On Asymptotic Estimation of a Discrete Type \(C_0\)Semigroups on Dense Sets: Application to Neutral Type Systems
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Abstract
We consider an abstract Cauchy problem for a certain class of linear differential equations in Hilbert space. We obtain a criterion for stability on some dense subsets of the state space of the \(C_0\)semigroups in terms of location of eigenvalues of their infinitesimal generators (socalled polynomial stability). We apply this result to analysis of stability and stabilizability of special class of neutral type systems with distributed delay.
Keywords
Polynomial stability Asymptotic behavior of solutions Delay systems Neutral type systemsMathematics Subject Classification
34K40 34K20 47D03 47D061 Introduction
An important problem in the theory of differential equations is to determine the asymptotic behavior of solutions. One of the main issues in this topic concern stability. In the case of finite dimensional linear systems (exponential or asymptotic) stability of a system is equivalent to the fact that all eigenvalues are in the open left halfplane. For linear equations in infinitedimensional space the problem of stability is much more complicated. In particular, a system can be asymptotically stable even if it possesses a point of spectrum on the imaginary axis (see Arendt and Batty [1], Lyubich and Phong [8], Sklyar and Shirman [17]). On the other hand, the system may be unstable even in the case when it is spectrum is contained in the open left halfplane, while the eigenvalues approach imaginary axis. Such a situation may occur for hyperbolic equations or delay equations of neutral type (e.g. Rabah et al. [13, 14]).
Definition 1.1
In this paper we conduct the analysis of polynomial stability in case of certain class of not necessarily bounded discrete semigroups acting in Hilbert space extending the results of [2, 5] to this class. Namely we consider the the class of semigroups whose generator has spectrum splitted into a family of finite separated sets and corresponding eigenspaces are finite dimensional and form a Riesz basis. We give an estimation of asymptotic behavior of these semigroups on dense sets (like \(D(A^{\alpha })\)) depending on asymptotic closeness of the eigenvalues to the vertical line \(\text{ Re }\lambda =\omega _0\). In particular, in the case of bounded semigroups of our class we show that the condition (A) is equivalent to (iii). The class of semigroups mentioned above was considered earlier in the paper of Miloslavskii [9], where the estimation of the semigroup norm was obtained (see [9, Theorem 1]).

(B1) \(\sigma (\mathcal {A})= \bigcup _{k \in \mathbb {Z}} \sigma _k\), and \(\inf \{\lambda \mu :\lambda \in \sigma _i,\mu \in \sigma _j, i\ne j \}=d>0;\)

(B2) \(\mathrm{dim}\,\mathcal {P}_k\mathcal {H}\le N, k\in \mathbb {Z}\), where \(\mathcal {P}_k\) is a spectral projection corresponding to \(\sigma _k\);

(B3) subspaces \(V_k:=\mathcal {P}_k\mathcal {H}, k\in \mathbb {Z}\), constitute Riesz basis of subspaces.
(B3’) the span over the (generalized) eigenvectors of \(\mathcal {A}\) is dense in \(\mathcal {H}\).
The main goal of our paper is to extend polynomial stability analysis to the mentioned above class of \(C_0\)semigroups. We obtain a spectral criterion for polynomial stability of not necessarily bounded semigroups generated by the operators satisfying (B1)–(B3). In particular, we describe the asymptotic behavior of the semigroups restricted to some dense, nonclosed subsets in terms of location of the spectrum. Thus we obtain
Theorem 1.1
Basing on this Theorem and results from [2] we obtain the following:
Theorem 1.2
Let \(\mathcal {A}:D(\mathcal {A})\subset \mathcal {H} \rightarrow \mathcal {H}\) generate \(C_0\)semigroup T(t) on \(\mathcal {H}\) and satisfies assumptions (B1)–(B3) and let \(\rho (\mathcal {A})\subset \mathbb {C}^{}\). Then semigroup T(t) is polynomially stable if and only if condition (A) holds for some positive constants \(\gamma ,\alpha \) and small values of \(\mathrm{Re}\,\lambda \).
We also use Theorem 1.1 to describe the asymptotic behavior of solutions of neutral type equations (Theorem 3.1). This theorem complements our previous results [16] concerning the behavior of the norm of semigroups corresponding to equations (3).
