^{1}

^{2}

The use of functions, expressible in terms of Lucas polynomials of the second kind, allows us to write down the solution of linear dynamical systems—both in the discrete and continuous case—avoiding the Jordan canonical form of involved matrices. This improves the computational complexity of the algorithms used in literature.

Even in recent books (see e.g. [

In order to avoid this serious problem, we propose here an alternative method, based on recursion, using the

After recalling the

In Section 3, we prove our main results, relevant to an alternative method for the solution of linear dynamical systems, both in the discrete and continuous time case and via the Riesz-Fantappiè formula, also known in literature as the Dunford-Schwartz formula [

Some concrete examples of computation are presented in Section 4, showing the more simple complexity of our procedure with respect to the traditional algorithms, as they appear in the above mentioned books.

We want to remark explicitly that, in our article, by using the

Consider the

Supposing the coefficients vary, its solution is given by every bilateral sequence

that

A basis for the r-dimensional vectorial space

Since

Therefore, assuming the initial conditions

For further considerations, relevant to the classical method for solving the recurrence (1.1), see [

An important result, originally stated by É Lucas [

showing that all

Therefore, we assume the following

Definition 1-The bilateral sequence

is called the fundamental solution of (1.1) (“fonction fondamentale” by É. Lucas [

For the connection with Chebyshev polynomials of the second kind in several variables, see [

In preceding articles [

Theorem 1 Given an

its characteristic polynomial (or possibly its minimal polynomial, if this is known), the matrix powers

where the functions

Moreover, if

It is worth to recall that the knowledge of eigenvalues is equivalent to that of invariants, since the second ones are the elementary symmetric functions of the first ones.

Remark 1 Note that, as a consequence of the above result, the higher powers of matrix

It is well known that an analytic function f of a matrix

is assumed for defining (and computing)

Let

the polynomial interpolating

If the eigenvalues are all distinct,

This avoids the use of higher powers of

A classical result is as follows:

Theorem 2 Under the hypotheses and definitions considered above, the resolvent matrix

Then, by the Riesz-Fantappiè formula, we recover the classical result:

Theorem 3 If

In particular:

Remark 2 If the eigenvalues of

As a consequence of the above recalled results, we can prove our main results both in the discrete and continuous time case.

Theorem 4 Consider the dynamical problem for the homogeneous linear recurrence system

where

Let

denote by

Define the vector

and the matrix

then, the solution of problem (3.1) can be written

That is, for the components:

Proof It is well known that the solution of problem (3.1) is given by

From the results about matrix powers, it follows that

Then, taking into account the above definitions of vectors

Remark 3 Note that, even if this is unrealistic, solution (3.2) still holds for negative values of n, assuming definition (1.2) for the

Theorem 5 Consider the Cauchy problem for the homogeneous linear differential system

where

Let

denote by

Introduce the matrix

then, the solution of problem (3.3) can be written

Proof-It is well known that the solution of problem (3.3) is given by

From the results about matrix exponential, it follows that

where

so that Equation (3.5) becomes

and taking into account the above positions, it follows

Then, Equation (3.4) immediately follows by introducing the vector function

Remark 4 Note that the convergence of the vectorial series in any compact set K of the space

By using the Riesz-Fantappiè it is possible to avoid series expansions. Indeed, we can prove the following result.

Theorem 6 The solution of the Cauchy problem (3.3) can be found in the form

where we denoted by

Proof It is a straightforward application of the Riesz-Fantappiè formula, taking into account the definition of

We show that the above results are easier with respect to the methods usually presented in literature ( [

We consider the

with matrix

The invariants of

We will consider, the initial conditions:

Then, as a consequence, we have:

and

Starting from the initial conditions:

and by means of the recurrence relation

with

The (4.4) coincides with the following solution of the problem (4.1)-(4.2) obtained with the classical method of eigenvalues

We consider the

with matrix

The invariants of

We will consider, the Cauchy problem with initial conditions:

Then, as a consequence, we have:

and

Starting from the initial conditions (4.3) and by means of the recurrence relation

with

Here we compute an approximation of the solution of the Cauchy problem obtained by a suitable truncation of order N of the Taylor expansion

The exact solution of the Cauchy problem (4.5)-(4.6) is

such that we can compute, by using a Mathematica program, the approximation error obtained, for some values of N, in a fixed points t of the real axes. For example for

We consider the

with matrix

The invariants of

We will consider, the Cauchy problem with initial conditions:

Then, as a consequence, we have:

and

Starting from the initial conditions:

and by means of the recurrence relation

with

Here we compute an approximation of the solution of the Cauchy problem obtained by a suitable truncation of the Taylor expansion

Consider the problem

with matrix

Characteristic polynomial

Matrix eigenvalues

Matrix invariants

From the initial condition

we find

Riesz-Fantappiè formula

i.e.

Integrals computation (using the Residue Theorem).

Solution of the problem

i.e.

Checking our result

We have recalled that the exponential

By using the functions

Furthermore, the use of the Riesz-Fantappiè formula (Sections 3.3 and 4.4) reduces to a finite computation the algorithms used in literature.

Therefore, the methods considered in this article are more convenient, with respect to those usually found in literature, for solving linear dynamical systems.

Pierpaolo Natalini,Paolo E. Ricci, (2016) Solution of Linear Dynamical Systems Using Lucas Polynomials of the Second Kind. Applied Mathematics,07,616-628. doi: 10.4236/am.2016.77057