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Some Schrödingertype inequalities for stabilization of discrete linear systems associated with the stationary Schrödinger operator
Journal of Inequalities and Applications volume 2016, Article number: 247 (2016)
Abstract
By applying some Schrödingertype inequalities developed by Huang (Int. J. Math. 27(2):1650009, 2016), we are concerned with stabilization of discrete linear systems associated with the Schrödinger operator. Our first aim is to prove a statedependent switching law associated with the Schrödinger operator, which is based on a convex combination. Next, we derive sufficient conditions associated with the Schrödinger operator that guarantee the uniform exponential stability of the system. Finally, we propose a necessary and sufficient condition for the stability of a system with two Schrödinger subsystems.
Introduction
A switched system is composed of several subsystems and a decision rule that orchestrates switching between these subsystems. Due to their wide existence in engineering applications, this kind of dynamical systems has attracted considerable attention during the past decades. In the real word, many quantities are nonnegative [2–4]. As a special case of switched systems, a switched positive system with all subsystems associated with the Schrödinger operator being positive systems is very common in communication systems associated with the Schrödinger operator, formation flying, and rival mutation treatment system [5].
As we all know, it is necessary to address the stability and stabilization issues for switched positive systems associated with the Schrödinger operator. Generally, the synthesis problem of switched systems associated with the Schrödinger operator primarily concerns on two topics: stabilization by feedback controllers and stabilization by feedback switching law. The former topic is studied mainly based on state feedback or output feedback controller designation [6, 7], whereas the latter topic is proved by feedback switching law [8, 9]. Furthermore, there are many switched systems associated with the Schrödinger operator whose subsystems associated with the Schrödinger operator are not stable in real control systems associated with the Schrödinger operator [1, 10]. To the best of our knowledge, existing stabilization results on switched positive linear systems associated with the Schrödinger operator have little results referring to the statedependent switching law, which motivates the research of this paper.
This paper investigates the boundary value problems of discrete linear systems associated with the Schrödinger operator in discrete time context, which contain unstable subsystems associated with the Schrödinger operator. The state feedback switching law is designed to guarantee the stabilization of the switched positive linear systems associated with the Schrödinger operator. The result is then applied to systems associated with the Schrödinger operator with two subsystems associated with the Schrödinger operator, and sufficient and necessary condition is also derived. The rest of paper is organized as follows. Section 2 gives introduces some preliminaries. In Section 3, we give the main results. Section 4 concludes this paper.
Throughout the paper, \(\Re^{n}\) and \(\Re^{n\times n}\) represent the vectors of ntuples of real numbers and the space of \(n\times n\) matrices with real entries, respectively, and \(\mathbb{N}\) is the set of nonnegative integers. For v in \(\Re^{n}\), \(v_{i}\) is the ith component of v. The notation \(v\succ0 \) (⪰0) means that all components of v are positive (nonnegative), that is, \(v_{i}>0 \) (≥0), and \(\overline{\lambda}_{v}\) and \(\underline{\lambda}_{v}\) stand for the maximal and minimal components of v, respectively. For a matrix A in \(\Re^{n\times m}\), \(a_{ij}\) is the element in the ith row of the jth column of A, and \(A\succ0 \) (⪰0) means that all elements of A are positive (nonnegative), that is, \(a_{ij}>0\) (≥0).
Preliminaries
Consider the following discretetime switched linear system:
where \(\sigma(k)\) is a piecewise constant switching signal taking values in the finite set \(S=\{1,2,\ldots,N \}\), and \(A_{i}\in\Re ^{n\times n} \) (\(i\in S\)) are system matrices.
Assumption 1
For system (2.1), \(A_{i}\succeq0\) for each \(i\in S\).
Definition 1
A switching signal σ is said to be a statefeedback switching law if it depends on system states and its past value, that is, \(\sigma(t^{+})=\sigma(x(t),\sigma(t^{}))\) for system (2.1), where \(\sigma(k+1)=\sigma(x(k),\sigma(k1))\).