The work is organized as follows. First we give the proof of Theorem 1.1 preceded by several technical results. Next section is devoted to the analysis of stability of neutral type equations (3) and regular feedback stabilizability of these equations [14]. In the appendix we give two simple statements about complex matrices, which are used in our work.
2 Proof of the Main Results
In the beginning we give some technical results which will be used in the proof of Theorem 1.1.
Lemma 2.1
Proof
Lemma 2.2
Let \(\mathcal {A}:D(\mathcal {A})\subset \mathcal {H} \rightarrow \mathcal {H}\), be a generator of \(C_0\)semigroup \(e^{\mathcal {A}t}\) satisfying assumptions (B1)–(B3). In addition to this, we assume
Proof
Theorem 2.1
Proof
Without loss of generality we assume that each \(\mathcal {A}_k\) has \(n_k\le N\) different eigenvalues and no rootvectors. Indeed, if this assumption is not satisfied then for any \(\varepsilon _1>0\) we can find the operator \(\mathcal {A}_{\varepsilon }\) close to \(\mathcal {A}\) e.i. satisfying condition \(\Vert \mathcal {A}_{\varepsilon }\mathcal {A}\Vert \le \varepsilon _1\) and \(V_k'=V_k, k\in \mathbb {Z}\), and \(\mathcal {A}_{\varepsilon }\) has only simple, different eigenvalues. If assertion is true for any operator \(\mathcal {A}_{\varepsilon },\,\varepsilon >0\) then it is also true for \(\mathcal {A}\).
Lemma 2.3
Proof
Now we are ready to prove the main theorem.
Proof
Proof
Theorem 1.1 shows that condition (A) is sufficient for polynomial stability. Necessity. Let T(t) be polynomially stable. We choose one eigenvalue from each family \(\sigma _k\) (say \(\lambda _k\)) and corresponding eigenvectors \(\phi _k\). Consider subspace \(S=\overline{\text{ span }\,\{\phi _k:k\in \mathbb {Z}\}}\). It is easy to see that subspace S is Tinvariant and the semigroup T(t) is bounded on S. Applying the results of Sect. 3 from [2] for semigroup T(t) restricted to S we see that family \(\{\lambda _k\}_{k\in \mathbb {Z}}\) satisfies condition (A) with some positive constants \(\gamma , \alpha \).
Since eigenvalues \(\lambda _k\) was chosen arbitrarily, then whole spectrum satisfies condition (A) with some positive constants \(\gamma , \alpha \). \(\square \)
3 Stability and Stabilizability of Neutral Type Equations
Following [13], we consider the delay systems of neutral type of the form (3), which can be represented in the operator form (4), with generator \(\mathcal {A}\) given by (5)–(6). Our goal is to investigate the asymptotic behavior of the solutions of above equation, in particular its stability. The stability is closely related to the location of the spectrum of the operator \(\mathcal {A}\) thus we recall some important properties of \(\mathcal {A}\) (for more details see [13]).
The spectrum of \(\mathcal {A}\) consist of eigenvalues only. Almost all of them lie close to \(\tilde{\lambda }_m^{(k)}\). More precisely, for k large enough they are contained in the discs \(L_m^{(k)}\) centered at \(\tilde{\lambda }_m^{(k)}\) of radii \(r_k \rightarrow 0\) (see [14, Theorem 4]). The sum of multiplicities of eigenvalues of \( \mathcal {A}\) lying in each disc centered at \(\tilde{\lambda }_m^{(k)}\) equals the multiplicity of \(\tilde{\lambda }_m^{(k)}\) and \(\mu _m\), that is \(p_m\). We denote eigenvalues of the operator \(\mathcal {A}\) by \(\lambda _{m,i}^{(k)}, k\in \mathbb {Z}; m=1,\ldots ,\ell \), and we have \(\{\lambda _{m,i}^{(k)}\}_{i=1}^{p_m}\subset L_m^{(k)}, k>N; m=1,\ldots ,\ell \). If there exist eigenvalues of \(\mathcal {A}\) with real part \(\omega =\sup _{\lambda \in \sigma (\mathcal {A})} \mathrm{Re}\,\lambda \) we denote their maximal multiplicity by \(p_0\) and we set \(p_0=0\) if there are no such eigenvalues. Let us denote \(\mathcal {A}\)invariant subspaces \(V_m^{(k)}=P_m^{(k)}\mathcal {M}_2\), where \(P_m^{(k)}x=\frac{1}{2\pi i} \int _{L_m^{(k)}} R(\mathcal {A},\lambda )xd\lambda \) are Riesz projections, \(m=1,\ldots ,\ell , k \in \mathbb {Z}\). The sequence of \(p_m\)dimensional subspaces \(V_m^{(k)}, m=1,\ldots ,\ell , k \ge N \), and some \(2(N+1)n\)dimensional subspace \(W_N\) constitute an \(\mathcal {A}\)invariant Riesz basis of space \(\mathcal {M}_2\).
