Linking Controllability to the Sturm–Liouville Problem in Ordinary Time-Varying Second-Order Differential Equations

Abstract

This paper establishes some links between Sturm–Liouville problems and the well-known controllability property in linear dynamic systems, together with a control law design that allows any prefixed arbitrary final state finite value to be reached via feedback from any given finite initial conditions. The scheduled second-order dynamic systems are equivalent to the stated second-order differential equations, and they are used for analysis purposes. In the first study, a control law is synthesized for a forced time-invariant nominal version of the current time-varying one so that their respective two-point boundary values are coincident. Afterward, the parameter that fixes the set of eigenvalues of the Sturm–Liouville system is replaced by a time-varying parameter that is a control function to be synthesized without performing, in this case, any comparison with a nominal time-invariant version of the system. Such a control law is designed in such a way that, for given arbitrary and finite initial conditions of the differential system, prescribed final conditions along a time interval of finite length are matched by the state trajectory solution. As a result, the solution of the dynamic system, and thus that of its differential equation counterpart, is subject to prefixed two-point boundary values at the initial and at the final time instants of the time interval of finite length under study. Also, some algebraic constraints between the eigenvalues of the Sturm–Liouville system and their evolution operators are formulated later on. Those constraints are based on the fact that the solutions corresponding to each of the eigenvalues match the same two-point boundary values.

1. Introduction

Many problems appearing in physics are mathematically described by second-order differential equations with prefixed boundary values on finite intervals. A wide class of such problems are indistinctly referred to as Sturm–Liouville problems or as Sturm–Liouville systems. Within such a class, they are, for instance, the well-known Bessel’s and Legendre’s equations and the simple second-order ordinary differential equations whose solutions are harmonic oscillations. Another typical Sturm–Liouville system is the Schrödinger differential equation, which is a basic theoretical support in quantum theory. Further examples are, for instance, the vibrating string, the boundary conditions being the fixed values at the ends, and the problem of heat conduction in a rod where the temperature distribution is suited to have prefixed values at certain points.

The main characteristic of Sturm–Liouville systems is that the solution trajectory of the differential system achieves the given two-boundary for a countable set of values of a constant real parameter $\lambda$ that parameterizes the differential equation. Such a set of values is referred to as the set of eigenvalues of the Sturm–Liouville system [1,2]. Typically, an eigen solution of a Sturm–Liouville system is associated with its set of countably many eigenvalues, which are taken by the above-mentioned λ-parameter.

On the other hand, the so-called controllability property is a well-established property of controlled dynamic systems, which is of high interest in many applications [3,4,5,6,7,8,9]. Basically, a system is said to be controllable if there is a control function able to match any arbitrary prefixed finite value of its state solution in prefixed finite time for any given finite initial conditions [3,4,5]. In other words, the state trajectory solution reaches any prefixed suitable value in a finite time, irrespective of the given finite initial conditions, for some control function that exists if the system is controllable. This mentioned controllability property is, by nature, stronger than just that of the solution of the differential equation of order n being able to reach prefixed values since its first (n − 1) time-derivatives also reach prefixed values. This is due to the fact that the controllability property is associated with the differential system of equations, namely, with the equivalent set of n first-order differential equations to the given differential equation of order n.

This paper discusses some very close relationships between Sturm–Liouville systems borrowed from the theory of differential equations and controllability properties and associated control synthesis of linear dynamic systems borrowed from control theory [9,10,11,12,13]. The considered dynamic systems are of second order and equivalent to the primary stated second-order differential equations. In this context, the second-order differential equations are decomposed into an equivalent system of two first-order differential equations. Basically, the constant parameter, whose set of values in the differential equation is the set of eigenvalues of the Sturm–Liouville system, is replaced by a time-varying function that plays the role of a control function to be synthesized. Such a control is designed so that, for given arbitrary and finite initial conditions of the differential system, the prescribed final conditions along a finite length time interval are achieved by the injection of the control law as a result of the system’s controllability. Then, the solution of the dynamic system, and thus that of the differential equation counterpart, satisfies prefixed two-point boundary values at the initial and at the final time instants of the time interval of finite length under study. The advantageous idea to be addressed is that based on the integration of the Sturm–Liouville theory with controllability properties, it is possible to synthesize a control function that jointly prefixes both initial and final values of the state solution trajectory, that is, two boundary values, of the problem at hand. The control function role is taken by the real parameter whose values are the eigenvalues of the Sturm–Liouville problem, which is now time-varying in general. It has to be emphasized that the obtained results are not reduced to a reinterpretation of previous results in the Sturm–Liouville theory since the usual $\lambda$-parameter, which represents the eigenvalues of the Sturm–Liouville system, is replaced here by a time-varying control used to address the tracking of the prescribed boundary value problems in the solution of the differential system. Such a function is synthesized in an analytic way in order to achieve the mentioned objective.

The paper is organized as follows. Section 2 briefly describes two standard Sturm–Liouville time-invariant and time-varying systems given by ordinary second-order differential equations. The conditions for the two-point boundary value values are parameterized through “ad hoc” constraints and referred to as the initial conditions. Section 3 considers a forced version of the time-invariant differential equation, which is equivalently expressed as a second-order differential system to facilitate the analysis purposes [3,4,5]. The controllability of this system is first studied, and it is seen that the system is trivially controllable. Then, the forcing control is synthesized via feedback in such a way that, for any given finite arbitrary initial conditions, the final conditions in the final point of the prefixed time interval are also arbitrarily prefixed to finite values. The control synthesis is performed in an analytic way and expressed via a closed formula. In this way, a two-boundary value problem is solved for arbitrarily defined values of such boundary values. It turns out that the solution trajectory is defined by a forcing control for given initial conditions and any given value of the parameterizing parameter in the differential equation in such a way that the resulting system satisfies prefixed boundary values at the end point of the considered interval where the solution is calculated. In that way, the forced system is not a Sturm–Liouville one. Section 4 is devoted to the characterization of the time-varying dynamic system so that its solution trajectory has prefixed finite boundary values. For such a purpose, the parameterizing functions of the differential equations are monitored. In this section, there is no comparison of the solution trajectories of the time-varying system and the forced time-invariant one. Section 5 synthesizes a time-varying $\lambda$-parameter, playing itself the role of a control function of the differential system, in order to achieve prefixed boundary value problems of the solution trajectory of the time-varying differential system. Section 6 discusses some algebraic concerns of the relations among the distinct constant eigenvalues of a given Sturm–Liouville system, all of them being associated with the same two-point boundary values and the various algebraic constraints to be fulfilled by their associated evolution operators. The stated constraints are based on the fact that the set of eigenvalues corresponds to the same Sturm–Liouville system so that the respective solutions have to achieve the same two-point boundary values. Finally, the conclusions end the paper.

Notation and Glossary of Symbols

$I_n$ is the $n$-th identity matrix

$$\overline{n}=\left\{1,2,\cdots ,n\right\}$$

the superscript $T$ denotes transposition

$$e_1={\left(1,0\right)}^T; e_2={\left(0,1\right)}^T$$

if $M\in {\mathit{R}}^{n\times n}$ then $M\succ 0$ (respectively, $M⪰0$) denotes that $M$ is positive definite (respectively, semidefinite positive). Also, $M\prec 0$ (respectively, $M⪯0$) denotes that $M$ is negative definite (respectively, semidefinite negative).

$p_0>0$, ${\omega }_0>0$, $q_0$ are parameters of a second-order time-invariant differential equation.

${\lambda }_0$ and $\lambda$ are the eigenvectors of the time-invariant and second-order time-varying differential equation.

$\lambda$(t) defines the time-dependent $\lambda$-parameter when it becomes updated through time to play the role of a feedback control designed to accomplish the two-point boundary conditions.

$p\left(t\right)$, $q\left(t\right)$, and $\omega \left(t\right)$ are the functions that parameterize a second-order time-varying differential equation.

A Sturm–Liouville system is that built by the decomposition of the second-order differential equations into equivalent differential systems of first-order differential equations with defined two-point boundary conditions at the time instants a, b > a.

The real function y(t) and the real vector function x(t) are, respectively, the solutions of the second-order differential equation and that of the equivalent system of first-order equations formed by the solution and its first-order time derivative. Subscripts “0” stand for their nominal time-invariant counterpart versions.

2. Sturm–Liouville Type Basic Systems Associated with Ordinary Differential Equations

The following linear time-invariant differential equation:

$$p_0{y}_0^{″}\left(t\right)+\left(q_0+{\lambda }_0{\omega }_0\right)y_0\left(t\right)=0$$

(1)

as a nominal version of the more general following current time-varying differential equation:

$${\left(p\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }+\left(q\left(t\right)+\lambda \omega \left(t\right)\right)y\left(t\right)=0$$

(2)

where the following is true:

$p_0>0$, ${\omega }_0>0$, $q_0$, ${\lambda }_0$, and $\lambda$ are real numbers; $p\left(t\right)$, $q\left(t\right)$, and $\omega \left(t\right)$ are real functions defined in the real interval $\left[a,b\right]$; and piecewise-continuous in $\left(a,b\right)$ and, $p\left(t\right)$ is differentiable in $\left(a,b\right)$ and $p\left(t\right)>0$ and $\omega \left(t\right)>0$ for $t\in \left[a,b\right]$.

Remark 1.

Note that, since$q_0$ and ${\lambda }_0$ have no specific sign, and $p_0$ and ${\omega }_0$ are positive, it turns out that the constraints $p_0>0$ and ${\omega }_0>0$ can be signed reversed in (1) without altering it or its solutions. In the same way, the positive conditions of $p\left(t\right)$ and $\omega \left(t\right)$ in $\left[a,b\right]$ can be changed by negativity ones without altering Equation (2) and its solutions. The particular sense of referring to the differential Equation (1) as being a nominal version of the current differential Equation (2) is that the second one can typically include time-varying effects or parametrical errors that are not considered in the first one. In other words, Equation (1) is a simplified modelling version of Equation (2). Through the main article body, a relevant problem to be solved is that both differential equations, whose state-trajectory solutions are not point-to-point identical, might reach the same final conditions for given identical initial conditions in the context of the two-point boundary value of the Sturm–Liouville theory. Such a goal is achieved by the synthesis of an appropriate time-varying $\lambda$(t) control action.

The above differential equations are so-called regular Sturm–Liouville systems [1,2] for the respective infinite countable sets of real constants ${\lambda }_0$ and $\lambda$ generating non-trivial solutions in $\left[a,b\right]$ under the respective sets of two-point boundary value constraints:

$${\alpha }_{01}{y}_0\left(a\right)+{\beta }_{01}{y}_0^{\prime }\left(a\right)=0$$

(3)

$${\alpha }_{02}{y}_0\left(b\right)+{\beta }_{02}{y}_0^{\prime }\left(b\right)=0$$

(4)

for Equation (1), and

$${\alpha }_{1}y\left(a\right)+{\beta }_1{y}^{\prime }\left(a\right)=0$$

(5)

$${\alpha }_{2}y\left(b\right)+{\beta }_2{y}^{\prime }\left(b\right)=0$$

(6)

for Equation (2), where ${\alpha }_{0i}$, ${\beta }_{0i}$, ${\alpha }_i$, and ${\beta }_i$ are real constants with $\left|{\alpha }_{0i}\right|+\left|{\beta }_{0i}\right|\ne 0$ and $\left|{\alpha }_i\right|+\left|{\beta }_i\right|\ne 0$ for $i=1,2,3,4$. Equations (3)–(6) can be recombined for an initial value problem by parameterizing the final conditions at $t=b$ related to the initial conditions at $t=a$ as follows. Thus, we write:

$$y_0\left(b\right)={\lambda }_{01}{y}_0\left(a\right)+{\lambda }_{02}{y}_0^{\prime }\left(a\right)$$

(7)

$$y_0^{\prime }\left(b\right)={\lambda }_{03}{y}_0\left(a\right)+{\lambda }_{04}{y}_0^{\prime }\left(a\right)$$

(8)

and

$$y\left(b\right)={\lambda }_{1}y\left(a\right)+{\lambda }_2{y}^{\prime }\left(a\right)$$

(9)

$$y^{\prime }\left(b\right)={\lambda }_{3}y\left(a\right)+{\lambda }_4{y}^{\prime }\left(a\right)$$

(10)

with $\left|{\lambda }_{01}\right|+\left|{\lambda }_{02}\right|\ne 0$, $\left|{\lambda }_{03}\right|+\left|{\lambda }_{04}\right|\ne 0$, $\left|{\lambda }_1\right|+\left|{\lambda }_2\right|\ne 0$, and $\left|{\lambda }_3\right|+\left|{\lambda }_4\right|\ne 0$. The constraints on the final condition can be related to those on the initial conditions in the two-boundary value problem. Thus, one obtains the following by combining Equations (3)–(10):

$$M_0{\left[y_0\left(a\right),y_0^{\prime }(a)\right]}^T=0$$

(11)

and

$$M{\left[y\left(a\right),y^{\prime }(a)\right]}^T=0$$

(12)

, respectively, where

$$M_0=M_0\left({\alpha }_{01},{\alpha }_{02},{\beta }_{01},{\beta }_{02},{\lambda }_{01},{\lambda }_{02},{\lambda }_{03},{\lambda }_{04}\right)=\left[\begin{array}{cc}{\alpha }_{01}& {\beta }_{01}\\ {\alpha }_{02}{\lambda }_{01}+{\beta }_{02}{\lambda }_{03}& {\alpha }_{02}{\lambda }_{02}+{\beta }_{02}{\lambda }_{04}\end{array}\right]$$

(13)

and

$$M=M\left({\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2,{\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4\right)=\left[\begin{array}{cc}{\alpha }_1& {\beta }_1\\ {\alpha }_2{\lambda }_1+{\beta }_2{\lambda }_3& {\alpha }_2{\lambda }_2+{\beta }_2{\lambda }_4\end{array}\right]$$

(14)

In summary, the subsequent result holds:

Proposition 1.

