Stability and Asymptotic Behavior of Systems with Multi-time

Abstract

Stability and asymptotic behavior of linear multi-time systems $\frac {\partial }{\partial t_{i}} u(\textbf {t})= A_{i} u (\textbf {t})$ +f _i (t) is considered, where A _i are generators of commuting C ₀-semigroups on a Banach space. We formulate and prove versions of the results on exponential stability to homogeneous linear systems with multi-time. For non-homogeneous systems, we introduce the compatibility condition and obtain the variation-of-constants formula. Using simulta neous solutions of Sylvester equations, we obtain results on asymptotic almost periodicity of solutions.

Appendix:

In this Appendix, we present a proof of Proposition 1.

Proposition 1

Let T _i (t) (1≤i≤k) be commuting C ₀ -semigroups, with generators A _i . Then the operators A _i are commuting, i.e., $A_{i}A_{j}x = A_{j}A_{i}x (\forall x \in D(A_{i}A_{j}) \cap D(A_{j}A_{i}))$ , the operator A=A ₁ +⋯+A _k is closable, densely defined, and its closure $\overline {A}$ is the generator of the semigroup T(t)=T ₁ (t)⋯T _k (t). Moreover, the linear manifold $D_{2}(\mathcal {A}) :=~\cap _{i,j=1}^{k} D(A_{i}A_{j})$ also is dense.

First, recall the following well-known facts. If T(t) (t≥0) is a one-parameter C ₀ semigroup in a Banach space $\mathcal E$, then its generator A is a closed, densely defined linear operator. Moreover, the following hold:

• (i) If x∈D(A), then T(t)x∈D(A) and $\frac {d}{dt}T(t)x = AT(t)x = T(t)Ax$;

• (ii) For all $x \in \mathcal E$ we have ${{\int }_{0}^{t}} T(s)xds \in D(A)$ and $A{{\int }_{0}^{t}} T(s)xds = T(t)x - x$; if, in addition, x∈D(A), then ${{\int }_{0}^{t}} T(s)Axds = T(t)x - x$;

• (iii) $\lim _{t~\to ~0}\frac {1}{t}{{\int }_{0}^{t}} T(s)xds~=~x \ (\forall x~\in ~E)$.

Lemma 4

If x ∈ D(A _i ), then $\mathcal T(\mathbf {t})x~\in ~D(A_{i})\ (\forall \mathbf {t}~\in ~\mathbb R_{+}^{k})$.

Proof

Let x ∈ D(A _i ). We have

$$\lim_{h\to0} \frac{1}{h}\left[T_{i}(h)T_{j}(t)x-T_{j}(t)x\right]=T_{j}(t)\lim_{h\to0} \frac{1}{h}\left[T_{i}(h)x-x\right]= T_{j}(t)A_{i}x, $$

which implies that T _j (t)x ∈ D(A _i ) and A _i T _j (t)x = T _j (t)A _i x. Hence, $\mathcal T(\textbf {t})x~\in ~D(A_{i})$ and $A_{i}\mathcal {T}(\textbf {t})x~=~\mathcal {T}(\textbf {t})A_{i}x \ (\forall \textbf {t}~\in ~\mathbb {R}_{+}^{k})$. □

Proof of Proposition 1

By Fubini’s Theorem, we have

$${\int}_{[\textbf{0},\textbf{T}]} \mathcal T(\textbf{t})xd\textbf{t}~=~{\int}_{0}^{t_{k}}T_{k}(s_{k}){\int}_{0}^{t_{k-1}} T_{k-1}(s_{k-1}) {\cdots} {\int}_{0}^{t_{1}} T_{1}(s_{1})x ds_{1} {\cdots} ds_{k} \quad (\forall x\in \mathcal{E}). $$

Since $x_{1}~:=~{\int }_{0}^{t_{1}} T_{1}(s_{1})x ds_{1}~\in ~D(A_{1})$, we have, by (ii) and Lemma 4

$$x_{2}:= {\int}_{0}^{t_{2}} T_{2}(s_{2}){\int}_{0}^{t_{1}} T_{1}(s_{1})x ds_{1} ds_{2}={\int}_{0}^{t_{2}} T_{2}(s_{2})x_{1} ds_{2}\in D(A_{1})\cap D(A_{2}). $$

Continuing this process, we eventually obtain that

$${\int}_{[\textbf{0},\textbf{T}]} \mathcal T(\textbf{t})xd\textbf{t} \in \cap_{i=1}^{k} D(A_{i}). $$

Since, for every x∈E, we have

$$ \lim_{h\to0} \frac{1}{h^{k}} {\int}_{[0,h]^{k}} \mathcal T(\textbf{t})xd\textbf{t} =x, $$

(36)

where by [0,h]^k we denote the generalized cube $\underbrace {[0,h]\times {\cdots } \times [0,h]}_{k \text { times}}$, we conclude that $D_{1}(\mathcal A)~=~\cap _{i=1}^{k} D(A_{i})$ is dense in $\mathcal E$. Hence, the operator A = A ₁+⋯+A _k is densely defined. Consider now the product semigroup $\mathcal T(t\textbf {1})~=~T_{1}(t){\cdots } T_{k}(t)$. Since $\mathcal T(t\textbf {1})$ is a strongly continuous one-parameter semigroup, its generator, denoted (temporarily) by $\widetilde {A}$, is closed and densely defined. Moreover, for every $x~\in ~D_{1}(\mathcal A)$, $\mathcal T(t\textbf {1})x$ is differentiable and

$$\begin{array}{@{}rcl@{}} \frac{d}{dt} \mathcal T(t\textbf{1})x&=& \frac{d}{dt} T_{1}(t){\cdots} T_{k}(t)x= T_{1}(t){\cdots} T_{k}(t)(A_{1}+\cdots+A_{k})x\\ &=&T_{1}(t){\cdots} T_{k}(t)Ax= T_{1}(t){\cdots} T_{k}(t) \widetilde{A}x. \end{array} $$

This implies that $Ax~=~\widetilde {A}x (\forall x~\in ~D_{1}(\mathcal A))$, hence the operator A is closable and $\overline {A}~=~\widetilde {A}$.

Next we show that $\cap _{i,j=1}^{k}D(A_{i}A_{j})$ is dense. Let $x~\in ~D_{1}(\mathcal A)$. We have, as shown above

$$\begin{array}{@{}rcl@{}} y:={\int}_{[\textbf{0},\textbf{T}]} \mathcal T(\textbf{t})xd\textbf{t} \in D_{1}(\mathcal A), \end{array} $$

and

$$\begin{array}{@{}rcl@{}} A_{i}y=A_{i}{\int}_{[\textbf{0},\textbf{T}]} \mathcal T(\textbf{t})xd\textbf{t}= {\int}_{[\textbf{0},\textbf{T}]} \mathcal T(\textbf{t})A_{i}xd\textbf{t} \in D_{1}(\mathcal A). \end{array} $$

This implies that $y~\in ~\cap _{i,j=1}^{k} D(A_{j}A_{i})$. Now the identity (36) implies that $\cap _{i,j=1}^{k} D(A_{j}A_{i})$ is dense $D_{1}(\mathcal A)$, hence also dense in $\mathcal E$. □