 (a)
\(\omega >\tilde{\omega }\), which implies that \(p_0>0\);
 (b)
\(\omega =\tilde{\omega }\) and \(p_0=0\).
 (c)
\(\omega =\tilde{\omega }\) and \(0<p_0<q\), where \(q\ge 1\) is maximal size of Jordan block of matrix \(A_{1}\) corresponding to the eigenvalue \(\mu _1\);
 (d)
\(\omega =\tilde{\omega }\) and \(q \le p_0<p_1\), where \(p_1\) is the multiplicity of \(\mu _1\);
 (e)
\(\omega =\tilde{\omega }\) and \(p_0\ge p_1\).
Theorem 3.1
Proof of Theorem 1.2
The operator \(\mathcal {A}\) satisfies the assumptions (B1)–(B3) and (A), so the proof of theorem follows directly from Theorem 1.1. \(\square \)
Corollary 3.1
Statement 3.2
 (a)
spectrum of \(\mathcal {A}\) consist of simple eigenvalues only, say \(\sigma (\mathcal {A})=\{\lambda _k:k\in \mathbb {Z}\}\),
 (b)
linear span of eigenvectors of \(\mathcal {A}\) is dens in \(\mathcal {H}\),
 (c)
eigenvalues \(\lambda _k\) lie in disjoint balls \(B(x_k,r_k)\) centered at the following points of imaginary axis \(x_k=i(kd+d_0), 0\le d_0<d\) and radii \(r_k\) satisfies \(\sum r_k^2<\infty \),
 (d)
vector \(b\in \mathcal {H}\) is not orthogonal to eigenvectors \(\phi _k\) of the operator \(\mathcal {A}^*\).
 (i)
stable, that is \(\Vert e^{(\mathcal {A}+b\mathcal {F})t}\Vert <M\) for some constant \(M>0\),
 (ii)
polynomially stable, that is \(\Vert e^{(\mathcal {A}+b\mathcal {F})t}\mathcal {A}^{1}\Vert \le M_{\alpha }t^{\frac{1}{\alpha }}\) for some constant \(M_{\alpha }>0\).
Proof of Theorem 1.2
By [14, Theorem 8,Lemma 13] there exist feedback \(\mathcal {F}\), which shifts the eigenvalues \(\lambda _k\) to the points \(\hat{\lambda }=r'_k+(kd+d_0)i\), where \(r'_k=\max \{r_k, Ck^{\alpha }\}\). Then all \(\hat{\lambda }_k\) are in the open left halfplane and assertion (i) follows from [9, Theorem 1]. To prove (ii) we check that eigenvalues \(\hat{\lambda }_k\) satisfies condition (A’) with constants \(\omega _0=0, \alpha ,\gamma =Cd^{\alpha }\) and use Theorem 1.1. \(\square \)
For the case of nonsingle eigenvalues \(\mu _m, m=1,\ldots , \ell _0\) of matrix \(A_{1}\), even if eigenvalues of \(\mathcal {A}\) can be moved to the open left halfplane, then, in general, stability can not be obtained because the corresponding group can be unbounded (see [16]). Although if we assume that we are able to move eigenvalues in each circle using proper feedback (27) the same way like in the case of single eigenvalues, then using Theorem 1.1 we can obtain polynomial stability of a corresponding group. To illustrate this idea we focus on some special class of equations (26).
Statement 3.3
Notes
Acknowledgments
The authors would like to thank the anonymous referee for the comments, which helped to improve the presentation of the paper and also would like to thank PROMEP (Mexico) via “Poyecto de Redes” for partial financial support of this research.
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