The following properties hold:

(i) 

The two-point boundary value problem of the parameterized conditions Equations (7) and (8) is equivalent to the problem of initial conditions ${\left[y_0\left(a\right),y_0^{\prime }\left(a\right)\right]}^T\in Ker\left(M_0\right)$, with $M_0$ defined in (13), subject to $\left|{\alpha }_{0i}\right|+\left|{\beta }_{0i}\right|\ne 0$; $i=1,2,3,4$, $\left|{\lambda }_{01}\right|+\left|{\lambda }_{02}\right|\ne 0$ , and $\left|{\lambda }_{03}\right|+\left|{\lambda }_{04}\right|\ne 0$.

(ii) 

The two-point boundary value problem of the parameterized conditions Equations (9) and (10) is equivalent to ${\left[y\left(a\right),y^{\prime }\left(a\right)\right]}^T\in Ker\left(M\right)$, with $M$ defined in Equation (14), subject to $\left|{\alpha }_i\right|+\left|{\beta }_i\right|\ne 0$; $i=1,2,3,4$, $\left|{\lambda }_1\right|+\left|{\lambda }_2\right|\ne 0$ and $\left|{\lambda }_3\right|+\left|{\lambda }_4\right|\ne 0$.

It can be pointed out that the given constraints include and might combine, in general, Dirichlet-type and von Neumann-type initial conditions so that they are of a mixed type of both kinds of initial conditions. Note also that Sturm–Liouville systems fix the trajectory solutions of their associated differential systems to the same prefixed two-point boundary values for, in general, infinitely many countable values of such constant parameters, which are referred to as the eigenvalues of the Sturm–Liouville system. The subsequent sections do not study the above differential equations in the Sturm–Liouville context since either the ${\lambda }_0$ and $\lambda$ parameters, often referred to as the eigenvalues of the Sturm–Liouville systems Equations (1) and (2), are fixed and either a forcing control is injected into the differential systems or such parameters are replaced by time-varying functions that exert control functions. The set of eigenvalues of a Sturm–Liouville system can be an infinite countable real set. It can also be pointed out that if a Sturm–Liouville system Equation (2) is singular, that is if $\omega \left(t\right)$ can be zero for points in $\left[a,b\right]$ and $p\left(a\right)=p\left(b\right)=0$, then the number of eigenvalues is not necessarily infinity.

3. Controllability of a Forced Version of the Nominal Time-Invariant Differential Equation

First, assume that (1) is modified with a forcing real control function $u\left(t\right)$ as follows:

$$p_0{y}_0^{″}\left(t\right)+\left(q_0+{\lambda }_0{\omega }_0\right)y_0\left(t\right)=b_{02}u\left(t\right)$$

(15)

where $u\left(t\right)$ is a continuous scalar control function on $\left[a,b\right]$ and $b_{02}$ is a nonzero real constant. Thus, Equation (15) becomes an inhomogeneous differential equation as a result of injecting the forcing function [3,6]. For controllability discussion and control synthesis, it is less involved for related analysis to write the second-order differential equations equivalently as dynamic systems of second order. Thus, Equation (1) can be rewritten as a second-order linear time-invariant differential system of first-order differential equations by defining $x_0\left(t\right)={\left(y_0\left(t\right),y_0^{\prime }\left(t\right)\right)}^T$ as follows:

$$x_0^{\prime }\left(t\right)=A_{0{\lambda }_0}{x}_0\left(t\right)+b_{0}u\left(t\right); t\in \left[a,b\right]$$

(16)

where

$$A_{0{\lambda }_0}=\left[\begin{array}{cc}0& 1\\ -p_0^{-1}\left(q_0+{\lambda }_0{\omega }_0\right)& 0\end{array}\right]; b_0=\left[\begin{array}{c}0\\ b_{02}\end{array}\right]=b_{02}{e}_2$$

(17)

with $e_2={\left(0,1\right)}^T$. The equilibrium point of the unforced system Equation (16), that is, for identically zero control, is $x_{eq}={\left(0,0\right)}^T$. The eigenvalues of $A_{0{\lambda }_0}$ are ${\rho }_{1,2}=\pm \mathit{i}\sqrt{p_0^{-1}\left(q_0+{\lambda }_0{\omega }_0\right)}$, with $\mathit{i}=\sqrt{-1}$ being the complex unit, if ${\lambda }_0>-q_0/{\omega }_0$ so that the equilibrium point of the unforced system is a center. The eigenvalues of $A_{0{\lambda }_0}$ are ${\rho }_{1,2}=\pm \sqrt{p_0^{-1}\left|q_0+{\lambda }_0{\omega }_0\right|}$ if ${\lambda }_0<-q_0/{\omega }_0$ so, the equilibrium point is a saddle point. If ${\lambda }_0=-q_0/{\omega }_0$, then ${\rho }_{1,2}=0$, so the equilibrium point is non-hyperbolic.

The differential system (16) and (17) is said to be controllable if, for any real $a\ge 0$, any nonzero finite $T=b-a>0$ and any given initial conditions $x_0\left(a\right)={\left(y_0\left(a\right),y_0^{\prime }\left(a\right)\right)}^T$, there is some control $u:\left[a,a+T\right]\to {\mathit{R}}^2$ such that $x_0\left(b\right)=x_{0b}^{*}$ for any prefixed finite $x_{0b}^{*}$. Since the differential system (16) and (17) is linear and time-invariant, the controllability property is independent of the interval $\left[a,b\right]$. Also, since the differential system of first-order Equations (16) and (17) is equivalent to the second-order differential Equation (15), its controllability issues can also be attributed to the differential equation. The controllability problems of major interest, in practice, are (a) that of fixing the trajectory solution of the differential system from a given $x\left(a\right)$ at $t=a$ to any prefixed $x\left(b\right)\ne 0$ at $t=b$ by means of the injection of appropriate control $u\left(t\right)$ on $\left[a,b\right]$. This property is often referred to as the state reachability, and (b) that of fixing the trajectory solution $x\left(b\right)=0$ at $t=b$ for any given $x\left(a\right)\ne 0$ through the injection of the appropriate control $u\left(t\right)$ on the time interval $\left[a,b\right]$, which is often referred to as the controllability to the origin.

The following related result holds:

Theorem 1 (controllability of the system (16) and (17)).

The following properties hold:

(i) 

The system of Equations (16) and (17) is controllable (in short, the pair $\left(A_{0{\lambda }_0},b_0\right)$ is controllable, or equivalently, the differential Equation (15) is controllable) if and only if the controllability matrix $\left[b_0,A_{0{\lambda }_0}{b}_0\right]$ is full rank, which always holds since, for $b_{02}\ne 0$,

$$rank\left[b_0,A_{0{\lambda }_0}{b}_0\right]=rank\left[\begin{array}{cc}0& b_{02}\\ b_{02}& 0\end{array}\right]=2$$

(18)

(ii) 

The control law

$$u\left(\tau \right)=b_0^T{e}^{A_{0{\lambda }_0}^T\left(b-\tau \right)}{v}_0\left(a\right)=b_{02}\left({\Psi }_{012}\left(b-\tau \right)v_{01}\left(a\right)+{\Psi }_{022}\left(b-\tau \right)v_{02}\left(a\right)\right); \tau \in \left[a,b\right]$$

(19)

drives the solution trajectory of Equation (16) from an arbitrary $x_0\left(a\right)$ to a given prefixed $x_0\left(b\right)=x_{0b}^{*}$, where ${\Psi }_0\left(t\right)=\left({\Psi }_{0_{ij}}\left(t\right)\right)=e^{A_{0{\lambda }_0}t}$; $i,j=1,2$; $t\ge 0$ is the fundamental matrix function of the unforced differential system Equation (16).

Proof. 

Property (i) is direct from the controllability matrix being full rank. To prove Property (ii), generate a control of the form

$$u\left(\tau \right)=b_0^T{e}^{A_{0{\lambda }_0}^T\left(b-\tau \right)}{v}_0=b_{02}\left({\Psi }_{012}\left(b-\tau \right)v_{01}+{\Psi }_{022}\left(b-\tau \right)v_{02}\right), \tau \in \left[a,b\right]$$

(20)

for some $v_0=v_0\left(a\right)={\left(v_{01}\left(a\right),v_{02}\left(a\right)\right)}^T\in {\mathit{R}}^2$ to be determined later on. Thus, from Equations (16)–(19), the solution trajectory of Equation (16), subject to the control, Equation (19), is unique and given by

$$\begin{array}{c}{x}_0\left(t\right)=e^{A_{0{\lambda }_0}\left(t-a\right)}{x}_0+{\displaystyle {\int }_a^t{e}^{A_{0{\lambda }_0}\left(t-\tau \right)}}{b}_{0}u\left(\tau \right)d\tau \\ =e^{A_{0{\lambda }_0}\left(t-a\right)}{x}_0+\left({\displaystyle {\int }_a^t{e}^{{A\lambda }_{0{\lambda }_0}\left(t-\tau \right)}}{b}_0{b}_0^T{e}^{A_{0{\lambda }_0}^T\left(t-\tau \right)}d\tau \right)v_0\\ ={\Psi }_0\left(t-a\right)x_0+{\Omega }_0\left(a,t\right)v_0\end{array}$$

(21)

Now, since $rank\left[b_0,A_{0{\lambda }_0}{b}_0\right]=2$, then, equivalently, the controllability gramian

$$\begin{gathered} {\Omega }_0\left(a,t\right)={\displaystyle {\int }_a^t{e}^{A_{0{\lambda }_0}\left(t-\tau \right)}}{b}_0{b}_0^T{e}^{A_{0{\lambda }_0}^T\left(t-\tau \right)}d\tau =b_{02}^2{\displaystyle {\int }_a^t\left[\begin{array}{cc}{\Psi }_{012}^2\left(t-\tau \right)& {\Psi }_{012}\left(t-\tau \right){\Psi }_{022}\left(t-\tau \right)\\ {\Psi }_{012}\left(t-\tau \right){\Psi }_{022}\left(t-\tau \right)& {\Psi }_{022}^2\left(ta-\tau \right)\end{array}\right]}d\tau ; \\ t\in \left[a,b\right]. \end{gathered}$$

(22)

is non-singular on a time interval $\left[a,t\right]$ for any $t\in \left[a,b\right]$, since $(A_{0{\lambda }_0},b_0)$ is controllable, then $rank\left[b_0,A_{0{\lambda }_0}{b}_0\right]=2$ if and only if the controllability gramian Equation (22) is nonsingular.

Thus, for any given prefixed finite, $x_0\left(b\right)=x_{0b}^{*}$, if

$$v_0={\Omega }_0^{-1}\left(a,b\right)\left(x_{0b}^{*}-{\Psi }_0\left(t-a\right)x_0\left(a\right)\right)$$

(23)

so that one has from Equation (20), by taking into account Equations (21)–(23), that

$$\begin{gathered} u\left(\tau \right)=b_{02}^{-1}\left({\Psi }_{012}\left(b-\tau \right),{\Psi }_{022}\left(b-\tau \right)\right){\left({\displaystyle {\int }_a^b\left[\begin{array}{cc}{\Psi }_{012}^2\left(b-\tau \right)& {\Psi }_{012}\left(b-\tau \right){\Psi }_{022}\left(b-\tau \right)\\ {\Psi }_{012}\left(b-\tau \right){\Psi }_{022}\left(b-\tau \right)& {\Psi }_{022}^2\left(b-\tau \right)\end{array}\right]}d\tau \right)}^{-1}\left(x_{0b}^{*}-{\Psi }_0\left(b-a\right)x_0\left(a\right)\right); \\ \tau \in \left[a,b\right] \end{gathered}$$

(24)

Then, Equation (21) becomes specified as follows after using Equations (20) and (23):

$$x_0\left(t\right)={\Psi }_0\left(t-a\right)x_0\left(a\right)+{\Omega }_0\left(a,t\right){\Omega }_0^{-1}\left(a,b\right)\left(x_{0b}^{*}-e^{A_{0{\lambda }_0}\left(b-a\right)}{x}_0\left(a\right)\right); t\in \left[a,b\right]$$

(25)

Note that, for $t=b$, and arbitrarily prefixed $x_0\left(b\right)=x_{0b}^{*}$, one directly obtains an identity in Equation (25), associated with the controllability property. Property (ii) has been proved. Property (iii) is a direct consequence of Property (ii), the fact that the initial and final points of the forced system Equations (16) and (17) are the reduction in the two-point boundary value problem to an equivalent one of initial conditions, and the fact that $M_0$ parameterizes all the initial conditions that are equivalent to a two-point boundary value problem. Note that to see it that $x_0\left(a\right)\in S_a$ implies that ${\left(x_0\left(a\right),x_{0b}^{*}\right)}^T$ is a two-point boundary value problem of Equations (16) and (17), with $x_0\left(b\right)$ satisfying Equations (7) and (8) for some admissible parameterization. □

Remark 2.

