A singularly perturbed linear time-dependent controlled system with multiple point-wise and distributed state delays is considered. The delays in the fast state variable are small of order of the small positive multiplier for a part of the derivatives in the system, which is a parameter of the singular perturbation. The delays in the slow state variable are non-small. Two types of the original singularly perturbed system, standard and nonstandard, are analyzed. For each type, two much simpler parameter-free subsystems (the slow and fast ones) are associated with the original system. It is established in the paper that the approximate state-space controllability of the slow and fast subsystems yields the approximate state-space controllability of the original system robustly with respect to the parameter of singular perturbation for all its sufficiently small values. Illustrative examples are presented.
The system under the consideration is
where for any , , , (u is a control); and are integers; is a small parameter; and are given constants independent of ε; , , , , , are matrices of corresponding dimensions; for any given , the matrix-valued functions , and , are bounded in the interval , while the matrix-valued functions , and , are bounded in the domains and , respectively; for any given , the control function ; denotes the real q-dimensional Euclidean space.
The system (1.1)–(1.2) is functional-differential. Therefore, it is infinite dimensional. The state variables of this system have the form , , , . In what follows, for any given and , we consider the pairs and in the spaces and , respectively, where . The components and of the state variables are called their Euclidean parts, while the components and are called the functional parts of the respective state variables.
Due to the smallness of the parameter ε, the system (1.1)–(1.2) is singularly perturbed [20,29,41]. The state variable , is called a slow state variable, and the equation (1.1) is called a slow mode of this system. The state variable , and the equation (1.2) are called a fast state variable and a fast mode of (1.1)–(1.2), respectively. It is important to note that the system (1.1)–(1.2) has the delays of two scales, the nonsmall delays in the slow state variable and the small delays (of order of ε) in the fast state variable.
Singularly perturbed controlled systems are extensively investigated in the literature, because they appear in various real-life problems and have a considerable theoretical meaning. Study of singularly perturbed controlled systems without delays can be found, for instance, in [8,12,27,29,43] and references therein, while analysis of such systems with delays can be found in [8,33,43] and references therein. Also, it should be noted that singularly perturbed systems with two scales of state delays (nonsmall in the slow state variable and small of order of ε in the fast state variable) were considered in a number of works in the literature. Thus, in [9], an estimate of the fundamental matrix solution was obtained and the exponential stability of the trivial solution was analyzed for a standard linear system. Asymptotic behavior of solution of an initial-value problem for a class of nonlinear systems was studied in [1]. Asymptotic behaviour of the stability radius for a linear time-invariant system was analyzed in [22]. In [10], the robust sampled-data -control for a singularly perturbed linear uncertain system was studied. For the slow sampling in the slow state variable and the fast sampling in the fast state variable, this study was reduced to analysis of the corresponding singularly perturbed system with point-wise time-dependent delays of two scales in the state variables, nonsmall in the slow one and small (of order of ε) in the fast one. In [16], parameter-free conditions of the complete Euclidean space controllability were derived for a linear time-dependent system. In [18], parameter-free -stabilizability conditions for a system with the Stieltjes integral form of the delays were obtained.
In the present paper, we study the approximate state-space controllability of the system (1.1)–(1.2). Controllability of a system is its important property, meaning the ability to transfer the system from any position of a given set of initial positions to any position of a given set of terminal positions in a finite time by a proper choice of the control function. Different types of controllability for various finite-dimensional and infinite-dimensional systems were extensively studied in the literature (see e.g. [2,4,23–26] and references therein). To check a proper type of controllability for a given singularly perturbed system, corresponding controllability conditions can be directly applied for any specified value of a small parameter of singular perturbation. However, the stiffness, as well as a possible high dimension of the singularly perturbed system, can considerably complicate this application. Moreover, such an application depends on the value of ε, i.e., it is not robust with respect to this parameter, while in most of real-life problems this value is unknown. Therefore, controllability analysis of singularly perturbed systems requires another approach, independent of the parameter of singular perturbation. One of such approaches is the approach based on the separation of time-scales concept (see e.g. [29]). Using this concept, the complete controllability of some linear and nonlinear systems without delays was analyzed in the works [28,38,39]. In [34], a linear time-invariant undelayed system with multipliers for the derivatives, which are different positive integer powers of a small positive parameter, was considered. A generalization of the separation of time-scales concept was proposed for the complete controllability analysis of this system. In [31], a linear standard singularly perturbed time-invariant system with a single point-wise delay in the state variables was considered. For this system, based on the separation of time-scales concept, the robust complete Euclidean space controllability was studied in the case of nonsmall delay. In [13,14], using this concept, parameter-free conditions of complete Euclidean space controllability, robust with respect to ε, were obtained for linear standard singularly perturbed systems with point-wise and distributed small delays (of order of ε) in the state variables. In [15], such a result was obtained for nonstandard singularly perturbed systems with multiple point-wise and distributed small delays in the state variables. In [19], a singularly perturbed linear time-dependent controlled system with a small point-wise delay in state and control variables was considered. Applying the separation of time-scales concept, parameter-free conditions of the complete Euclidean space controllability were established for both, standard and nonstandard, types of this system. In [17], a linear singularly perturbed system with small state delays (multiple point-wise and distributed) was considered. The separation of time-scales concept is not applicable for its controllability analysis. The parameter-free complete Euclidean space controllability conditions for this system were derived using the first-order asymptotic expansion of the controllability matrix. In [32], the defining equations approach (see e.g. [11]) was applied for analysis of the complete Euclidean space controllability of a linear singularly perturbed neutral type system with a single nonsmall point-wise delay.