Lemma 1
([2])
A discretetime system
is positive if and only if \(A\succeq0\).
Lemma 2
Let \(A\succeq0\). Then the following conditions are equivalent:

(i)
A is a Schur matrix;

(ii)
There exists vector \(v\succ0\) in \(\Re^{n}\) with \((AI)v\prec0\).
Proof
(i) For any \(i\in E,i\geq3\),
In the following, we compute \(E^{1}\{e^{\lambda\sigma_{1}}\}\). Consider the coordinate process \(W(s)\) on excursion space \((U,\mathcal {U})\). For any \(i\in E\), we define
and
Obviously, \(C_{0}, C_{1}\in\mathcal{U}\) and \(C_{0}\cup C_{1}=U\). Let \(\tau=\inf\{t\beta_{t}>\sigma_{2}\}\) for any \(t>0\), and
Then we have
So we know that τ is an exponential random variable with mean 1. Hence, we have
From the computation of a Poisson point process we know that
which yields that
Hence, we have
Therefore, if \(0<\lambda< \frac{1}{\sum_{n=3}^{\infty} (q_{n}q_{2})^{}1}\wedge q_{2}\), then we have \(E^{1}[e^{\lambda \sigma_{2}}]<\infty\). So \(E^{i}\{e^{\lambda \sigma_{1}}\}<\infty\). It is easy to see that \(p_{ij}(t)\) is exponentially ergodic.
(ii) By Lemma 1 and the previous results we have that if \(i\geq3\), then \(\tau_{2}^{+}=\sigma_{2} \). Let λ satisfy \(0<\lambda<\frac{1}{\sum_{n=3}^{\infty} (q_{n}q_{1})^{}1}\wedge q_{2}\). Then, for any \(i\in E \), we have
Put \(m=2\) in Theorem 1. By using a computable method of (2.1) we know that
So Lemma 2 is proved. □
For system (2.2), if there exists a vector \(v\succ0\) with \(v\in \Re^{n}\) such that \((AI_{n})v\prec0\), then \(V=v^{T}x\) is said to be a linear copositive Lyapunov function of system (2.2).
Next, we give an example to introduce our results.
Example 1
Consider system (2.1) with two subsystems associated with the Schrödinger operator described by
For the first subsystem matrix \(A_{1}\), there does not exist a vector \(v_{1}\succ0\) such that \((A_{1}I)v_{1}\prec0\). Similarly, there does not exist a vector \(v_{2}\succ0\) such that \((A_{2}I)v_{2}\prec0\).
It is easy to verify that these two subsystem matrices are not Schur matrices. Despite this disadvantage, we find that there exist some combinations \(A_{0}\) of \(A_{1}\) and \(A_{2}\) that are positive and Schur matrices, that is, \(A_{0}=\lambda_{1}A_{1}+\lambda _{2}A_{2}\) is a positive Schur matrix, where \(\lambda_{1},\lambda _{2}\in(0,1)\) and \(\lambda_{1}+\lambda_{2}=1\).
For example, choose \(\lambda_{1}=0.5\) and \(\lambda_{2}=0.5\). We obtain
Note that there exists a vector \(v\succ0\) in \(\Re^{n}\) with \((A_{0}I)v\prec0\). Thus, \(A_{0}\) is a positive and stable matrix.
This example contains an instantaneous state.
Suppose that \(q_{2},q_{3},\ldots \) are sequences of positive real numbers and consider the following Qmatrix:
where \(\sum_{i=2}^{\infty}{q_{i}}^{1}<\infty\). This matrix is called the Kolmogorov matrix. There exist infinitely many dishonest processes with this Qmatrix. In [9, 10], it is shown that the process with following resolvent is the only honest one:
where \(\lambda>0\). Let the state space be \(E=\{1,2,3,\ldots\}\). Obviously, the transition function \(p_{ij}(t)\) that corresponds to the resolvent is the only honest one. Though this chain is weakly symmetric, so far its convergence rate is still unknown because of its instantaneous state.