Note that the controllability property of Theorem 1 holds for any nonzero $b_{02}$ so that it can be fixed to unity with no loss in generality.

On the other hand, note also that the main difference between the controllability property of Theorem 1 and a Sturm–Liouville system is that the second one transfers the solution trajectory from $x\left(a\right)$ to $x\left(b\right)$ of fixed values under the generic constraints of Proposition 1(i) for an infinite countable set of values of ${\lambda }_0$. However, the controllability property of Theorem 1 is addressed under a forcing control function for a given value of ${\lambda }_0$.

Later on, we reformulate controllability in the following sense. Assume that $x_0\left(a\right)$ is chosen within a set $S_a=S_a\left(M_0\right)\subset Ker{M}_0$ subject to the constraints of Proposition 1(i). Then, the pair ${\left(x_0\left(a\right),x_0\left(b\right)\right)}^T$ for any given prefixed $x_0\left(b\right)=x_{0b}^{*}$ is a two-point boundary value problem of Equations (16) and (17), with $x_0\left(b\right)$ satisfying (7) and (8) since $x_0\left(a\right)\in S_a$, such that the unforced system Equation (2) subject to Equations (5) and (6) is a Sturm–Liouville system.

It is known [1,2] that, in general, the forced time-varying differential Equation (2) and the time-invariant Equation (15) can be related to more familiar respective forms as described as follows. Consider the subsequent forced second-order differential equations:

$$a_2\left(t\right)y^{″}\left(t\right)+a_1\left(t\right)y^{\prime }\left(t\right)+a_0\left(t\right)y\left(t\right)=f\left(t\right)$$

$${\left(p\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }+\left(q\left(t\right)+\lambda \omega \left(t\right)\right)y\left(t\right)=F\left(t\right)$$

The first one is a standard one, while the second one is a forced version of the usual Sturm–Liouville unforced one, Equation (2). It is direct to verify, by direct substitutions in them of the functions that follow, that the above equations are equivalent under the following relations from the second one to the first one:

$$p\left(t\right)=e^{{\displaystyle {\int }_0^t\left(a_1\left(\tau \right)/a_2\left(\tau \right)\right)d\tau }}\Rightarrow p^{\prime }\left(t\right)=\left(a_1\left(t\right)/a_2\left(t\right)\right)p\left(t\right)$$

$$q\left(t\right)=\left(a_0\left(t\right)/a_2\left(t\right)\right)p\left(t\right)-\lambda \omega \left(t\right)=\left(a_0\left(t\right)/a_2\left(t\right)\right)e^{{\displaystyle {\int }_0^t\left(a_1\left(\tau \right)/a_2\left(\tau \right)\right)d\tau }}-\lambda \omega \left(t\right)$$

$$F\left(t\right)=\left(f\left(t\right)/a_2\left(t\right)\right)p\left(t\right)=\left(f\left(t\right)/a_2\left(t\right)\right)e^{{\displaystyle {\int }_0^t\left(a_1\left(\tau \right)/a_2\left(\tau \right)\right)d\tau }}$$

and also under the inverse relations from the first one to the second one, one obtains

$${\displaystyle \frac{a_2\left(t\right)}{a_0\left(t\right)}}={\displaystyle \frac{p\left(t\right)}{q\left(t\right)+\lambda \omega \left(t\right)}}$$

$${\displaystyle \frac{a_1\left(t\right)}{a_0\left(t\right)}}={\displaystyle \frac{p^{\prime }\left(t\right)}{q\left(t\right)+\lambda \omega \left(t\right)}}$$

$${\displaystyle \frac{f\left(t\right)}{a_0\left(t\right)}}={\displaystyle \frac{F\left(t\right)a_2\left(t\right)}{p\left(t\right)a_0\left(t\right)}}={\displaystyle \frac{F\left(t\right)}{q\left(t\right)+\lambda \omega \left(t\right)}}$$

In the time invariant case, one has that $p\left(t\right)=p_0$, $q\left(t\right)=q_0$, and $\omega \left(t\right)={\omega }_0$, $a_i\left(t\right)=a_{i0}$; $i=0,1,2$, which lead to the relations

$$p_0=1, a_{10}=0, {\displaystyle \frac{a_{20}}{a_{00}}}={\displaystyle \frac{1}{q_0+{\lambda }_0{\omega }_0}}, {\displaystyle \frac{f\left(t\right)}{a_{00}}}={\displaystyle \frac{F\left(t\right)}{q_0+{\lambda }_0{\omega }_0}}$$

to be replaced in the differential system.

4. Controllability of the Time-Invariant Forced System Versus an Equivalent Parameterized Solution Trajectory of the Time-Varying Current One

By putting $p\left(t\right)=p_0+\tilde{p}\left(t\right)$, $q\left(t\right)=q_0+\tilde{q}\left(t\right)$, $\omega \left(t\right)={\omega }_0+\tilde{\omega }\left(t\right)$, and $\lambda ={\lambda }_0+\tilde{\lambda }$ of $\lambda$, it can be interpreted that (1) is a nominal time-invariant differential equation, which is also a particular case of the general current one, Equation (2), if the parametrical errors are null, that is, if $\tilde{p}\left(t\right)=\tilde{q}\left(t\right)=\tilde{\omega }\left(t\right)=0$ for $t\in \left[a,b\right]$, and $\tilde{\lambda }=0$. This section considers controllability issues through the selection of a control function $u\left(t\right)$ of a forced version of Equation (1) to fix prefixed final conditions of the solution at $t=b$ for any given nontrivial initial conditions at $t=a$ in the case when $\left(b-a\right)$ is finite. The objective is the synthesis of a control function for the forced nominal system so that the boundary values of the current differential system coincide with those of the forced nominal time-invariant one.

The question to be addressed in this section is the way of achieving the same solutions for the forced time-invariant differential controllable Equation (15) and specific parameterizations of $q\left(t\right)$, $p\left(t\right)$, and $\omega \left(t\right)$ of the unforced time-varying differential Equation (2). In other words, it is now addressed if there is a control in Equation (15), which has been proved to be trivially controllable in Theorem 1, such that some special parameterization of the time-varying differential Equation (2) might be made equivalent to Equation (15) in the sense that their respective solutions have identical two-point boundary values at $t=a$ and $t=b$. It is assumed in the sequel that $b_2=1$ with no loss in generality (see Remark 2).

Firstly, Equation (2) is rewritten as the unforced second-order time-differential system:

$$x^{\prime }\left(t\right)=A_{\lambda }\left(t\right)x\left(t\right)=\left(A_{0{\lambda }_0}+{\tilde{A}}_{\lambda }\left(t\right)\right)x\left(t\right)$$

(26)

where $x\left(t\right)={\left(y\left(t\right),y^{\prime }\left(t\right)\right)}^T$, and

$$A_{\lambda }(t)=\left[\begin{array}{cc}0& 1\\ -\frac{q(t)+\lambda \omega (t)}{p(t)}& -\frac{p^{\prime }(t)}{p(t)}\end{array}\right];{\tilde{A}}_{\lambda }(t)=\left[\begin{array}{cc}0& 0\\ {\tilde{\Delta }}_{\lambda }(t)& -\frac{p^{\prime }(t)}{p(t)}\end{array}\right]$$

(27)

where

$${\tilde{\Delta }}_{\lambda }\left(t\right)=p_0^{-1}\left(q_0+{\lambda }_0{\omega }_0\right)-p^{-1}\left(t\right)\left(q\left(t\right)+\lambda \omega \left(t\right)\right).$$

(28)

Now, the differential system Equations (26)–(28) is equivalent to the differential Equation (15) and, equivalently, to the differential system Equations (16) and (17), on $\left[a,b\right]$ for the same initial conditions $y\left(\mathrm{a}\right)=y_0(\mathrm{a})$; $y^{\prime }\left(a\right)=y_0^{\prime }\left(a\right)$ if the control is generated such that, since the control vector is ${\left(0,b_2\right)}^T=e_2={\left(0,1\right)}^T$, then Equation (26) is rewritten as a controlled differential system:

$$x^{\prime }\left(t\right)=A_{\lambda }\left(t\right)x\left(t\right)=A_{0{\lambda }_0}x\left(t\right)+e_{2}u\left(t\right)$$

(29)

where the control u(t) is given by

$$u\left(t\right)={\tilde{\Delta }}_{\lambda }\left(t\right)y\left(t\right)-\left(p^{\prime }\left(t\right)/p\left(t\right)\right)y^{\prime }\left(t\right).$$

(30)

Then, the time-varying differential Equation (2) is rewritten as follows by taking into account Equations (27)–(30):

$$\begin{gathered} y^{″}\left(t\right)=e_2^T{x}^{\prime }\left(t\right)=e_2^T{A}_{\lambda }\left(t\right)x\left(t\right) \\ =e_2^T{A}_{0{\lambda }_0}x\left(t\right)+e_2^T{e}_{2}u\left(t\right)=e_2^T{A}_{0{\lambda }_0}x\left(t\right)+u\left(t\right) \\ =e_2^T{A}_{0{\lambda }_0}x\left(t\right)+\left({\tilde{\Delta }}_{\lambda }\left(t\right)y\left(t\right)-\left(p^{\prime }\left(t\right)/p\left(t\right)\right)y^{\prime }\left(t\right)\right) \\ =-p_0^{-1}\left(q_0+{\lambda }_0{\omega }_0\right)y\left(t\right)+\left({\tilde{\Delta }}_{\lambda }\left(t\right)y\left(t\right)-\left(p^{\prime }\left(t\right)/p\left(t\right)\right)y^{\prime }\left(t\right)\right). \end{gathered}$$

(31)

It turns out that, if the control satisfies jointly Equations (24) and (30), then the solution trajectory of Equation (29), subject to Equations (27) and (28), is the same as that of the forced time-invariant nominal system Equations (16) and (17) on $\left[a,b\right]$ under the same initial conditions. It is now seen that to make identical the control Equations (30) and (24) implies special structures of the time-varying parameterization of the time-varying system (2) involving the functions $q\left(t\right)$, $p\left(t\right)$, and $\omega \left(t\right)$. Now, for Equation (30) to be identical to Equation (24), the subsequent constraint has to hold:

$$\begin{array}{l}u\left(\tau \right)={\tilde{\Delta }}_{\lambda }\left(\tau \right)y\left(\tau \right)-\left(p^{\prime }\left(\tau \right)/p\left(\tau \right)\right)y^{\prime }\left(\tau \right)=b_{02}^{-1}\left({\Psi }_{012}\left(b-\tau \right),{\Psi }_{022}\left(b-\tau \right)\right)\\ \times {\left({\displaystyle {\int }_a^b\left[\begin{array}{cc}{\Psi }_{012}^2\left(b-\tau \right)& {\Psi }_{012}\left(b-\tau \right){\Psi }_{022}\left(b-\tau \right)\\ {\Psi }_{012}\left(b-\tau \right){\Psi }_{022}\left(b-\tau \right)& {\Psi }_{022}^2\left(b-\tau \right)\end{array}\right]}d\tau \right)}^{-1}\left(x_{0b}^{*}-{\Psi }_0\left(b-a\right)x_0\left(a\right)\right)\end{array}$$

(32)

Since $p\left(t\right)$ is positive in $\left[a,b\right]$ and differentiable in $\left(a,b\right)$, it is possible to define a function $\alpha :\left(a,b\right)\to \mathit{R}$ such that $p^{\prime }\left(\tau \right)=\alpha \left(\tau \right)p\left(\tau \right)$ for $\tau \in \left(a,b\right)$, that is $\alpha \left(\tau \right)=p^{\prime }\left(\tau \right)/p\left(\tau \right)$ for $\tau \in \left(a,b\right)$. Then,

$$p\left(\tau \right)=e^{{\displaystyle {\int }_a^{\tau }\alpha \left(\sigma \right)d\sigma }}p\left(a\right)=e^{{\displaystyle {\int }_a^{\tau }\left(p^{\prime }\left(\sigma \right)/p\left(\sigma \right)\right)d\sigma }}p\left(a\right); \tau \in \left[a,b\right]$$