In the present paper, we study a more general type of controllability than the Euclidean space controllability for the system (1.1)–(1.2). Namely, we study the approximate state-space controllability of this system. This study is carried out for the standard and nonstandard cases of (1.1)–(1.2) by application of the separation of time-scales concept. Namely, for each case, two much simpler parameter-free subsystems (the slow and fast ones) are associated with the original system. It is established that the approximate state-space controllability of the slow and fast subsystems yields the approximate state-space controllability of the original system. This controllability is based on parameter-free conditions, while it is valid for all sufficiently small values of the parameter of singular perturbation ε, i.e., robustly with respect to this parameter.
The paper is organized as follows. In the next section, an asymptotic decomposition of the original system (1.1)–(1.2) into the slow and fast subsystems is carried out, main definitions are presented, and objective of the paper is stated. Some auxiliary results are obtained in Section 3. Section 4 is devoted to main results of the paper. Illustrative examples are placed in Section 5. Section 6 is devoted to concluding remarks. Technically complicated proof of one of the main theorems is presented in Appendix.
Completing the Introduction, let us note that the mentioned above set is a real linear space of the pairs , where , . For any two pairs and , the inner product in this space is defined as:
where the superscript “T” denotes the transposition.
For any element , its norm is defined as:
The linear space with the inner product and the norm, defined by (1.3) and (1.4), is a Hilbert space.
Asymptotic decomposition of the system (1.1)–(1.2) and main definitions
For the sake of further analysis, we decompose asymptotically the original singularly perturbed system (1.1)–(1.2) into two much simpler ε-free subsystems, the slow and fast ones.
Slow subsystem
The slow subsystem is obtained by setting formally in (1.1)–(1.2) which yields
where for any , , , ( is a control);
The slow subsystem (2.1)–(2.2) is a differential-algebraic system with state delays, and it is ε-free. If
the slow subsystem can be converted to the following differential equation with state delays
where
Note that the equation (2.5) also called the slow subsystem, associated with the system (1.1)–(1.2).
Fast subsystem
The fast subsystem, obtained by the asymptotic decomposition of (1.1)–(1.2), is derived from the fast mode (1.2) in the following way: (i) the terms containing the slow state variable , are removed from (1.2); (ii) the transformation of the variables , , is made in the resulting system, where ξ is a new independent variable, , , is any fixed time instant.
Thus, we obtain the system
Formal setting in the latter and replacing with t yield the fast subsystem
where t is a parameter; for any , , ( is a control).
The new independent variable ξ, called the stretched time, is expressed by the original one t in the form . For any , as . The fast subsystem is a differential equation with state delays. It is of a lower dimension than the original system (1.1)–(1.2), and it is ε-free.
Definitions of controllability for the systems (1.1)–(1.2), (2.1)–(2.2), (2.5) and (2.9)
First of all let us note that, due to the results of [6], for any given , , , and , the system (1.1)–(1.2) subject to the initial conditions
has the unique absolutely continuous solution , with the derivatives and belonging to and , respectively. It is clear that, for any fixed , the following inclusions are valid: , . The similar statements are valid for the slow and fast subsystems (2.5) and (2.9).
Let be a given time instant, satisfying the inequality
Using the results of [6], we introduce the following definitions.
For a given , the system (1.1)–(1.2) is said to be approximately state-space controllable at the time instant if for any , , and , there exists a sequence of control functions from the space such that the solution of the system (1.1)–(1.2) for the control and subject to the initial conditions (2.10)–(2.11) satisfies the limit equalities
Subject to (2.4), the slow subsystem (2.5) is said to be approximately state-space controllable at the time instant if for any and , there exists a sequence of control functions from the space such that the solution of the equation (2.5) for the control and the initial conditions
satisfies the limit equality
The slow subsystem (2.1)–(2.2) is said to be impulse-free approximately state-space controllable with respect toat the time instant if for any and , there exists a sequence of control functions from the space such that the initial-value problem (2.1)–(2.2), (2.15) for each control has an impulse-free solution , satisfying the limit equality
For a given , the fast subsystem (2.9) is said to be approximately state-space controllableif for any and there exist a number , independent of and , and a sequence of control functions from the space such that the solution of the equation (2.9) for the control and subject to the initial conditions
satisfies the limit equality
Objective of the paper
Our objective in this paper is the following. Based on the approximate state-space controllability of the systems (2.5) and (2.9), as well as on the impulse-free approximate state-space controllability of the system (2.1)–(2.2) and the approximate state-space controllability of the system (2.9), to establish the approximate state-space controllability of the original singularly perturbed system (1.1)–(1.2) robust with respect to for all its sufficiently small values.
Auxiliary results
Approximate state-space controllability of the system (1.1)–(1.2): Necessary and sufficient conditions
For any given , consider the block vector , and the following block matrices:
Using the above introduced vector and matrices, we can rewrite the system (1.1)–(1.2) in the following equivalent form for all and :
Let for any given and , the -matrix-valued function be the solution of the initial-value problem
Let us partition the matrix into blocks as follows:
where the blocks , , and are of the dimensions , , and , respectively.