Next, we discusses the stability for system (2.1). Define the following stable convex combination of the system matrices:
where
Assumption 2
There exists a vector \(\overline{v}\in\Re^{n}\) with \(\overline{v}\succ0\) such that
Without loss of generality, let
where \(\overline{\mathbf{e}}\in\Re^{n}\) with \(\overline{\mathbf{e}}\succ0\), and \(\frac{\underline{\lambda}_{\overline{\mathbf{e}}}}{\overline{\lambda }_{\overline{v}}}\leq1\).
Denote
Combing (2.3), (2.4), (2.5), (2.6), and (2.7), we get
Switching rule 1

(i)
For any initial state \(x(k_{0})=x_{0}\), we select
$$\sigma(k_{0})=\operatorname*{\arg\min}_{i\in S} \bigl\{ x_{0}^{T}\overline{\mathbf{\ell}}_{i} \bigr\} . $$ 
(ii)
The first switching time instant is selected as
$$k_{1}=\inf \bigl\{ k\geq k_{0} x(k)^{T} \overline{\mathbf{\ell}}_{\sigma (k_{0})}>r_{\sigma(k_{0})}x(k)^{T} \overline{\mathbf{e}} \bigr\} . $$Thus, the switching index can be determined by
$$\sigma(k_{1})=\operatorname*{\arg\min}_{i\in S} \bigl\{ x(k_{1})^{T}\overline {\mathbf{\ell}}_{i} \bigr\} . $$ 
(iii)
The switching index sequences are defined by
$$\begin{aligned}& k_{j+1}=\inf \bigl\{ k\geq k_{j} x(k)^{T} \overline{\mathbf{\ell}}_{\sigma (k_{j})}>r_{\sigma(k_{j})}x(k)^{T} \overline{\mathbf{e}} \bigr\} , \\& \sigma(k_{j+1})=\operatorname*{\arg\min}_{i\in S} \bigl\{ x(k_{j+1})^{T}\overline{\mathbf{\ell}}_{i} \bigr\} , \end{aligned}$$where \(r_{\sigma(k_{j})}\in(0, 1)\) and \(j\in\mathbb{N}\).
Main results
Theorem 1
Under Assumption 2 and Switching rule 1, system (2.1) is uniformly exponentially stable.
Proof
Choose \(V(k, x(k))=x(k)^{T}\overline{v}\). For any \(k\in[k_{m}, k_{m+1})\), by system (2.1) it follows that
which, together with (2.1) and (3.1), gives that
By (3.2) and (iii) of Switching rule 1 we know that
Using (3.3) and (iii) of Switching rule 1, we can obtain the following inequality
Since \(\frac{\underline{\lambda}_{\overline{M}}}{\overline{\lambda }_{\overline{v}}}\leq1\) and \(r_{\sigma(k_{m})}\in(0, 1)\), we have that
With (3.5) in mind, using (3.6), we get that
Repeating the above steps in (3.4)(3.7), it straightforwardly follows that
From (3.8) and (3.7) it follows that
Choose \(\eta=\max_{i=0, 1, \ldots, m}\{1\frac{r_{\sigma (k_{i})}\underline{\lambda}_{\overline{\mathbf{e}}}}{\overline{\lambda }_{\overline{v}}} \}\). So
which immediately gives that
So
where \(\alpha=\frac{\overline{\lambda}_{\overline{v}}}{\underline{\lambda }_{\overline{v}}}\).
Thus, system (2.1) is uniformly exponentially stable. □
Remark 1
By the properties of discretetime systems associated with the Schrödinger operator and (ii) and (iii) of Switching rule 1, it is not hard to find that the dwell time in each time interval is at least greater than 1. So there exist finite switchings in any finite time interval. Thus, Switching rule 1 is well defined.