(33)

$$p^{\prime }\left(\tau \right)/p\left(\tau \right)=p^{-1}\left(a\right)p\left(\tau \right)\alpha \left(\tau \right)e^{-{\displaystyle {\int }_a^{\tau }\alpha \left(\sigma \right)d\sigma }}=p^{-1}\left(a\right)p^{\prime }\left(\tau \right)e^{-{\displaystyle {\int }_a^{\tau }\left(p^{\prime }\left(\sigma \right)/p\left(\sigma \right)\right)d\sigma }}; \tau \in \left(a,b\right)$$

(34)

Thus, the control function is generated as follows for each $\tau \in \left(a,b\right)$ by equalizing the identities of Equation (32):

Case a: If $y\left(\tau \right)\ne 0$, then one obtains the following from Equation (28) and the first identity of Equation (32), by taking into account Equation (34):

$$\begin{array}{l}{\tilde{\Delta }}_{\lambda }(\tau )=p_0^{-1}(q_0+{\lambda }_0{\omega }_0)-p^{-1}(\tau )(q(\tau )+\lambda \omega (\tau ))\\ =y^{-1}(\tau )(u(\tau )+(p^{\prime }(\tau )/p(\tau ))y^{\prime }(\tau ))\\ =y^{-1}(\tau )(u(\tau )+y^{\prime }(\tau )p^{-1}(a)p^{\prime }(\tau )e^{-{\int }_a^{\tau }(p^{\prime }(\sigma )/p(\sigma ))d\sigma })\end{array}$$

(35)

, that is, the following constraint has to be satisfied by $q\left(\tau \right)$, $p\left(\tau \right)$ and $\omega \left(\tau \right)$ by equalizing Equation (35) to Equation (28):

$$q\left(\tau \right)+\lambda \omega \left(\tau \right)=p\left(\tau \right)\left[p_0^{-1}\left(q_0+{\lambda }_0{\omega }_0\right)-y^{-1}\left(\tau \right)\left(u\left(\tau \right)+y^{\prime }\left(\tau \right)p^{-1}\left(a\right)p^{\prime }\left(\tau \right)e^{-{\displaystyle {\int }_a^{\tau }\left(p^{\prime }\left(\sigma \right)/p\left(\sigma \right)\right)d\sigma }}\right)\right]$$

(36)

with $u\left(\tau \right)$ being generated from Equation (24), that is, by the second equality of Equation (32).

Case b: If $y\left(\tau \right)=0$ and $y^{\prime }\left(\tau \right)\ne 0$, then the term $\tilde{\Delta }\left(\tau \right)y\left(\tau \right)$ does not contribute to the control function, and, from Equations (2), (30) and (34), one has with $u\left(\tau \right)$ being generated from Equation (24):

$$u\left(\tau \right)=y^{″}\left(\tau \right)=-\left(p^{\prime }\left(\tau \right)/p\left(\tau \right)\right)y^{\prime }\left(\tau \right)=-y^{\prime }\left(\tau \right)p^{-1}\left(a\right)p^{\prime }\left(\tau \right)e^{-{\displaystyle {\int }_a^{\tau }\left(p^{\prime }\left(\sigma \right)/p\left(\sigma \right)\right)d\sigma }}$$

(37)

which is achievable with a bounded possibly discontinuous value of $p^{\prime }\left(t\right)$ for $t=\tau$ according to:

$$p^{\prime }\left(\tau \right)=p^{\prime }\left({\tau }^{+}\right)=-u\left(\tau \right)p\left(a\right){y^{\prime }}^{-1}\left(\tau \right)e^{{\displaystyle {\int }_a^{\tau }\left(p^{\prime }\left(\sigma \right)/p\left(\sigma \right)\right)d\sigma }}$$

(38)

Note that, in view of Equation (34), an eventual negativity of $p^{\prime }\left(\tau \right)$ does not affect the positivity of $p\left(\tau \right)$ since $p\left(a\right)>0$.

Case c: If $y\left(\tau \right)=y^{\prime }\left(\tau \right)=0$ then $y^{″}\left(\tau \right)=0$ and: (a) either $\tau =b$ with $y\left(b\right)=y^{\prime }\left(b\right)=0$; or (b) $\tau \in \left(a,b\right)$ and $y\left(t\right),y^{\prime }\left(t\right),y^{″}\left(t\right)$ are zero for $t\in \left[\tau ,\mathrm{b}\right)$, which is the trivial solution. It turns out that the boundary conditions at $\tau =b$ are $y\left(b\right)=y^{\prime }\left(b\right)=0$ in both cases and the control function value on $\left[\tau ,b\right]$ is zero.

The subsequent summarized result follows from the above Cases a, b, and c whose practical proposal suggested is to generate the function $p\left(t\right)$ for Equation (2) from an “ad hoc” first-order differential equation, with eventual isolated discontinuities of its time-derivative, for given initial positive conditions for given functions $\omega \left(t\right)$ and $q\left(t\right)$, which involves the contribution of the synthesized control of the forced nominal system, which prefixes the values $y\left(b\right)$ and $y^{\prime }\left(b\right)$ to prescribed values at $t=b$ for any given $y\left(a\right)$ and $y^{\prime }\left(a\right)$ at $t=a$ with $\left|y_0\left(a\right)\right|+\left|y_0^{\prime }\left(a\right)\right|\ne 0$.

Theorem 2 (two-point boundary value conditions by monitoring $p\left(t\right)$).

Assume that $u:\left[a,b\right]\to \mathit{R}$ is given by Equation (24) for the time-invariant forced differential system Equation (15) in order to satisfy given prefixed two-point boundary value conditions $y_0\left(a\right)$, $y_0^{\prime }\left(a\right)$, $y_0\left(b\right)$ and $y_0^{\prime }\left(b\right)$ with $\left|y_0\left(a\right)\right|+\left|y_0^{\prime }\left(a\right)\right|\ne 0$. Then, the unforced time-varying differential system Equation (2) satisfies the same two-point boundary value conditions as the above forced time-invariant one, that is, $y\left(a\right)=y_0\left(a\right)$, $y^{\prime }\left(a\right)=y_0^{\prime }\left(a\right)$, $y\left(b\right)=y_0\left(b\right)$ and $y^{\prime }\left(b\right)=y_0^{\prime }\left(b\right)$ if, for given $\lambda \in \mathit{R}$, $p\left(a\right)>0$ $q:\left[a,b\right]\to \mathit{R}$ and $\omega :\left[a,b\right]\to {\mathit{R}}_{+}$, $p\left(t\right)$ is the solution of the following differential equation on $\left[a,b\right]$: 

$$\begin{gathered} p^{\prime }\left(t\right)=p^{\prime }\left(t^{-}\right)=\left[p_0^{-1}\left(q_0+\lambda {\omega }_0\right)-y^{-1}\left(t\right)u\left(t\right)-p^{-1}\left(t\right)\left(q\left(t\right)+\lambda \omega \left(t\right)\right)\right]e^{{\displaystyle {\int }_a^t\left(p^{\prime }\left(\sigma \right)/p\left(\sigma \right)\right)d\sigma }}y\left(t\right)y^{-1}\left(t\right)p\left(a\right) \\ \mathrm{if} y\left(t\right)\ne 0 \end{gathered}$$

(39)

and, if  $y\left(t\right)=0,y^{\prime }\left(t\right)\ne 0$,

$$\begin{gathered} p^{\prime }\left(t^{-}\right)=\left[p_0^{-1}\left(q_0+\lambda {\omega }_0\right)-y^{-1}\left(t\right)u\left(t\right)-p^{-1}\left(t\right)\left(q\left(t\right)+\lambda \omega \left(t\right)\right)\right]e^{{\displaystyle {\int }_a^t\left(p^{\prime }\left(\sigma \right)/p\left(\sigma \right)\right)d\sigma }}y\left(t\right)y^{-1}\left(t\right)p\left(a\right) \\ {p}^{\prime }\left(t\right)=p^{\prime }\left(t^{+}\right)=-e^{{\displaystyle {\int }_a^t\left(p^{\prime }\left(\sigma \right)/p\left(\sigma \right)\right)d\sigma }}u\left(t\right)y^{-1}\left(t\right)p\left(a\right) \end{gathered}$$

(40)

for $t\in \left(a,b\right)$ with $u\left(t\right)$ being calculated from Equation (24) for $t\in \left(a,b\right)$.

$\mathrm{If} y\left(t\right)=y^{\prime }\left(t\right)=y^{″}\left(t\right)=0$ then $t=b$ and $y_0\left(b\right)=y_0^{\prime }\left(b\right)=y\left(b\right)=y^{\prime }\left(b\right)=0$.

Theorem 2 has monitored $p\left(t\right)$ if the remaining parameterizing functions are given. Alternatively, the subsequent parallel result monitors $q\left(t\right)$ and $p\left(t\right)$ in a combined way depending on the zeros of $y\left(t\right)$ and $y^{\prime }\left(t\right)$ for given $\omega :\left[a,b\right]\to {\mathit{R}}_{+}$ and $\lambda \in \mathit{R}$.

Corollary 1 (two-point boundary value conditions by monitoring $q\left(t\right)$ and $p\left(t\right)$).

Assume that $u:\left[a,b\right]\to R$ is given by Equation (24) for the time-invariant forced differential system Equation (15) in order to satisfy given prefixed two-point boundary value conditions $y_0\left(a\right)$, $y_0^{\prime }\left(a\right)$, $y_0\left(b\right)$ and $y_0^{\prime }\left(b\right)$. Then, the unforced time-varying differential system Equation (2) satisfies the same two boundary value conditions as the above time-invariant one if, for given $\lambda \in \mathit{R}$ and $\omega :\left[a,b\right]\to {\mathit{R}}_{+}$, $q:\left[a,b\right]\to {\mathit{R}}_{+}$ is piecewise-continuous in $\left(a,b\right)$ and given by the following expression:

$$\begin{gathered} q\left(t\right)=p\left(t\right)\left[p_0^{-1}\left(q_0+{\lambda }_0{\omega }_0\right)-y^{-1}\left(t\right)\left(u\left(t\right)+y^{\prime }\left(\tau \right)p^{-1}\left(a\right)p^{\prime }\left(t\right)e^{-{\displaystyle {\int }_a^t\left(p^{\prime }\left(\sigma \right)/p\left(\sigma \right)\right)d\sigma }}\right)\right]-\lambda \omega \left(t\right) \\ \mathrm{if} y\left(t\right)\ne 0 \end{gathered}$$

(41)

and $p\left(t\right)$ is continuous and differentiable for $t\in \left(a,b\right)$ generated from Equation (40) if $y\left(t\right)=0,y^{\prime }\left(t\right)\ne 0$, with $u\left(t\right)$ being calculated from Equation (24) for $\tau \in \left(a,b\right)$.

$\mathrm{If} y\left(t\right)=y^{\prime }\left(t\right)=y^{″}\left(t\right)=0$ then $t=b$ and $y_0\left(b\right)=y_0^{\prime }\left(b\right)=y\left(b\right)=y^{\prime }\left(b\right)=0$.

Also, the subsequent parallel result monitors $\omega \left(t\right)$ and $p\left(t\right)$ in a combined way depending on the zeros of $y\left(t\right)$ and $y^{\prime }\left(t\right)$ for given $q:\left[a,b\right]\to {\mathit{R}}_{+}$ and a nonzero $\lambda \in \mathit{R}$ by reversing the roles of $q\left(t\right)$ and $\omega \left(t\right)$ in Equation (41).

Corollary 2 (two-point boundary value conditions by monitoring $q\left(t\right)$ and $\omega \left(t\right)$).

Assume that $u:\left[a,b\right]\to R$ is given by Equation (24) for the time-invariant forced differential system Equation (15) in order to satisfy given prefixed two-point boundary value conditions $y_0\left(a\right)$, $y_0^{\prime }\left(a\right)$, $y_0\left(b\right)$ and $y_0^{\prime }\left(b\right)$. Then, the unforced time-varying differential system Equation (2) satisfies the same two boundary value conditions as the above time-invariant one if, for given nonzero $\lambda \in \mathit{R}$ and $q:\left[a,b\right]\to {\mathit{R}}_{+}$, $\omega :\left[a,b\right]\to {\mathit{R}}_{+}$ is piecewise-continuous in $\left(a,b\right)$ and given by the following expression, which is identical to Equation (41):

$$\begin{gathered} \omega \left(t\right)={\lambda }^{-1}\left(p\left(t\right)\left[p_0^{-1}\left(q_0+{\lambda }_0{\omega }_0\right)-y^{-1}\left(t\right)\left(u\left(t\right)+y^{\prime }\left(\tau \right)p^{-1}\left(a\right)p^{\prime }\left(t\right)e^{-{\int }_a^t\left(p^{\prime }\left(\sigma \right)/p\left(\sigma \right)\right)d\sigma }\right)\right]-q\left(t\right)\right) \\ \mathrm{if} y\left(t\right)\ne 0 \end{gathered}$$

(42)

and $p\left(t\right)$ is continuous and differentiable for $t\in \left(a,b\right)$ generated from Equation (40) $\mathrm{if} y\left(t\right)=0,y^{\prime }\left(t\right)\ne 0$ with $u\left(t\right)$ being calculated from Equation (24) for $\tau \in \left(a,b\right)$.