For any given , and , consider the linear bounded operators , , and , defined by the equations:
where , .
For any given , consider the linear space of vector-valued elements , where , . This space, being endowed with the inner product
becomes a Hilbert space. In (3.13), and are any elements of .
Now, let us consider the -dependent matrix-valued operator, mapping the space into the space ,
where .
For any given , and , the operator is linear bounded. Along with this operator, we consider its adjoint operator
having the form
where , and ; the operators , , and have the form:
, .
Based on the -dependent operators and , we construct the following operator:
where .
For any given and , the operator is linear, self-adjoint and bounded.
For a given, the system (
1.1
)–(
1.2
) is approximately state-space controllable at the time instantif and only if the operatoris positive definite.
Let be any given control function from the space . Let , , , be the state variable of the system (1.1)–(1.2) with this control function and subject to the initial conditions (2.10)–(2.11), For any given , the following inclusion is valid: . Using the results of [6], the equivalence of the systems (1.1)–(1.2) and (3.6), and the definition of the operator (see the equations (3.9)–(3.12), (3.14)), we can represent the evolution of the state variable in the form
where ; for any given , is some linear bounded operator mapping into itself; for any , the image of this operator is continuous with respect to ; for any given , the linear bounded operator is defined as:
For any given , the image of this operator is continuous with respect to .
Now, the statement of the lemma directly follows from Definition 1, the equations (3.15), (3.21), (3.22)–(3.23), and the results of [7]. □
For the sake of further analysis, we rewrite the operators and , appearing in the definition of the operator , in another form. To do this, we consider for any given , and the following two terminal-value problems with respect to -matrix-valued functions and , respectively:
In these problems, it is assumed that the blocks of the matrices , (), , () and satisfy the following equalities:
By virtue of the results of [20] (Section 4.3), we obtain that the problems (3.24) and (3.25) have the unique solutions and . Moreover, for any given ,
Let us partition the matrices and into blocks as follows:
where the blocks and are of the dimension ; the blocks and are of the dimension ; the blocks and are of the dimension ; the blocks and are of the dimension .
Using (3.27)–(3.30), we can rewrite the blocks of the operator (see (3.9)–(3.12), (3.14)) and the blocks of the operator (see (3.15)–(3.20)) in the following form:
where , .
As a direct consequence of Lemma 1, we have the following corollary.
For a given, the system (
1.1
)–(
1.2
) is approximately state-space controllable at the time instantif and only if the operator, given by the equation (
3.21
) and the equations (
3.14
), (
3.15
)–(
3.16
), (
3.31
)–(
3.38
), is positive definite.
Some estimates of solutions to singularly perturbed matrix differential equations with delays
In what follows, we assume:
The matrix-valued functions , , , are continuously differentiable in the interval .
The matrix-valued functions , are piece-wise continuous with respect to for each , and they are continuously differentiable with respect to uniformly in .
The matrix-valued functions , are piece-wise continuous with respect to for each , and they are continuously differentiable with respect to uniformly in .
All roots of the equation
satisfy the inequality for all , where is some constant.
Let the assumptions A1–A4 be valid. Then, there exists a positive numbersuch that, for all, the following inequalities are satisfied:where, () are the blocks of the solution to the initial-value problem (
3.7
);is some constant independent of ε.
We prove the inequalities for and . The other inequalities are proven similarly.
Substituting (3.1)–(3.5) and (3.8) into (3.7), we obtain the initial-value problem for and
Let for any given , be the -matrix-valued function satisfying the initial-value problem
Since the matrix-valued functions , () are bounded for , and the matrix-valued function is bounded for , then
where denotes the Euclidean norm of a matrix; is some constant.
Further, let for any given and , be the -matrix-valued function satisfying the initial-value problem
By virtue of the results of [3], we directly obtain the existence of a positive number such that the following estimate of is valid for all :
where is some constant independent of ε.
Applying the variation-of-constant formula (see e.g. [6,20]) to the problems (3.40) and (3.41), and using the matrix-valued functions and , we can rewrite these problems in the equivalent integral form
Substituting (3.45) into (3.44), changing the integration order and taking into account the initial conditions for , we obtain the following integral equation with respect to :
where
The estimates (3.42) and (3.43) yield the following inequality for all :
where is some constant independent of ε.
Now, applying the method of successive approximations with zero initial guess to the integral equation (3.46) and using the estimate (3.47), we directly obtain the existence of a positive number such that, for all , the following inequality is valid:
where is some constant independent of ε.
Finally, using the equation (3.45), and the estimates (3.43) and (3.48), we have immediately
where is some constant independent of ε. Thus, the lemma is proven. □
Let for a given , the matrix-valued function be the solutions of the following problem:
where, taking into account (3.26), the matrix-valued functions , and are given by (2.6) and (2.7), respectively.
Let the assumptions A1–A4 be valid. Then, there exists a positive numbersuch that, for alland, the following inequalities are satisfied:where, () are the corresponding blocks of the solution to the terminal-value problem (
3.24
),is some constant independent of η and ε.
Proof. First of all let us note that, due to the assumption A4, the inequality (2.4) is valid for all . Therefore, the matrix-valued function exists for any and all .
