1 Projector analysis of dynamic systems on time scales

Svetlin G. Georgiev Department of Mathematics, Sorbonne University, Paris, France

Abstract

This chapter presents a projector analysis of dynamic systems on time scales. We investigate the linear time-varying dynamic systems and classify them into those of the first, second, third, and fourth kind. The considered systems are investigated in the case when they are regular with tractability index 1. Then, we define jets of a function of one independent time scale variable and jets of a function of n independent real variables and one independent time scale variable. We introduce jet spaces and give some of their properties. In the chapter, we also define differentiable functions and total derivatives. We consider nonlinear dynamic systems on arbitrary time scales. We define properly involved derivatives, constraints, and consistent initial values for the considered equations. We introduce a linearization for nonlinear dynamic systems and investigate the total derivative for regular linearized equations with tractability index 1.

1.1 Linear time-varying dynamic-algebraic equations

This chapter is devoted to linear time-varying dynamic-algebraic equations. We classify them into those of the first, second, third and fourth kind. We investigate them in the case when they are regular with tractability index 1.

Suppose T is a time scale with forward jump operator and delta differentiation operator s and ?, respectively. Let I?T.

1.1.1 Linear time-varying dynamic-algebraic equations of the first kind

In this section, we will investigate the following linear time-varying dynamic-algebraic equation:

(1.1)As(t)(Bx)?(t)=Cs(t)xs(t)+f(t),t?I,

where A:IMn×m, B:IMm×n, C:IMn×n, and f:IRn are given. Here, with Mp×q we denote the set of all p×q real matrices.

Definition 1.1.

Equation (1.1) is said to be a linear time-varying dynamic-algebraic equation of the first kind.

We will consider the solutions of (1.1) within the space CB1(I). Below, we remove the explicit dependence on t for the sake of notational simplicity.

1.1.1.1 A particular case

Suppose that A,C:IMn×n. Consider the equation

(1.2)Asx?=Csxs+f.

We will show that equation (1.2) can be reduced to equation (1.1). Suppose that P is a C1-projector along kerAs. Then

AsP=As

and

Asx?=AsPx?=As(Px)?-AsP?xs.

Hence, equation (1.2) takes the form

As(Px)?-AsP?xs=Csxs+f,

or

As(Px)?=(AsP?+Cs)xs+f.

Set

C1s=AsP?+Cs.

Thus, (1.2) takes the form

(1.3)As(Px)?=C1sxs+f,

i.?e., equation (1.2) is a particular case of equation (1.1).

Example.

Let

(1.4)T=2N0,A(t)=1000-t1000,C(t)=-t1t012tt01,t?T.

We have

s(t)=2t,t?T,

and

As(t)=1000-2t1000,Cs(t)=-2t12t014t2t01,t?T.

We will find a vector

y(t)=y1(t)y2(t)y3(t),t?T,

so that

As(t)y(t)=000,t?T.

We have

000=1000-2t1000y1(t)y2(t)y3(t)=y1(t)-2ty2(t)+y3(t)0,t?T,

whereupon

y1(t)y2(t)y3(t)=012t,t?T,

and the null projector to As(t), t?T, is

Q(t)=000001002t,t?T.

Hence,

P(t)=I-Q(t)=100010001-000001002t=10001-1001-2t,t?T,

is a projector along kerAs. Note that

P?(t)=00000000-2,C1s(t)=As(t)P?(t)+Cs(t)=1000-2t100000000000-2+-2t12t014t2t01=00000-2000+-2t12t014t2t01=-2t12t014t-22t01.

Equation (1.2) can be written as follows:

1000-2t1000x1?(t)x2?(t)x3?(t)=-2t12t014t2t01x1s(t)x2s(t)x3s(t)+f1(t)f2(t)f3(t),t?T,

or

x1?(t)=-2tx1s(t)+x2s(t)+2tx3s(t)+f1(t),-2tx2?(t)+x3?(t)=x2s(t)+4tx3s(t)+f2(t),0=2tx1s(t)+x3s(t)+f3(t),t?T.

This system, using (1.3), can be rewritten in the form

1000-2t100010001-1001-2tx1(t)x2(t)x3(t)?=-2t12t014t-22t01x1s(t)x2s(t)x3s(t)+f1(t)f2(t)f3(t),t?T,

or

1000-2t1000x1(t)x2(t)-x3(t)(1-2t)x3(t)?=-2tx1s(t)+x2s(t)+2tx3s(t)+f1(t)x2s(t)+(4t-2)x3s(t)+f2(t)2tx1s(t)+x3s(t)+f3(t),t?T,

or

1000-2t1000x1?(t)x2?(t)-x3?(t)(1-4t)x3?(t)-2x3(t)=-2tx1s(t)+x2s(t)+2tx3s(t)+f1(t)x2s(t)+(4t-2)x3s(t)+f2(t)2tx1s(t)+x3s(t)+f3(t),t?T,

or

x1?(t)=-2tx1s(t)+x2s(t)+2tx3s(t)+f1(t),-2t(x2?(t)-x3?(t))+(1-4t)x3?(t)-2x3(t)=x2s(t)+(4t-2)x3s(t)+f2(t),0=-2tx1s(t)+x3s(t)+f3(t),t?T,

or

x1?(t)=-2tx1s(t)+x2s(t)+2tx3s(t)+f1(t),-2tx2?(t)+(1-2t)x3?(t)=x2s(t)+(4t-2)x3s(t)+2x3(t)+f2(t),0=2tx1s(t)+x3s(t)+f3(t),t?T.

1.1.1.2 Standard form index 1 problems

In this section, we will investigate the equation

(1.5)As(Px)?=Csxs+f,

where kerA is a C1-space, C?C(I), P is a C1-projector along kerA. Then

AP=A.

Assume in addition that

Q=I-P

and

(B1)

the matrix

A1=A+CQ

is invertible.

Definition 1.2.

Equation (1.5) is said to be regular with tractability index 1.

We will start our investigations with the following useful lemma.

Lemma 1.1.

Suppose that (B1) holds. Then

A1-1A=P

and

A1-1CQ=Q.

Proof.

We have

A1P=(A+CQ)P=AP+CQP=A.

Since Q=I-P and kerP=kerA, we have imQ=kerA and

AQ=0.

Then

...