$\mathrm{If} y\left(t\right)=y^{\prime }\left(t\right)=y^{″}\left(t\right)=0$ then $t=b$ and $y_0\left(b\right)=y_0^{\prime }\left(b\right)=y\left(b\right)=y^{\prime }\left(b\right)=0$.

5. Controllability of the Time-Varying System Through a Time-Varying Parametrized Function $\mathit{\lambda }\mathbf{\left(}\mathit{t}\mathbf{\right)}$ and Feedback Control Law

It is now discussed that a slight modification of the time-differential Equation (2), or equivalently, a related ad-hoc modification of the differential system Equation (26), subject to Equations (27) and (28), can be controlled by a time-varying parameter $\lambda \left(t\right)$ that replaces the constant $\lambda$ in Equation (2). Also, a control law is explicitly formulated that accomplishes the achievement of prefixed two-point boundary values of the state trajectory solution on a finite given time interval.

It has to be pointed out that, through this section, there are no comparisons of the time-varying differential equation with the nominal time-invariant one or comparisons between their respective associated second-order systems. Also, the time-varying functions $p\left(t\right)$, $q\left(t\right)$, and $\omega \left(t\right)$ are not updated but prefixed in $\left[a,b\right]$ as they were assumed in Section 3 in order to make the two-point boundary value of the current differential Equation (2) equivalent to that of the forced nominal differential Equation (15).

The differential system Equation (26), subject to Equations (27) and (28), can also be rewritten as follows provided that the constant real parameter $\lambda$ is replaced with a function $\lambda :\left[a,b\right]\to R$ to be then synthesized as a control function:

$$x^{\prime }\left(t\right)=A\left(t\right)x\left(t\right)+b\left(t\right)\lambda \left(t\right)=A_{0}x\left(t\right)+{\tilde{A}}_0\left(t\right)x\left(t\right)+b\left(t\right)\lambda \left(t\right)$$

(43)

where

$$A\left(t\right)=\left[\begin{array}{cc}0& 1\\ -p^{-1}\left(t\right)q\left(t\right)& -p^{-1}\left(t\right)p^{\prime }\left(t\right)\end{array}\right]; b\left(t\right)=\left[\begin{array}{c}0\\ p^{-1}\left(t\right)\omega \left(t\right)y\left(t\right)\end{array}\right]=\left[\begin{array}{c}0\\ 1\end{array}\right]p^{-1}\left(t\right)\omega \left(t\right)y\left(t\right)$$

(44)

$$A_0=\left[\begin{array}{cc}0& 1\\ -p_0^{-1}{q}_0& 0\end{array}\right]; {\tilde{A}}_0\left(t\right)=A\left(t\right)-A_0=\left[\begin{array}{cc}0& 1\\ p_0^{-1}{q}_0-p^{-1}\left(t\right)q\left(t\right)& -p^{-1}\left(t\right)p^{\prime }\left(t\right)\end{array}\right].$$

(45)

The unique solution of Equation (43) on $\left[a,b\right]$, subject to Equations (44) and (45), for any given initial conditions $y\left(a\right)=x_1\left(a\right)$, $y^{\prime }\left(a\right)=x_2\left(a\right)$ is

$$\begin{gathered} x\left(t\right)=\Psi \left(t,a\right)x\left(a\right)+{\displaystyle {\int }_a^t\Psi \left(t,\tau \right)}b\left(\tau \right)\lambda \left(\tau \right)d\tau =e^{A_0\left(t-a\right)}x\left(a\right)+{\displaystyle {\int }_a^t{e}^{A_0\left(t-\tau \right)}}{\tilde{A}}_0\left(\tau \right)x\left(\tau \right)d\tau +{\displaystyle {\int }_a^t{e}^{A_0\left(t-\tau \right)}b\left(\tau \right)}\lambda \left(\tau \right)d\tau ; \\ t\in \left[a,b\right] \end{gathered}$$

(46)

where $\Psi \left(t,\tau \right):\left[a,b\right]\times \left[a,t\right]\to {\mathit{R}}^2$ is the evolution operator of Equation (43) on $\left[a,b\right]$ which is the unique solution of the differential system ${\Psi }^{\prime }\left(t,\tau \right)=A\left(t\right)\Psi \left(t,\tau \right)$, subject to $\Psi \left(\tau ,\tau \right)=I_2$; $\tau \in \left[a,b\right]$, and $e^{A_{0}t}$ is a $C_0$-semigroup of infinitesimal generator $A_0$ for all $\tau \ge 0$, which in the control theory usual terminology is termed as the state transition matrix of $z^{\prime }\left(t\right)=A_{0}z\left(t\right)$ and which is, in fact, the fundamental matrix of such a differential system of first-order equations. The following result holds:

Theorem 3 (controllability result for the system (43)–(45)).

Assume that  $q:\left[a,b\right]\to \mathit{R}$ and $\omega :\left[a,b\right]\to {\mathit{R}}_{+}$ are piecewise-continuous, and that $p:\left[a,b\right]\to {\mathit{R}}_{+}$ is continuous and differentiable in $\left(a,b\right)$ with piecewise-continuous derivative in $\left[a,b\right]$. Assume also that $x\left(a\right)\ne 0$, equivalently, that $\left|y\left(a\right)\right|+\left|y^{\prime }\left(a\right)\right|\ne 0$.

Then, the differential system Equation (43), subject to Equations (44) and (45), is controllable in $\left[a,b\right]$ for any b>a with a bounded piecewise-continuous control function $\lambda :\left[a,b\right]\to \mathit{R}$. Then, there exists a control function $\lambda :\left[a,b\right]\to \mathit{R}$ such that the state-trajectory solution of the differential system Equation (43), subject to Equations (44) and (45), reaches any prefixed $x\left(b\right)$ at $t=b$ for any given $x\left(a\right)$. Equivalently, the solution of the differential system Equation (2) reaches any prefixed values  $y\left(b\right)$ and $y^{\prime }\left(b\right)$ for any given values $y\left(a\right)$ and $y^{\prime }\left(a\right)$.

Proof. 

One has from Equation (46) that

$${\int }_a^b{e}^{A_0\left(b-\tau \right)}b\left(\tau \right)\lambda \left(\tau \right)d\tau =x\left(b\right)-e^{A_0\left(b-a\right)}x\left(a\right)-{\int }_a^b{e}^{A_0\left(b-\tau \right)}{\tilde{A}}_0\left(\tau \right)x\left(\tau \right)d\tau$$

(47)

For each fixed $t\in \left[a,b\right]$, the controllability matrix function of the pair $\left(A_0,b\left(t\right)\right)$ is

$$C\left(A_0,b\left(t\right)\right)=\left[b\left(t\right),A_{0}b\left(t\right)\right]=p^{-1}\left(t\right)\omega \left(t\right)y\left(t\right)\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]; t\in \left[a,b\right].$$

(48)

Since $p\left(t\right)>0$ and $q\left(t\right)>0$ for $t\in \left[a,b\right]$ then $rank C\left(A_0,b\left(t\right)\right)=2$ for $t\in {\left[a,b\right]}_{y\left(t\right)\ne 0}\left(\equiv \left[a,b\right]{\backslash \mathrm{Z}}_{\mathrm{y}}\left[a,b\right]\right)$, where ${\mathrm{Z}}_{\mathrm{y}}\left[a,b\right]=\left\{t\in \left[a,b\right]:\mathrm{y}\left(t\right)=0\right\}$, then the pair $\left(A_0,b\left(t\right)\right)$ is controllable for the set of nonzero Lebesgue measure ${\left[a,b\right]}_{y\left(t\right)\ne 0}$ and it is uncontrollable on the (empty or nonempty) set of zero Lebesgue measure ${\mathrm{Z}}_{\mathrm{y}}\left[a,b\right]$. Let $\lambda :\left[a,b\right]\to \mathit{R}$ be defined as follows:

$$\lambda \left(\tau \right)=b^T\left(\tau \right)e^{A_0^T\left(b-\tau \right)}v\equiv \left\{\begin{array}{c}{b}^T\left(\tau \right)e^{A_0^T\left(b-\tau \right)}v \mathrm{if} \tau \in {\left[a,b\right]}_{y\left(t\right)\ne 0}\\ 0 \mathrm{if} \tau \in {\mathrm{Z}}_{\mathrm{y}}\left[a,b\right]\end{array}\right.$$

(49)

$\tau \in \left[a,b\right]$, which was replaced in Equation (47), yields

$$C{g}_0\left[a,b\right]v=x\left(b\right)-e^{A_{0}b}\left[x\left(a\right)-{\displaystyle {\int }_a^b{e}^{-A_0\tau }}{\tilde{A}}_0\left(\tau \right)x\left(\tau \right)\right]d\tau$$

(50)

where

$$C{g}_0\left[a,b\right]=\left({\displaystyle {\int }_a^b{e}^{A_0\left(b-\tau \right)}b\left(\tau \right)}{b}^T\left(\tau \right)e^{A_0^T\left(b-\tau \right)}d\tau \right)$$

(51)

where $C{g}_0\left[a,b\right]$ is the controllability gramian associated with the controllability matrix function $C\left(A_0,b\left(t\right)\right)$ of the pair $\left(A_0,b\left(t\right)\right)$ on $\left[a,b\right]$. It is well-known that if $1\le \mu \le 2$, with $n=2$ being the order of the differential system Equations (43)–(45), and then that of the square matrix $A_0$, and $\mu$ being the degree of the minimal polynomial of $A_0$, for any given integer $\overline{\mu }\ge \mu$, there exists a unique set of $\overline{\mu }$ linearly independent continuous functions ${\alpha }_k^{\overline{\mu }}:{\mathit{R}}_{0+}\to \mathit{R}$; $k=0,1,\mathrm{\dots },\overline{\mu }-1$ such that $e^{A_{0}t}={\displaystyle {\sum }_{k=0}^{\overline{\mu }-1}{\alpha }_k^{\overline{\mu }}}\left(t\right)A_0^k$ for $t\in {\mathit{R}}_{0+}$ [3,4]. Thus, by choosing, with no loss in generality, $\overline{\mu }$ equal to $n$ and simplifying the notation as ${\alpha }_k^1\left(t\right)\equiv {\alpha }_k\left(t\right)$; $k=0,1$, one has that Equation (50) becomes

$$\begin{gathered} C{g}_0\left[a,b\right]v=\left(C{g}_0\left[a,b\right]\right)v=\left({\displaystyle {\sum }_{j=0}^1{\displaystyle {\sum }_{k=0}^1{\displaystyle {\int }_a^b{\alpha }_j\left(b-\tau \right){\alpha }_k\left(b-\tau \right)C\left(A_0,b\left(\tau \right)\right)C^T\left(A_0,b\left(\tau \right)\right)d\tau }}}\right)\nu \\ =x\left(b\right)-e^{A_{0}b}\left[x\left(a\right)-{\displaystyle {\int }_a^b{e}^{-A_0\tau }}{\tilde{A}}_0\left(\tau \right)x\left(\tau \right)\right]d\tau \end{gathered}$$

(52)

since $rank C\left(A_0,b\left(\tau \right)\right)C^T\left(A_0,b\left(\tau \right)\right)=rank C\left(A_0,b\left(\tau \right)\right)=2$ for $\tau \in {\left[a,b\right]}_{y\left(t\right)\ne 0}$. Note that for any $t\in \left(a,b\right)$, the controllability gramian

$$\begin{gathered} C{g}_0\left[a,t\right]={\displaystyle {\int }_a^t{e}^{A_0\left(t-\tau \right)}b\left(\tau \right)}{b}^T\left(\tau \right)e^{A_0^T\left(t-\tau \right)}d\tau \\ ={\displaystyle {\int }_a^t\left[\begin{array}{cc}{\alpha }_0\left(t-\tau \right)& 0\\ 0& {\alpha }_1\left(t-\tau \right)\end{array}\right]C\left(A_0,b\left(\tau \right)\right)}{C}^T\left(A_0,b\left(\tau \right)\right)\left[\begin{array}{cc}{\alpha }_0\left(t-\tau \right)& 0\\ 0& {\alpha }_1\left(t-\tau \right)\end{array}\right]d\tau \end{gathered}$$