Let us denote
Using these notations, the terminal-value problems (3.24) and (3.49), and the equation (3.50), we obtain the terminal-value problem for the matrix-valued function
where the -matrix-valued function has the block form
and the -block is
Let us estimate . First of all note, that is bounded for uniformly in . Therefore, and are bounded for uniformly in . This observation, the assumptions A1, A3 and the fact that , for yield the existence of a positive number such that, for all , the following estimate is valid:
where is some constant independent of η and ε.
By application of the results of [20] (Section 4.3), we can rewrite the problem (3.53) in the equivalent integral form
where , is given by (3.7).
The equation (3.57), along with the block representation of (see the equation (3.8)), Lemma 2, the expression for (see the equation (3.53)), the block representation of (see the equation (3.54)), the estimate for (see the equation (3.56)) and the inequality , , , directly yields the following inequalities for all , and :
where and are the upper and lower blocks of the matrix of the dimensions and , respectively; are some constant independent of η and ε.
The latter inequalities, along with the notations (3.51) and (3.52), imply the validity of the inequalities stated in the lemma. □
Since , , and for , then the statements of Lemma 3 are valid for all .
Let the matrix-valued function be the solution of the following problem:
From the assumption A4 and the results of [21], we directly obtain
where is some constant.
Let the assumptions A1–A4 be valid. Then, there exists a positive numbersuch that, for alland, the following inequalities are satisfied:where, () are the corresponding blocks of the solution to the terminal-value problem (
3.25
);is some constant independent of ε and τ.
Since , for , and for , then the statements of Lemma 4 are valid for all .
Approximate state-space controllability of the slow (2.5) and fast (2.9) subsystems: Necessary and sufficient conditions
For a given , consider the linear bounded operator and its adjoint operator , given by the equations
where .
Also, for a given , we consider the linear bounded operator and its adjoint operator , given by the equations
where .
Based on the operators , , and , we construct two linear bounded operators and . These operators are:
Note that the first operator depends on , while the second operator is -dependent.
Similarly to Corollary 1, we have the following two assertions.
Let the inequality (
2.4
) be valid. Then, the slow subsystem (
2.5
) is approximately state-space controllable at the time instantif and only if the operatoris positive definite.
The fast subsystem (
2.9
) is approximately state-space controllable if and only if there exists a numbersuch that the operatoris positive definite.
Let us transform the control in the system (1.1)–(1.2) as:
where is a new control; and are any specified matrix-valued functions of corresponding dimensions, given for and ; for any given , the matrix-valued function is bounded in the interval , while the matrix-valued function is bounded in the domain .
Due to this transformation, for all , the system (1.1)–(1.2) becomes as:
where
For a given, the system (
1.1
)–(
1.2
) is approximately state-space controllable at the time instant, if and only if the system (
3.67
)–(
3.68
) is approximately state-space controllable at this time instant.
Necessity: Suppose that for some , the system (1.1)–(1.2) is approximately state-space controllable at . Let , , and be arbitrary given. Due to Definition 1, there exists a sequence of control functions from the space such that the solution of the system (1.1)–(1.2) for the control and subject to the initial conditions (2.10)–(2.11) satisfies the limit equalities (2.13)–(2.14).
For the above mentioned control , the system (1.1)–(1.2) can be rewritten in the equivalent form
Denote
Since , (), then
Now, using the equivalent form (3.73)–(3.74) of the system (1.1)–(1.2), one directly has that the sequence of the control functions , given by (3.75) and satisfying (3.76), generates the sequence of the solutions to the system (3.67)–(3.68) subject to the initial conditions (2.10)–(2.11), and this sequence of the solutions satisfies the limit equalities (2.13)–(2.14). The latter, along with Definition 1, implies the approximate state-space controllability of the system (3.67)–(3.68) at the time instant . Thus, the necessity is proven.
Sufficiency: The sufficiency is proven similarly to the necessity. □
Asymptotic decomposition of the transformed system (3.67)–(3.68)
Similarly to the asymptotic decomposition of the original system (1.1)–(1.2), we can decompose the transformed system (3.67)–(3.68). Thus the slow subsystem, associated with (3.67)–(3.68), is the differential-algebraic system with state delays
where for any , , , ( is a control);
If
the differential-algebraic system (3.77)–(3.78) can be reduced to the differential equation with state delays
where
The fast subsystem, associated with (3.67)–(3.68), is the differential equation with state delays
where t is a parameter; ξ is an independent variable; for any , , ( is a control).
The system (
2.1
)–(
2.2
) is impulse-free approximately state-space controllable with respect toat the time instant, if and only if the system (
3.77
)–(
3.78
) is impulse-free approximately state-space controllable with respect toat this time.
Let us make the following control transformation in the differential-algebraic system (2.1)–(2.2):
where is a new control, and
Using the equations (3.69)–(3.72), (3.79), (3.87), we directly obtain that the transformation (3.86) converts the system (2.1)–(2.2) to the system (3.77)–(3.78). Based on Definition 3, the rest of the proof is carried out similarly to the proof of Lemma 7. □
Let the inequality (
3.80
) be valid. Then, the system (
3.77
)–(
3.78
) is impulse-free approximately state-space controllable with respect toat the time instantif and only if the system (
3.81
) is approximately state-space controllable at this time instant.
