)] instead of on T is sufficient

By these assumptions, (25) with the initial condition Pρ(t0)(x(t0) − x0)= 0 has a unique

solution x(t ; t0, x0)given by (24), where Φ(t, t0)is given by (22)

For the sake of simplicity, we denote

Φ(t, t0) = Φ(t),

x q (t )=

 t

t0

Φ

t, σ (s)

P s + (P ρ(s) ) Δ T s Q s

G−1s q s Δs + T t Q t G−1t q t

Then

x(t ; t0, x0)= Φ(t)P ρ(t0)x0 + x q(t ). (27)

It is easy to see that x q (t )is a partial solution of (25) satisfying P ρ(t0) x q (t0)= 0

We are now in a position to find the initial condition x0such that x( ·; t0, x0)is the solution

of the dynamic equation (25) satisfying the boundary condition (26) Substituting (27) into (26) we obtain

D1

Φ(t0)x0 + x q (t0)



Φ(t1)x0 + x q (t1)

= b,

or



D1Φ(t0) + D2Φ(t1)