(53)

is positive semidefinite for all $t>a$ because of its structure. Furthermore, there exists some time instant $t\in \left(a,b\right)$ such that (a) both ${\alpha }_i\left(t-\tau \right)$ are non-identically zero while for all $a\le \tau \left(<t\right)\in \left(a,b\right)$ they also keep a constant sign in some subset $S_1=\left[t_{11},t_{12}\right]$ of $\left[a,t\right]$ being of nonzero Lebesgue measure since the fundamental matrix is everywhere non-singular and those functions are continuous; (b) $y\left(\tau \right)\ne 0$ in some subset $S_2=\left[t_{21},t_{22}\right]$ of $S_1$ which is of nonzero Lebesgue measure. Thus, $C{g}_0\left[t_{21},t_{22}\right]$ is positive definite. Then, for such an existing $t\in \left[a,b\right]$, $C{g}_0\left[a,t\right]$ is positive definite. Then, for any $t^{\prime }\in \left(t,b\right]$, $C{g}_0\left[a,t^{\prime }\right]$ is positive definite as well and, as a result, $C{g}_0\left[a,b\right]$ is positive definite too, so that one concludes that the controllability gramian $C{g}_0\left[a,b\right]$ is non-singular. Therefore, one obtains the following from Equation (52):

$$v=C_{g_0}^{-1}\left[a,b\right]\left(x\left(b\right)-e^{A_{0}b}\left[x\left(a\right)-{\displaystyle {\int }_a^b{e}^{-A_0\tau }}{\tilde{A}}_0\left(\tau \right)x\left(\tau \right)\right]d\tau \right)$$

(54)

and then, for any given two-point boundary values $x\left(a\right)$ and $x\left(b\right)$, the combination of Equation (49) with Equation (54) yields

$$\lambda \left(\tau \right)=p^{-1}\left(\tau \right)\omega \left(\tau \right)y\left(\tau \right)\left(0,1\right)e^{A_0^T\left(b-\tau \right)}{C}_{g_0}^{-1}\left[a,b\right]\left(x\left(b\right)-e^{A_{0}b}\left[x\left(a\right)-{\displaystyle {\int }_a^b{e}^{-A_0\tau }}{\tilde{A}}_0\left(\tau \right)x\left(\tau \right)\right]d\tau \right); \tau \in \left[a,b\right]$$

(55)

which is the applicable expression for $\tau \in {\left[a,b\right]}_{y\left(\tau \right)\ne 0}$, but which also applies directly on ${\mathrm{Z}}_{\mathrm{y}}\left[a,b\right]$ since $y\left(\tau \right)$ is a multiplicative factor. It turns out that above control function $\lambda \left(\tau \right)$ on $\left[a,b\right]$ fixes any state trajectory solution of Equation (43) to any prefixed $x\left(b\right)$ for $t=b$ for any given $x\left(a\right)$. □

Remark 3.

It turns out that the differential Equation (2), equivalently described by the differential system Equation (43), subject to Equations (44) and (45), is not a Sturm–Liouville system under Theorem 3 since the control Equation (55) is time-variant so that it is not associated with the infinitely many countable constant real eigenvalues of values $\lambda$ of a problem of two-point boundary value constraints of the differential Equation (2). Furthermore, the boundary value conditions at $t=b$ are fixed arbitrarily independent of those at $t=a$ while in the two-point boundary value conditions of the Sturm–Liouville problem, the two-point boundary value conditions have to satisfy mutually related constraints.

Remark 4.

In view of Equations (43)–(45), and since the evolution operator  $\Psi \left(t,\tau \right):\left[a,b\right]\times \left[a,t\right]\to {\mathit{R}}^{2\times 2}$ of Equation (43) satisfies  ${\Psi }^{\prime }\left(t,\tau \right)=A\left(t\right)\Psi \left(t,\tau \right)$, subject to $\Psi \left(t,t\right)=I_2$ for any $t\in \mathit{R}$, one obtains by directly solving the above differential system that

$$\Psi \left(t,a\right)=e^{A_0\left(t-a\right)}\left[I_2+{\displaystyle {\int }_a^t{e}^{A_0\left(a-\tau \right)}{\tilde{A}}_0\left(\tau \right)\Psi \left(t,\tau \right)d\tau }\right];t\in \left[a,b\right]$$

(56)

and, by replacing the value $x\left(\tau \right)=\Psi \left(\tau ,a\right)x\left(a\right)+{\displaystyle {\int }_a^{\tau }\Psi \left(\tau ,\sigma \right)}b\left(\sigma \right)\lambda \left(\sigma \right)d\sigma$ arising from the second right-hand side of Equation (46) into the third right-hand side of Equation (46), one also obtains since $b\left(\tau \right)=p^{-1}\left(\tau \right)\omega \left(\tau \right)y\left(\tau \right){\left(0,1\right)}^T$

$$\begin{gathered} x\left(t\right)=e^{A_0\left(t-a\right)}\left(I_2+{\displaystyle {\int }_a^t{e}^{A_0\left(a-\tau \right)}{\tilde{A}}_0\left(\tau \right)\Psi \left(t,\tau \right)d\tau }\right)x\left(a\right) \\ +{\displaystyle {\int }_a^t{e}^{A_0\left(t-\tau \right)}}{\tilde{A}}_0\left(\tau \right)\left({\displaystyle {\int }_a^{\tau }\Psi \left(\tau ,\sigma \right)p^{-1}\left(\sigma \right)\omega \left(\sigma \right)y\left(\sigma \right)\lambda \left(\sigma \right)d\sigma +p^{-1}\left(t\right)\omega \left(\tau \right)y\left(\tau \right)\lambda \left(\tau \right)}\right){\left(0,1\right)}^{T}d\tau ; \\ t\in \left[a,b\right] \end{gathered}$$

(57)

In particular, if  $\lambda \left(t\right)=\lambda$ is a real constant in $\left[a,b\right]$ then one has from Equation (57) that

$$\begin{gathered} x\left(b\right)=e^{A_0\left(b-a\right)}\left(I_2+{\displaystyle {\int }_a^b{e}^{A_0\left(a-\tau \right)}{\tilde{A}}_0\left(\tau \right)\Psi \left(b,\tau \right)d\tau }\right)x\left(a\right) \\ +\lambda {\displaystyle {\int }_a^b{e}^{A_0\left(b-\tau \right)}}{\tilde{A}}_0\left(\tau \right)\left({\displaystyle {\int }_a^{\tau }\Psi \left(\tau ,\sigma \right)p^{-1}\left(\sigma \right)\omega \left(\sigma \right)y\left(\sigma \right)d\sigma +p^{-1}\left(t\right)\omega \left(\tau \right)y\left(\tau \right)}\right){\left(0,1\right)}^{T}d\tau \end{gathered}$$

(58)

Note that $y\left(t\right)=y^{\prime }\left(t\right)=0$ can only occur at $t=b$ if $\left|y\left(a\right)\right|+\left|y^{\prime }\left(a\right)\right|\ne 0$ since if $y\left(t_0\right)=y^{\prime }\left(t_0\right)=0$ for some $t_0\in \left(a,b\right)$ then $y\left(t\right)=y^{\prime }\left(t\right)=0$ for $t\in \left[t_0,b\right]$.

On the other hand, note that Equation (55) cannot be generated in an analytical way since $\lambda \left(\tau \right)$ depends on its values for $\sigma >\tau$ through the sate $x\left(\sigma \right)$ on $\left[a,b\right]$. To overcome this drawback, it can be first noticed that a control being jointly dependent on $y\left(t\right)$ and its time-derivative $y^{\prime }\left(t\right)$, since both functions cannot be jointly null, can be reformulated from the first expression of Equation (43), subject to Equation (44), as follows. The solution to Equation (43) is given in Equation (46), that is

$$x\left(t\right)=\Psi \left(t,a\right)x\left(a\right)+{\displaystyle {\int }_a^t\Psi \left(t,\tau \right)}b\left(\tau \right)\lambda \left(\tau \right)d\tau ; t\in \left[a,b\right]$$

(59)

The control function $\lambda :\left[a,b\right]\to \mathit{R}$ is generated as follows:

$$\lambda \left(\tau \right)=b^T\left(\tau \right){\Psi }^T\left(b,\tau \right)g; \tau \in \left[a,b\right]$$

(60)

depending on a vector $g\in {\mathit{R}}^2$ to be specified later on. One obtains the following after replacing Equation (60) into Equation (59)

$$x\left(t\right)=\Psi \left(t,a\right)x\left(a\right)+\left({\displaystyle {\int }_a^t\Psi \left(t,\tau \right)}b\left(\tau \right)b^T\left(\tau \right){\Psi }^T\left(t,\tau \right)d\tau \right)g; t\in \left[a,b\right]$$

(61)

The pair $\left(A\left(t\right),b\left(t\right)\right)$ is controllable on $\left[a,b\right]$ if and only if the controllability gramian

$$Cg\left[a,b\right]=\left({\displaystyle {\int }_a^b\Psi \left(t,\tau \right)}b\left(\tau \right)b^T\left(\tau \right){\Psi }^T\left(t,\tau \right)d\tau \right)\succ 0$$

(62)

Note that because of its structure, the above controllability gramian is (at least) positive semidefinite [5], i.e.,

$$Cg\left[a,t\right]=\left({\displaystyle {\int }_a^t\Psi \left(t,\tau \right)}b\left(\tau \right)b^T\left(\tau \right){\Psi }^T\left(t,\tau \right)d\tau \right)⪰0; t\in \left(a,b\right).$$

(63)

The positive definiteness condition Equation (63) holds if and only if there is no $z\in {\mathit{R}}^2$ such that $z^T\Psi \left(b,\tau \right)b\left(\tau \right)\equiv 0$ for $\tau \in \left(a,b\right)$, equivalently, if for any given nonzero $\eta \in {\mathit{R}}^2$ and some $\tau =\tau \left(\eta \right)\in \left(a,b\right)$, ${\eta }^T\Psi \left(b,\tau \right)b\left(\tau \right)\ne 0$. This is seen as follows. Since the evolution operator is continuous, there is a real interval of nonzero Lebesgue measure $\left(\eta -\epsilon \eta +\epsilon \right)$ for some real constant $\epsilon >0$ such that $\tau -a<\epsilon <b-\tau$ if there is some nonzero $\eta \in \left(a,b\right)$ such that ${\eta }^T\Psi \left(b,\tau \right)b\left(\tau \right)\ne 0$. As a result,

$${\eta }^T\left(Cg\left[a,b\right]\right)\eta \ge {\eta }^T\left({\displaystyle {\int }_{\tau -a}^{b-\tau }\Psi \left(b,\sigma \right)}b\left(\sigma \right)b^T\left(\sigma \right){\Psi }^T\left(b,\sigma \right)d\sigma \right)\eta >0$$

(64)

for any $\eta \ne 0$ so that

$$Cg\left[a,b\right]={\displaystyle {\int }_a^b\Psi \left(b,\sigma \right)}b\left(\sigma \right)b^T\left(\sigma \right){\Psi }^T\left(b,\sigma \right)d\sigma ⪰{\displaystyle {\int }_{\tau -a}^{b-\tau }\Psi \left(b,\sigma \right)}b\left(\sigma \right)b^T\left(\sigma \right){\Psi }^T\left(b,\sigma \right)d\sigma \succ 0$$

(65)

which also implies that $Cg\left[a,b^{\prime }\right]\succ 0$ for $b^{\prime }>b$. Conversely, Equation (66) fails so that $\left(A\left(t\right),b\left(t\right)\right)$ is uncontrollable on $\left[a,b\right]$, if and only if there is some nonzero $\eta \in {\mathit{R}}^2$ such that ${\eta }^T\Psi \left(b,\tau \right)b\left(\tau \right)\equiv 0$ for $\tau \in \left(a,b\right)$, that is, ${\eta }^T{\Psi }^{\prime }\left(b,\tau \right)e_2={\eta }^{T}B\left(\tau \right)\Psi \left(b,\tau \right)e_2\equiv 0$; $\tau \in \left(a,b\right)$. If Equation (65) holds then $g$ and $\lambda \left(\tau \right)$ in Equation (60) become:

$$g=C{g}^{-1}\left[a,b\right]\left(x\left(b\right)-\Psi \left(b,a\right)x\left(a\right)\right)$$

(66)

$$\lambda \left(\tau \right)=p^{-1}\left(\tau \right)\omega \left(\tau \right)y\left(\tau \right)e_2^T{\Psi }^T\left(b,\tau \right)\left(C{g}^{-1}\left[a,b\right]\right)\left(x\left(b\right)-\Psi \left(b,a\right)x\left(a\right)\right); \tau \in \left[a,b\right]$$

(67)

Thus, the following result holds from the above reasoning:

Theorem 4 (feedback control law by generation of $\lambda \left(\tau \right)$).