Due to the inequality (3.80), the systems (3.77)–(3.78) and (3.81) are equivalent to each other with respect to . This means the following. If for a given the pair , is a solution of (3.77)–(3.78), then for this the first component of this pair is a solution of (3.81) in the interval . Vise versa, if for a given the vector-valued function , is a solution of (3.81), then for this the pair , where
is a solution of the system (3.77)–(3.78) in the interval .
Now, the above mentioned equivalence of the systems (3.77)–(3.78) and (3.81) directly yields the statement of the lemma. □
For a given, the system (
2.9
) is approximately state-space controllable if and only if the system (
3.85
) is approximately state-space controllable.
Using the control transformation in the system (2.9)
the lemma is proven similarly to Lemma 7. □
Parameter-dependent hybrid set of Riccati-type matrix equations
Let us denote
Using this notation, consider the following set consisting of one algebraic and two differential equations (ordinary and partial) with respect to matrices , , and :
where is a parameter; and are independent variables; is the Dirac delta-function.
The set of equations (3.90)–(3.92) is subject to the boundary conditions
In what follows, we assume:
A5. For all and any complex number λ with , the following equality is valid:
Let the assumption A5 be valid. Then, for each, the set of equations (
3.90
)–(
3.92
) subject to the boundary conditions (
3.93
) has the unique solutionsuch that the matrixdefines a linear bounded self-adjoint nonnegative operator mapping the spaceinto itself. The matrix-valued functionis piece-wise absolutely continuous inwith the bounded jumps at,, the matrix-valued functionis piece-wise absolutely continuous inand inwith the bounded jumps atand,.
Moreover, all rootsof the equationsatisfy the inequalitywhereis some function of.
The statements of the lemma directly follow from the results of [5] and [42]. □
[
15
]Let the assumption A1 with respect toand, and the assumptions A3 and A5 be valid. Then, the derivatives,,exist and are continuous functions ofuniformly in.
Let the assumption A1 with respect toand, and the assumptions A3 and A5 be valid. Then, there exists a positive numbersuch that all rootsof the equation (
3.94
) satisfy the inequality,.
Let us assume that the statement of the lemma is wrong. Due to the inequality (3.95), this means the existence of two sequences and satisfying the properties: (a) , ; (b) , and ; (c) the equation (3.94) is satisfied for any pair , .
Due to the property (a), there exists a convergent subsequence of . For the sake of simplicity (but without loss of generality), we assume that the sequence itself is convergent, and . Hence, .
The following two cases can be distinguished with respect to the sequence : (i) is bounded; (ii) is unbounded. We start with the first case. In this case, there exists a convergent subsequence of . For the sake of simplicity (but without loss of generality), we assume that the sequence itself is such a subsequence. Let . Due to the above mentioned property (b), . Also, let us note that, by virtue of Lemma 12, the matrix is a continuous function of , and the matrix is a continuous function of uniformly in .
Now, the substitution of into (3.94) and the calculation of the limit of the resulting equality for , yield:
The latter means that is a root of the equation (3.94) for . Thus, due to the inequality (3.95), , which contradicts the above obtained equality .
Proceed to the case (ii) where the sequence is unbounded. In this case, there exists a subsequence of , modules of elements of which tend to infinity. Similarly to the case (i), we assume that itself is such a subsequence, i.e., . Substituting into (3.94), dividing the resulting equality by and, then, calculating the limit of the last equality for , one obtains the contradiction .
The contradictions, obtained in the cases (i) and (ii), prove the lemma. □
Main results: Parameter-free conditions of the approximate state-space controllability
If the assumption A4 is valid, then does not satisfy the equation (3.39) for all . The latter means that the condition (2.4) holds for all . In the literature, singularly perturbed systems with such a feature are called standard (see e.g. [12,29]). Thus, subject to the validity of the assumption A4, the system (1.1)–(1.2) is standard.
Let the assumptions A1–A4 be valid. Let the slow subsystem (
2.5
) be approximately state-space controllable at the time instant. Let, for, the fast subsystem (
2.9
) be approximately state-space controllable. Then, there exists a positive numbersuch that, for all, the system (
1.1
)–(
1.2
) is approximately state-space controllable at the time instant.
The proof of the theorem is presented in Appendix.
Assumption A4 is not valid for the system (1.1)–(1.2)
In this subsection, in contrast with the previous subsection, we consider the case where for at least one value of the equation (3.39) has a root with nonnegative real part. If is such a root, then for this value of t the condition (2.4) does not hold. In the literature, singularly perturbed systems with such a feature are called nonstandard (see e.g. [12,29]).
Let the assumptions A1–A3 and A5 be valid. Let the slow subsystem (
2.1
)–(
2.2
) be impulse-free approximately state-space controllable with respect toat the time instant. Let, for, the fast subsystem (
2.9
) be approximately state-space controllable. Then, there exists a positive numbersuch that, for all, the system (
1.1
)–(
1.2
) is approximately state-space controllable at the time instant.
Let ε be any given positive number. For this ε, let us transform the control in the system (1.1)–(1.2) in accordance with (3.66) where and are chosen in the form
and and are components of the solution to the problem (3.90)–(3.93). This transformation yields the system (3.67)–(3.68). Due to Lemmas 11–12, the coefficients (3.69)–(3.72) of this system have the properties, similar to the assumptions (A1)–(A3) on the coefficients of the system (1.1)–(1.2).