Corollary 1
Suppose \(N=2\). Consider the stability of system (2.1) in the sense of Lyapunov function. Then system (2.1) is stable if and only if Assumption 2 holds.
Proof
Here we only give the proof of “only if”. The stability of system (2.1) implies that there exists an LCLF \(V(k,x(k))=x^{T}(k)\overline{v}\) such that
or
where \(\overline{v}\succ0, \overline{\mathbf{e}'}\succ0\), and ς̅ is a positive real number.
That is to say, \(x^{T}(k)(A^{T}_{1}I_{2})\overline{v}<\overline {\varsigma}x^{T}(k)\overline{\mathbf{e}'}\) whenever \(x^{T}(k)(A^{T}_{2}I_{2})\overline{v}\geq\overline{\varsigma }x^{T}(k)\overline{\mathbf{e}'}\), and \(x^{T}(k)(A^{T}_{2}I_{2})\overline{v}<\overline{\varsigma }x^{T}(k)\overline{\mathbf{e}'}\) whenever \(x^{T}(k)(A^{T}_{1} I_{2})\overline{v}\geq\overline{\varsigma}x^{T}(k)\overline{\mathbf {e}'}\). We only prove the first case since the second case can be derived as the first one. By the compactness theorem there exists a positive real number μ̅ such that \(x^{T}(k)(A^{T}_{1}I_{2})\overline{v}\overline{\varsigma }x^{T}(k)\overline{\mathbf{e}'}>\overline{\mu}\). Between any two consecutive switching instants, \(x(k)\) is bound. Thus, there exists a positive real number κ̅ such that
Choose \(\overline{\varepsilon}=\frac{\overline{\mu}}{\overline{\kappa}}\). We obtain that \(x^{T}(k)(A^{T}_{1}I_{2})\overline{v}\overline{\varsigma }x^{T}(k)\overline{\mathbf{e}'}\overline{\varepsilon }(x^{T}(k)(A^{T}_{2}I_{2})\overline{v}+\overline{\varsigma }x^{T}(k)\overline{\mathbf{e}'})>x^{T}A^{T}_{1}v\overline{\varsigma}x^{T} \mathbf{e}'\overline{\mu}>0\), which yields that
Define \(\overline{w}_{1}=\frac{1}{1+\overline{\varepsilon}}\) and \(\overline{w}_{2}=\frac{\overline{\varepsilon}}{1+\overline{\varepsilon }}\). We have \(x^{T}[A_{0}I_{2}]\overline{v}\) \(<\overline{\varsigma }x^{T}\overline{\mathbf{e}'}\), where \(A_{0}=\overline {w}_{1}A^{T}_{1}+\overline{w}_{2}A^{T}_{2}\). This inequality implies that Assumption 2 holds. □
Conclusions
In this paper, we consider the stabilization problem of discretetime switched positive linear systems associated with the Schrödinger operator that contain unstable subsystems associated with the Schrödinger operator. We present a statedependent switching law guaranteeing that the systems associated with the Schrödinger operator are uniformly exponentially stable. Furthermore, we establish a necessary and sufficient condition for the stabilization of the system with two subsystems associated with the Schrödinger operator.
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Acknowledgements
This work was supported by the Science and Technology Research Project of Henan Province (No. 152102310089)), the Key Scientific Research Projects for Colleges and Universities of Henan Province (No. 17A120006) and the Humanities and Social Sciences Research Project of Henan Provincial Department of Education (No. 2017ZZJH014).
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Jiang, Z. Some Schrödingertype inequalities for stabilization of discrete linear systems associated with the stationary Schrödinger operator. J Inequal Appl 2016, 247 (2016). https://doi.org/10.1186/s1366001611803
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DOI: https://doi.org/10.1186/s1366001611803
Keywords
 Schrödingertype inequalities
 stabilization
 stationary Schrödinger operator