Assume that, for any nonzero $\eta \in {\mathit{R}}^2$, there is some $\tau =\tau \left(\eta \right)\in \left(a,b\right)$ such that ${\eta }^T\Psi \left(b,\tau \right)e_2\ne 0$. Then, the system Equations (59) and (60) is controllable on $\left[a,b\right]$ and the feedback control law of Equation (67), subject to Equation (66), fixes the state-trajectory solution to any prefixed $x\left(b\right)=x_b^{*}$ at $t=b$ for any given boundary condition value $x\left(a\right)$ at $t=a$.

6. Some Links of the Sturm–Liouville Problem with Linear Algebra

It turns out that the solution of the differential system depends on $\lambda$ in view of Equation (2) since depends on $y\left(t\right)$ and thus on $\lambda$. In the following, we take account of this dependence in the notation unless no confusion is expected. Through this section, one assumes that the differential Equation (2) is a Sturm–Liouville system with $x\left(a\right)\ne 0$, $x\left(b\right)\ne 0$ and that $\lambda \in \mathit{R}$ is constant. The fact that $x\left(a\right)\ne 0$ implies that the associated solution of the differential Equation (2) is non-trivial. Note from Equation (59), since $b\left(t\right)=p^{-1}\left(t\right)\omega \left(t\right)y\left(t\right)e_2$, that for any two real constant eigenvalues $\lambda$ and ${\lambda }^{\prime }$, the solution trajectories of the differential system associated with (2) satisfy:

$$\begin{gathered} x_{\lambda }\left(t\right)=\Psi \left(t,a\right)x\left(a\right)+\lambda {\displaystyle {\int }_a^t\Psi \left(t,\tau \right)}{p}^{-1}\left(\tau \right)\omega \left(\tau \right)y_{\lambda }\left(\tau \right)e_{2}d\tau \\ =\Psi \left(t,a\right)x\left(a\right)+\lambda {\displaystyle {\int }_a^t\Psi \left(t,\tau \right)}{p}^{-1}\left(\tau \right)\omega \left(\tau \right)e_2{e}_1^T{x}_{\lambda }\left(\tau \right)d\tau \\ =x_{{\lambda }^{\prime }}\left(t\right)+{\displaystyle {\int }_a^t\Psi \left(t,\tau \right)}{E}_{21}{p}^{-1}\left(\tau \right)\omega \left(\tau \right)\left(\lambda x_{\lambda }\left(\tau \right)-{\lambda }^{\prime }{x}_{{\lambda }^{\prime }}\left(\tau \right)\right)d\tau \\ =x_{{\lambda }^{\prime }}\left(t\right)+{\displaystyle {\int }_a^t{\left[\begin{array}{cc}{\Psi }_{12}\left(t,\tau \right)& {\Psi }_{22}\left(t,\tau \right)\end{array}\right]}^T}{p}^{-1}\left(\tau \right)\omega \left(\tau \right)\left(\lambda y_{\lambda }\left(\tau \right)-{\lambda }^{\prime }{y}_{{\lambda }^{\prime }}\left(\tau \right)\right)d\tau ; t\in \left[a,b\right] \end{gathered}$$

(68)

with ${\Psi }^{\prime }\left(t,\tau \right)=A\left(t\right)\Psi \left(t,\tau \right)$ and $\Psi \left(\tau , \tau \right)=I_2$ for $\tau \left(\le t\right),t\in \left[a,b\right]$, where

$$E_{21}=e_2{e}_1^T=\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right]$$

implying that

$$\begin{gathered} x_{\lambda }\left(t\right)-x_{{\lambda }^{\prime }}\left(t\right)={\int }_a^t{\left[\begin{array}{cc}{\Psi }_{12}\left(t,\tau \right)& {\Psi }_{22}\left(t,\tau \right)\end{array}\right]}^T{p}^{-1}\left(\tau \right)\omega \left(\tau \right)\left(\lambda y_{\lambda }\left(\tau \right)-{\lambda }^{\prime }{y}_{{\lambda }^{\prime }}\left(\tau \right)\right)d\tau ; \\ t\in \left[a,b\right] \end{gathered}$$

(69)

From Equations (26) and (27), the solutions for real constant values $\lambda$ and ${\lambda }^{\prime }$ are also written equivalently as.

$$x_{\lambda }\left(t\right)={{\rm Z}}_{\lambda }\left(t,a\right)x\left(a\right), x_{{\lambda }^{\prime }}\left(t\right)={{\rm Z}}_{{\lambda }^{\prime }}\left(t,a\right)x\left(a\right); t\in \left[a,b\right]$$

where

$$Z_{\lambda }^{\prime }\left(t,\tau \right)=A_{\lambda }\left(t\right){{\rm Z}}_{\lambda }\left(t,\tau \right)$$

(70)

with ${{\rm Z}}_{\lambda }\left(\tau , \tau \right)={{\rm Z}}_{{\lambda }^{\prime }}\left(\tau , \tau \right)=I_2$ for $\tau \left(\le t\right),t\in \left[a,b\right]$, implying that if the two claimed boundary values are identical for $\lambda$ and ${\lambda }^{\prime }$, that is, $x_{\lambda }\left(a\right)=x_{{\lambda }^{\prime }}\left(a\right)=x\left(a\right)$ and $x_{\lambda }\left(b\right)=x_{{\lambda }^{\prime }}\left(b\right)=x\left(b\right)$, then

$$x\left(b\right)={{\rm Z}}_{\lambda }\left(b,a\right)x\left(a\right)=\Psi \left(b,a\right)x\left(a\right)+\lambda \left({\int }_a^b{p}^{-1}\left(\tau \right)\omega \left(\tau \right){\left[\begin{array}{cc}{\Psi }_{12}\left(b,\tau \right)& {\Psi }_{22}\left(b,\tau \right)\end{array}\right]}^T{e}_1^T{{\rm Z}}_{\lambda }\left(\tau ,a\right)d\tau \right)x\left(a\right)$$

(71)

and

$$\begin{gathered} 0=x\left(b\right)-x\left(b\right)=x_{\lambda }\left(b\right)-x_{{\lambda }^{\prime }}\left(b\right)=\left({{\rm Z}}_{\lambda }\left(b,a\right)-{{\rm Z}}_{{\lambda }^{\prime }}\left(b,a\right)\right)x\left(a\right) \\ =\left({\int }_a^b{p}^{-1}\left(\tau \right)\omega \left(\tau \right){\left[\begin{array}{cc}{\Psi }_{12}\left(b,\tau \right)& {\Psi }_{22}\left(b,\tau \right)\end{array}\right]}^T{e}_1^T\left(\lambda {{\rm Z}}_{\lambda }\left(\tau ,a\right)-{\lambda }^{\prime }{{\rm Z}}_{{\lambda }^{\prime }}\left(\tau ,a\right)\right)d\tau \right)x\left(a\right). \end{gathered}$$

(72)

Then, the above discussion leads to the subsequent result in view of Equations (71) and (72):

Theorem 5. 

Let  $Eig$ be the set of eigenvalues of the Sturm–Liouville system Equation (2) with boundary values $x\left(a\right)={\left(y\left(a\right),y^{\prime }\left(a\right)\right)}^T\ne 0$ at $t=a$ and $x\left(b\right)={\left(y\left(b\right),y^{\prime }\left(b\right)\right)}^T\ne 0$ at $t=b$. Then, the following properties hold:

(i) 

$\lambda \in Eig$ if and only if

$$x\left(a\right)\left(={{\rm Z}}_{\lambda }^{-1}\left(b,a\right)x\left(b\right)\right)\in Ker\left({{\rm Z}}_{\lambda }\left(b,a\right)-\Psi \left(b,a\right)-\lambda \left({\int }_a^b{p}^{-1}\left(\tau \right)\omega \left(\tau \right){\left[\begin{array}{cc}{\Psi }_{12}\left(b,\tau \right)& {\Psi }_{22}\left(b,\tau \right)\end{array}\right]}^T{e}_1^T{{\rm Z}}_{\lambda }\left(\tau ,a\right)d\tau \right)\right)$$

(ii) 

Assume that $\lambda \in Eig$. Then, ${\lambda }^{\prime }\in Eig$ if and only if 

$$\begin{gathered} x\left(a\right)\in Ker\left({{\rm Z}}_{\lambda }\left(b,a\right)-{{\rm Z}}_{{\lambda }^{\prime }}\left(b,a\right)\right) \\ \equiv Ker\left({\int }_a^b{p}^{-1}\left(\tau \right)\omega \left(\tau \right){\left[\begin{array}{cc}{\Psi }_{12}\left(b,\tau \right)& {\Psi }_{22}\left(b,\tau \right)\end{array}\right]}^T{e}_1^T\left(\lambda {{\rm Z}}_{\lambda }\left(\tau ,a\right)-{\lambda }^{\prime }{{\rm Z}}_{{\lambda }^{\prime }}\left(\tau ,a\right)\right)d\tau \right). \end{gathered}$$

Now, note from Equation (27) that for $\lambda ,{\lambda }^{\prime }\in Eig$, one has that

$$A_{{\lambda }^{\prime }}\left(t\right)=A_{\lambda }\left(t\right)+\left(\lambda -{\lambda }^{\prime }\right)p^{-1}\left(\tau \right)\omega \left(\tau \right)E_{21}; t\in \left[a,b\right]$$

(73)

and

$${\dot{x}}_{\lambda }\left(t\right)=A_{\lambda }\left(t\right)x\left(t\right); t\in \left[a,b\right]$$

(74)

$${\dot{x}}_{{\lambda }^{\prime }}\left(t\right)=A_{{\lambda }^{\prime }}\left(t\right)x_{{\lambda }^{\prime }}\left(t\right)=A_{\lambda }\left(t\right)x_{{\lambda }^{\prime }}\left(t\right)+\left(\lambda -{\lambda }^{\prime }\right)p^{-1}\left(\tau \right)\omega \left(\tau \right)E_{21}{x}_{{\lambda }^{\prime }}\left(t\right); t\in \left[a,b\right]$$

(75)

Thus,

$$x_{\lambda }\left(t\right)={{\rm Z}}_{\lambda }\left(t,a\right)x_{\lambda }\left(a\right); t\in \left[a,b\right]$$

(76)

$$\begin{gathered} x_{{\lambda }^{\prime }}\left(t\right)={{\rm Z}}_{{\lambda }^{\prime }}\left(t,a\right)x_{{\lambda }^{\prime }}\left(a\right)={{\rm Z}}_{\lambda }\left(t,a\right)x_{{\lambda }^{\prime }}\left(a\right)+\left(\lambda -{\lambda }^{\prime }\right){\int }_a^t{p}^{-1}\left(\tau \right)\omega \left(\tau \right){{\rm Z}}_{\lambda }\left(t,\tau \right)E_{21}{x}_{{\lambda }^{\prime }}\left(\tau \right)d\tau \\ =\left({{\rm Z}}_{\lambda }\left(t,a\right)+\left(\lambda -{\lambda }^{\prime }\right){\int }_a^t{p}^{-1}\left(\tau \right)\omega \left(\tau \right){{\rm Z}}_{\lambda }\left(t,\tau \right)E_{21}{{\rm Z}}_{{\lambda }^{\prime }}\left(\tau ,a\right)d\tau \right)x_{{\lambda }^{\prime }}\left(a\right); t\in \left[a,b\right] \end{gathered}$$

(77)

where ${{\rm Z}}_{\lambda }\left(t,\tau \right)=A_{\lambda }\left(t\right){{\rm Z}}_{\lambda }\left(t,\tau \right)$ and ${{\rm Z}}_{{\lambda }^{\prime }}^{\prime }\left(t,\tau \right)=A_{{\lambda }^{\prime }}\left(t\right){{\rm Z}}_{{\lambda }^{\prime }}\left(t,\tau \right)$ for $t,\tau \left(\le t\right)\in \left(a,b\right)$ subject to ${{\rm Z}}_{\lambda }\left(t,t\right)={{\rm Z}}_{{\lambda }^{\prime }}\left(t,t\right)=I_2$. Note that $x_{\lambda }\left(a\right)=x_{{\lambda }^{\prime }}\left(a\right)=x\left(a\right)$ and $x_{\lambda }\left(b\right)=x_{{\lambda }^{\prime }}\left(b\right)=x\left(b\right)\lambda ,{\lambda }^{\prime }\in Eig$, then ${{\rm Z}}_{{\lambda }^{\prime }}\left(b,a\right)={{\rm Z}}_{\lambda }\left(b,a\right)$. Thus, one has:

$$x\left(b\right)=\left({{\rm Z}}_{\lambda }\left(b,a\right)+\left(\lambda -{\lambda }^{\prime }\right){\int }_a^b{p}^{-1}\left(\tau \right)\omega \left(\tau \right){{\rm Z}}_{\lambda }\left(b,\tau \right)E_{21}{{\rm Z}}_{{\lambda }^{\prime }}\left(\tau ,a\right)d\tau \right)x\left(a\right)={{\rm Z}}_{\lambda }\left(b,a\right)x\left(a\right)={{\rm Z}}_{{\lambda }^{\prime }}\left(b,a\right)x\left(a\right)$$