The slow subsystem, associated with (3.67)–(3.68), has the form (3.77)–(3.78), where
By virtue of Lemma 11, is not a root of the equation (3.94) for any . Therefore, , given by (4.1), satisfies the inequality (3.80) for . Hence, the slow differential-algebraic subsystem (3.77)–(3.78) can be reduced to the slow differential subsystem (3.81). Moreover, since the slow subsystem (2.1)–(2.2) is impulse-free approximately state-space controllable with respect to at the time instant , then, due to Lemmas 8 and 9, the slow subsystem (3.81) is approximately state-space controllable at this time instant.
The fast subsystem, associated with (3.67)–(3.68), is (3.85), where
By virtue of Lemma 13, the fast subsystem (3.85) with the coefficients (4.2) satisfies the stability property similar to the assumption A4 on the fast subsystem (2.9).
Since, for , the fast subsystem (2.9) is approximately state-space controllable, then due to Lemma 10, the fast subsystem (3.85), (4.2) with is approximately state-space controllable.
Thus, the system (3.67)–(3.68) satisfies all the conditions of Theorem 1, meaning the existence of a positive number such that, for all , this system is approximately state-space controllable at the time instant . Finally, the application of Lemma 7 to the system (3.67)–(3.68) yields the approximate state-space controllability of the system (1.1)–(1.2) at the time instant for all . This completes the proof of the theorem. □
As a direct consequence of Theorem 2, we have the following assertion.
Let the assumptions A1–A3 and A5 be valid. Let the condition (
2.4
) hold for all. Let the slow subsystem (
2.5
) be approximately state-space controllable at the time instant. Let, for, the fast subsystem (
2.9
) be approximately state-space controllable. Then, there exists a positive numbersuch that, for all, the system (
1.1
)–(
1.2
) is approximately state-space controllable at the time instant.
Examples
In this section, we consider three examples, illustrating the above obtained theoretic results.
Example 1
Consider the following system, a particular case of (1.1)–(1.2),
where , , and are scalars, i.e., , , , , .
In this example, we study the approximate state-space controllability of the system (5.1)–(5.3) at the time instant for all sufficiently small . For this purpose, let us decompose this system into the slow and fast subsystems.
The slow subsystem has the form
From the equation (5.6) it is seen that in this example the matrix (see (3.79)) becomes the scalar . Hence, exists for all , i.e., the system (5.1)–(5.3) is standard. Therefore, the slow subsystem (5.4)–(5.6) can be converted to the form (2.5), i.e.,
where , ,
The fast subsystem, associated with (5.1)–(5.3), has the form
where t is a parameter.
The assumptions A1–A3 are satisfied for the system (5.1)–(5.3). Let us show the validity of the assumption A4. The equation (3.39) becomes
Thus, for , we obtain
meaning that all roots of the equation (5.10) satisfy the inequality , . Hence, for the system (5.1)–(5.3) and , the assumption A4 is valid with .
Based on the validity of the assumptions A1–A4 for the systems (5.1)–(5.3) and , let us apply Theorem 1 to the further controllability analysis of this system. For this purpose, the approximate state-space controllability of the slow subsystem (5.7) at the time instant , as well as the approximate state-space controllability of the fast subsystem (5.9) for , should be proven. It should be noted that both subsystems are linear constant coefficients differential-difference equations. The coefficients for the controls in these equations are nonzero. For such equations, one can try to use the results of [35] and [36]. Let us start with the slow subsystem (5.7). First of all, let us observe that . Hence, due to the results of the above mentioned works, the fulfilment of either the conditions of Theorem 4.3 [36] or the conditions of Theorem 4.5 [36] guarantees the approximate state-space controllability of the slow subsystem (5.7) at the time instant . Let us apply Theorem 4.5 [36] to check up this controllability. By virtue of this theorem, we should verify the fulfilment the following conditions:
In (5.12), the expression , where A is an matrix, denotes the image of the linear operator generated by A.
Using (5.8), we directly calculate that
implying the fulfilment of the conditions (5.11)–(5.13). Hence, the slow subsystem (5.7) is approximately state-space controllable at the time instant . Similarly, one can show the approximate state-space controllability of the fast subsystem (5.9) with . The number , mentioned in Definition 4, is any satisfying the inequality .
Thus, all the conditions of Theorem 1 are fulfilled for the system (5.1)–(5.3), meaning its approximate state-space controllability at the time instant robustly with respect to for all sufficiently small values of this parameter.
Example 2
Consider the following system, a particular case of (1.1)–(1.2),
where , , are scalars, and , .
In this example, like in Example 1, we study the approximate state-space controllability of the original system (5.14)–(5.15) at the time instant for all sufficiently small .
Decomposing asymptotically the system (5.14)–(5.15) and taking into account that this system is standard (), we obtain the slow and fast subsystems, associated with this system,
The assumptions A1–A3 are satisfied for the system (5.14)–(5.15). Let us find out whether the assumption A4 is satisfied for this system. The equation (3.39) becomes
It is verified directly that, for , the equation (5.18) has the purely imaginary roots and . Hence, the assumption A4 is not satisfied for (5.14)–(5.15). Therefore, Theorem 1 is not applicable to this system.
Let us show that the assumption A5 is satisfied for the system (5.14)–(5.15). The matrix, appearing in this assumption, becomes
To show the fulfilment of the assumption A5, one should show that for all complex λ with and all . It is clear that for all . Let us show that for all complex λ with . To show the latter, it is necessary and sufficient to prove that the equation
has no roots with nonnegative real parts.
Let and be the real and imaginary parts of a complex number λ. Using these notations, we can rewrite the equation (5.20) in the equivalent form
We assume that there exists a solution of (5.21)–(5.22) with and analyze this system in the two cases: (1) ; (2) . In the first case, we obtain from (5.21): , k is any integer. For these and , the equation (5.22) yields the contradiction . Proceed to the second case. In this case, we have . The latter, along with the equation (5.22), yields , meaning that . Due to this inequality, the left-hand side of the equation (5.21) is positive, which contradicts to the negativeness of its right-hand side. The contradictions, obtained in the analysis of both cases, mean that the system (5.21)–(5.22) and, therefore, the equation (5.20) do not have a solution with . Thus, for all complex λ with , yielding the fulfilment of the assumption A5.
Using the fulfilment of the assumptions A1–A3, A5 and the condition (2.4) for the system (5.14)–(5.15) and , let us try to apply Corollary 2 to the further controllability analysis of this system. The approximate state-space controllability of the slow subsystem (5.16) at is shown similarly to such a controllability of the slow subsystem (5.7) in the previous example. Proceed to the fast subsystem (5.17) for . This subsystem is scalar, the coefficients for and are constant nonzero, while the coefficient for equals zero. Therefore, by virtue of Theorems 3.3 and 4.3 from [36], the fast subsystem (5.17) for is approximately state-space controllable, and . Thus, all the conditions of Corollary 2 are fulfilled for the system (5.14)–(5.15), meaning the approximate state-space controllability of this system at the time instant robustly with respect to for all its sufficiently small values.
Example 3
Consider the following particular case of the system (1.1)–(1.2),
where , , are scalars, and , .
In this example, like in Examples 1 and 2, we study the approximate state-space controllability of the original system (5.23)–(5.24) at the time instant for all sufficiently small .
Asymptotic decomposition of the system (5.23)–(5.24) yields its slow and fast subsystems, respectively,
and
The assumptions A1–A3 are satisfied for the system (5.23)–(5.24), while the assumption A4 is not. Indeed, the equation (3.39) becomes . For , this equation has the root . Moreover, in this example the condition (2.4) also is not valid, because . Therefore, the slow subsystem (5.25)–(5.26) cannot be converted to the differential equation in the interval by eliminating .
Let us show that the system (5.23)–(5.24) satisfies the assumption A5. The matrix, appearing in this assumption, becomes
It is clear that for all . Moreover, since , then for all complex λ with . Hence, (5.23)–(5.24) satisfies the assumption A5.
Now, based on the fulfilment of the assumptions A1–A3 and A5 for (5.23)–(5.24), we will show the applicability of Theorem 2 to the further controllability analysis of the system (5.23)–(5.24). First, let us show that the slow subsystem (5.25)–(5.26) is impulse-free approximately state-space controllable with respect to at the time instant . For this purpose, we transform the control in this subsystem as , where is a new control. This transformation allows to eliminate from the resulting system, yielding the controlled differential equation
For this equation, all the conditions of Theorem 4.3 from [36] are fulfilled. Hence, the equation (5.29) is approximately state-space controllable at . Therefore, by virtue of Lemmas 8 and 9, the slow subsystem (5.25)–(5.26) is impulse-free approximately state-space controllable with respect to at . The approximate state-space controllability of the fast subsystem (5.27) for is shown similarly to such a controllability of the fast subsystem (5.9) in Example 1 using Theorem 4.5 [36]. Thus, all the conditions of Theorem 2 are valid for the system (5.23)–(5.24), implying its approximate state-space controllability at the time instant robustly with respect to for all sufficiently small values of this parameter.
Conclusions
In this paper, a singularly perturbed linear time-dependent controlled system with time delays (point-wise and distributed) in the state variables was considered. These delays are of the following two scales. In the fast state variable, they are proportional to the small positive multiplier ε for a part of the derivatives in the considered system, while in the slow state variable the delays are non-small. For this system, the approximate state-space controllability at a given time instant, robust with respect to ε, was studied. This study is based on the asymptotic decomposition of the original system into two ε-free subsystems, slow and fast ones. The slow subsystem is a differential-algebraic time delay system, while the fast subsystem is a differential time delay system. In the case of the standard original system, the slow subsystem can be converted to a differential time delay system.
Several auxiliary results were obtained, such as: (i) necessary and sufficient condition for the approximate state-space controllability of a linear singularly perturbed time-dependent system with small delays in the fast state variable and non-small delays in the slow state variable; (ii) block-wise estimates of solutions to singularly perturbed linear matrix differential equations with the small and non-small delays; (iii) the invariance of the controllability properties for the original system, as well as for the slow and fast subsystems, with respect to a linear control transformation; (iv) the smoothness of the solution to a hybrid set of Riccati-type matrix equations with respect to a parameter, varying in a closed bounded interval.