(78)

what implies that

$$\begin{gathered} {\int }_a^b{p}^{-1}\left(\tau \right)\omega \left(\tau \right){{\rm Z}}_{\lambda }\left(b,\tau \right)E_{21}{{\rm Z}}_{{\lambda }^{\prime }}\left(\tau ,a\right)d\tau ={\int }_a^b{p}^{-1}\left(\tau \right)\omega \left(\tau \right)\left[\begin{array}{cc}{{\rm Z}}_{{\lambda }_{12}}\left(b,\tau \right)& 0\\ {{\rm Z}}_{{\lambda }_{22}}\left(b,\tau \right)& 0\end{array}\right]\left[\begin{array}{cc}{{\rm Z}}_{{\lambda }_{11}^{\prime }}\left(\tau ,a\right)& {{\rm Z}}_{{\lambda }_{12}^{\prime }}\left(\tau ,a\right)\\ {{\rm Z}}_{{\lambda }_{21}^{\prime }}\left(\tau ,a\right)& {{\rm Z}}_{{\lambda }_{22}^{\prime }}\left(\tau ,a\right)\end{array}\right]d\tau \\ ={\int }_a^b{p}^{-1}\left(\tau \right)\omega \left(\tau \right)\left[\begin{array}{cc}{{\rm Z}}_{{\lambda }_{12}}\left(b,\tau \right){{\rm Z}}_{{\lambda }_{11}^{\prime }}\left(\tau ,a\right)& {{\rm Z}}_{{\lambda }_{12}}\left(b,\tau \right){{\rm Z}}_{{\lambda }_{12}^{\prime }}\left(\tau ,a\right)\\ {{\rm Z}}_{{\lambda }_{22}}\left(b,\tau \right){{\rm Z}}_{{\lambda }_{11}^{\prime }}\left(\tau ,a\right)& {{\rm Z}}_{{\lambda }_{22}}\left(b,\tau \right){{\rm Z}}_{{\lambda }_{12}^{\prime }}\left(\tau ,a\right)\end{array}\right]d\tau =0. \end{gathered}$$

(79)

On the other hand, it follows from Equations (72) and (73), that if $\lambda \in Eig$ then its associate evolution operator satisfies:

$${{\rm Z}}_{\lambda }\left(b,a\right)=\Psi \left(b,a\right)+\lambda \left({{\int }_a^b{p}^{-1}\left(\tau \right)\omega \left(\tau \right)\left[\begin{array}{cc}{\Psi }_{12}\left(b,\tau \right)& {\Psi }_{22}\left(b,\tau \right)\end{array}\right]}^T{e}_1^T{{\rm Z}}_{\lambda }\left(\tau ,a\right)d\tau \right)$$

(80)

Note also that the operator evolution ${{\rm Z}}_{\lambda }\left(.,.\right)$ related to the operator evolution ${{\rm Z}}_{{\lambda }^{\prime }}\left(.,.\right)$ satisfy reciprocal counterpart conditions so that the subsequent relations hold:

$${{\rm Z}}_{{\lambda }^{\prime }}\left(t,a\right)={{\rm Z}}_{\lambda }\left(t,a\right)+\left(\lambda -{\lambda }^{\prime }\right){\int }_a^t{p}^{-1}\left(\tau \right)\omega \left(\tau \right){{\rm Z}}_{\lambda }\left(t,\tau \right)E_{21}{{\rm Z}}_{{\lambda }^{\prime }}\left(\tau ,a\right)d\tau ; t\in \left[a,b\right]$$

(81)

$${{\rm Z}}_{\lambda }\left(t,a\right)={{\rm Z}}_{{\lambda }^{\prime }}\left(t,a\right)+\left({\lambda }^{\prime }-\lambda \right){\int }_a^t{p}^{-1}\left(\tau \right)\omega \left(\tau \right){{\rm Z}}_{{\lambda }^{\prime }}\left(t,\tau \right)E_{21}{{\rm Z}}_{\lambda }\left(\tau ,a\right)d\tau ; t\in \left[a,b\right]$$

(82)

$${{\rm Z}}_{{\lambda }^{\prime }}\left(b,a\right)-{{\rm Z}}_{\lambda }\left(b,a\right)={\int }_a^b{p}^{-1}\left(\tau \right)\omega \left(\tau \right)\left[\begin{array}{cc}{{\rm Z}}_{{\lambda }_{12}}\left(b,\tau \right){{\rm Z}}_{{\lambda }_{11}^{\prime }}\left(\tau ,a\right)& {{\rm Z}}_{{\lambda }_{12}}\left(b,\tau \right){{\rm Z}}_{{\lambda }_{12}^{\prime }}\left(\tau ,a\right)\\ {{\rm Z}}_{{\lambda }_{22}}\left(b,\tau \right){{\rm Z}}_{{\lambda }_{11}^{\prime }}\left(\tau ,a\right)& {{\rm Z}}_{{\lambda }_{22}}\left(b,\tau \right){{\rm Z}}_{{\lambda }_{12}^{\prime }}\left(\tau ,a\right)\end{array}\right]d\tau =0$$

(83)

$$\begin{gathered} {{\rm Z}}_{\lambda }\left(b,a\right)-{{\rm Z}}_{{\lambda }^{\prime }}\left(b,a\right)=-\left({{\rm Z}}_{{\lambda }^{\prime }}\left(b,a\right)-{{\rm Z}}_{\lambda }\left(b,a\right)\right) \\ ={\int }_a^b{p}^{-1}\left(\tau \right)\omega \left(\tau \right)\left[\begin{array}{cc}{{\rm Z}}_{{\lambda }_{12}^{\prime }}\left(b,\tau \right){{\rm Z}}_{{\lambda }_{11}}\left(\tau ,a\right)& {{\rm Z}}_{{\lambda }_{12}^{\prime }}\left(b,\tau \right){{\rm Z}}_{{\lambda }_{12}}\left(\tau ,a\right)\\ {{\rm Z}}_{{\lambda }_{22}^{\prime }}\left(b,\tau \right){{\rm Z}}_{{\lambda }_{11}}\left(\tau ,a\right)& {{\rm Z}}_{{\lambda }_{22}^{\prime }}\left(b,\tau \right){{\rm Z}}_{{\lambda }_{12}}\left(\tau ,a\right)\end{array}\right]d\tau =0. \end{gathered}$$

(84)

By inspecting Equation (80) and Equations (83) and (84), and by using the mean value theorem for integrals of continuous integrands, the subsequent result follows immediately:

Theorem 6.

Let  $Eig$ be the set of eigenvalues of the Sturm–Liouville system Equation (2) with boundary values $x\left(a\right)={\left(y\left(a\right),y^{\prime }\left(a\right)\right)}^T\ne 0$ at $t=a$ and $x\left(b\right)={\left(y\left(b\right),y^{\prime }\left(b\right)\right)}^T\ne 0$ at $t=b$. Then, the following properties hold:

(i) 

Assume that $\lambda ,{\lambda }^{\prime }\in Eig$. Then, there exist real constants  ${\theta }_i\in \left(a,b\right)$; $i\in \overline{8}$, of which at most four of them are distinct, such that 

$${{\rm Z}}_{{\lambda }_{12}}\left(b,{\theta }_1\right){{\rm Z}}_{{\lambda }_{11}^{\prime }}\left({\theta }_1,a\right)-{{\rm Z}}_{{\lambda }_{12}}\left(b,{\theta }_5\right){{\rm Z}}_{{\lambda }_{11}}\left({\theta }_5,a\right)=0$$

(85)

$${{\rm Z}}_{{\lambda }_{12}}\left(b,{\theta }_2\right){{\rm Z}}_{{\lambda }_{12}^{\prime }}\left({\theta }_2,a\right)-{{\rm Z}}_{{\lambda }_{12}}\left(b,{\theta }_6\right){{\rm Z}}_{{\lambda }_{12}}\left({\theta }_6,a\right)=0$$

(86)

$${{\rm Z}}_{{\lambda }_{22}}\left(b,{\theta }_3\right){{\rm Z}}_{{\lambda }_{11}^{\prime }}\left({\theta }_3,a\right)-{{\rm Z}}_{{\lambda }_{22}}\left(b,{\theta }_7\right){{\rm Z}}_{{\lambda }_{11}}\left({\theta }_7,a\right)=0$$

(87)

$${{\rm Z}}_{{\lambda }_{22}}\left(b,{\theta }_4\right){{\rm Z}}_{{\lambda }_{12}^{\prime }}\left({\theta }_4,a\right)-{{\rm Z}}_{{\lambda }_{22}}\left(b,{\theta }_8\right){{\rm Z}}_{{\lambda }_{12}}\left({\theta }_8,a\right)=0$$

(88)

(ii) 

For any $\lambda \in Eig$, there exist non-necessarily distinct real constants ${\varsigma }_i\in \left(a,b\right)$; $i\in \overline{4}$, such that

$${{\rm Z}}_{{\lambda }_{11}}\left(b,a\right)-{\Psi }_{11}\left(b,a\right)-\left(b-a\right){\Psi }_{12}\left(b,{\varsigma }_1\right){{\rm Z}}_{{\lambda }_{11}}\left({\varsigma }_1,a\right)=0$$

(89)

$${{\rm Z}}_{{\lambda }_{12}}\left(b,a\right)-{\Psi }_{12}\left(b,a\right)-\left(b-a\right){\Psi }_{12}\left(b,{\varsigma }_2\right){{\rm Z}}_{{\lambda }_{12}}\left({\varsigma }_2,a\right)=0$$

(90)

$${{\rm Z}}_{{\lambda }_{21}}\left(b,a\right)-{\Psi }_{21}\left(b,a\right)-\left(b-a\right){\Psi }_{22}\left(b,{\varsigma }_3\right){{\rm Z}}_{{\lambda }_{11}}\left({\varsigma }_3,a\right)=0$$

(91)

$${{\rm Z}}_{{\lambda }_{22}}\left(b,a\right)-{\Psi }_{22}\left(b,a\right)-\left(b-a\right){\Psi }_{22}\left(b,{\varsigma }_4\right){{\rm Z}}_{{\lambda }_{12}}\left({\varsigma }_4,a\right)=0$$

(92)

Proof. 

The fact that there are ${\theta }_i\in \left(a,b\right)$ for $i\in \overline{8}$ follows by inspecting Equations (83) and (84) by noting that ${{\rm Z}}_{\lambda }\left(b,a\right)-{{\rm Z}}_{{\lambda }^{\prime }}\left(b,a\right)=-\left({{\rm Z}}_{{\lambda }^{\prime }}\left(b,a\right)-{{\rm Z}}_{\lambda }\left(b,a\right)\right)$ and using the mean value theorem for integrals of continuous integrands, The fact that at least four of the above ${\theta }_{\left(.\right)}$ are repeated follows from the fact that at least one of each pair of factors being zero in each equation of the set of Equations (84)–(87) has to be zero in order for the corresponding product to be zero. Property (i) has been proved. Property (ii) follows from Equation (80) under similar arguments as those used in the proof of Property (i). □

7. Conclusions

Some links between Sturm–Liouville problems arising in second-order differential equations with two-point boundary values and the controllability property in linear dynamic systems have been discussed in this paper. Control law designs have been established so that any prefixed arbitrary final finite state value can be reached via feedback from arbitrarily defined finite initial conditions. The differential equations are described equivalently as second-order dynamic systems involving two first-order differential equations.

In particular, a control law has been synthesized for a forced time-invariant nominal version of the current time-varying system so that their respective two-point boundary values are coincident. The parameter that typically defines the set of eigenvalues of the Sturm–Liouville system is replaced by a time-varying function. Such a function is synthesized as a control law that has the objective of achieving finite arbitrarily prefixed final conditions from any given finite initial ones in a pre-defined finite time interval.

As a result, the trajectory solution of the dynamic system, and thus that of its differential equation counterpart, is subject to arbitrarily prefixed two-point boundary values being allocated at the initial time instant and at the final one of the defined time interval of finite length. Finally, some algebraic constraints between the constant eigenvalues of the Sturm–Liouville system and their respective associated evolution operators are formulated. Those constraints rely on the fact that all the differential equations, each one being parameterized by an eigenvalue of the Sturm–Liouville system, have solutions with the same two-point boundary values.

It is foreseen to extend the given formal results in potential future research to the case of the presence of delays and, eventually, unmodeled dynamics in the dynamics of the differential equation. Also, it is foreseen to extend the results to differential systems of order higher than two and to those that have more than two-point boundary values.