Based on the above mentioned asymptotic decomposition of the original singularly perturbed system and using the auxiliary results, ε-free sufficient conditions for the approximate state-space controllability of this system were established. These conditions were obtained for the cases of the standard and nonstandard original system. Moreover, being ε-free, these conditions provide the approximate state-space controllability of the original singularly perturbed system for all sufficiently small values of , i.e., robustly with respect to this parameter of singular perturbation.
Footnotes
Proof of Theorem 1
First of all let us note that, due to the assumption A4, the inequality (2.4) is valid for all . Therefore, the system (2.5) exists for these values of t.
Due to Corollary 1 and the results of [40], in order to prove the theorem it is sufficient to show that the following inequality is valid for any small enough :
where is an arbitrary element of the space ; is some constant independent of ; is the norm in generated by the inner product . Note, that κ may, in general, be dependent on ε.
Using the expression of the operator (see the equation (3.21)), the independence of on and the fact that is the adjoint operator to the operator , we can represent the inner product in the left-hand side of the inequality (A.1) as follows:
For a given , let be the linear space of vector-valued elements , where , . For this ε and a given , consider the following two matrix-valued operators: and : , defined by the equations
Thus, due to the equations (3.16), (3.35)–(3.38) and (A.3)–(A.4), the operator , appearing in (A.2), can be represented as:
Substituting (A.5) into (A.2), we obtain
where
Let us analyze these function of ε. We start with .
Denote
Substitution of (3.5) and (A.3) into (A.7) yields
where
Remember that is defined in (3.89).
Let us estimate the functions , for all sufficiently small . Using Lemma 3, Remark 1 and the equations (2.8), (3.89), (A.10), (A.12), we obtain after some algebra the following inequality:
where
is some constant independent of ε and .
Furthermore, using Lemmas 3, 4, Remarks 1, 2, the equations (3.58), (A.13)–(A.15) and the inequality (3.59), we have
where
, and are some constants independent of ε and , .
Let us analyze the function . Substitution of (3.5) and (A.3)–(A.4) into (A.8) yields
where
The functions , are estimated similarly to the functions , . Namely, using Lemma 3, the equations (2.8), (3.89), (A.10), (A.22), and the Cauchy–Bunyakovsky–Schwarz inequality for integrals [30], we obtain after some rearrangement the following inequality:
where
is some constant independent of ε, and ; denotes the norm in a corresponding -space.
Furthermore, using Lemmas 3 and 4, the equations (3.58), (A.13)–(A.15), the inequality (3.59), and the Cauchy–Bunyakovsky–Schwarz inequality, we have
where
, , and are some constants independent of ε and , , , .
Proceed to analysis of the function . Substitution of (3.5) and (A.4) into (A.9) yields
where
Functions , () are estimated similarly to the functions , (). Thus, we have
where
is some constant independent of ε and .
Furthermore,
where
, and are some constants independent of ε and , .
Consider the value
Using the equations (A.17), (A.28) and (A.40), this value can be rewritten in the form , where the operator is defined by the equations (3.60)–(3.61), (3.64); .
Since the slow subsystem (2.5) is approximately state-space controllable at the time instant , then the operator is positive definite (see Lemma 5). Therefore, there exists a positive number such that the following inequality is valid for all :
Due to the equation (A.45) and the inequalities (A.16), (A.27), (A.39), (A.46), there exists a positive number such that, for all and all , , the following inequality is satisfied:
where is some constant independent of ε and , .
Now, consider the value
where the operator is defined by (3.62), (3.63), (3.65); is any given element of , i.e., , , .
Since the fast subsystem (2.9) is approximately state-space controllable, then by virtue of Lemma 6, the following inequality is valid for all :
where and are some numbers independent of and . Moreover, due to the definitions of the operator and the value , we have
Using the equations (3.62), (3.63), (3.65), and the transformation techniques similar to (A.2)–(A.9), we can represent (A.48) in the form:
Calculating the limit for of the expression in the right-hand side of (A.51), and taking into account the inequalities (A.49) and (A.50), we obtain the following inequality for all and :
Note, that due to the inequality (3.59), the improper integrals in the left-hand side of this inequality exist.
Let, for a given , be any function from and is given by (A.33). We have
Let us choose in (A.52) . Then, using (A.20), (A.32), (A.44) and (A.53), we obtain the inequality
The latter, along with (A.19), (A.31), (A.43), directly yields the existance of a positive number such that, for all and all , , the following inequality is satisfied:
where is some constant independent of ε and , .
Now, using the equations (A.6), (A.11)–(A.15), (A.21)–(A.26), (A.34)–(A.38) and the inequalities (A.18), (A.29), (A.41), (A.42), (A.47), (A.54), we obtain
The expression in the right-hand side of this inequality is the quadratic form for the variables , , , with the following symmetric matrix of coefficients:
The leading principal minor of the first order of the matrix is . Moreover, by a routine algebra, it is shown the existence of a positive number such that, for all , the rest of the leading principal minors , () of satisfy the inequalities , where is some positive number. Thus, by virtue of the Silvester’s criterion [37], the matrix is positive definite for all . This observation, along with the inequality (A.55), yields the validity of the inequality (A.1) for all and all . This completes the proof of the theorem.
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