Well-posedness of infinite-dimensional non-autonomous passive boundary control systems

TL;DR

This paper investigates the well-posedness of non-autonomous boundary control systems governed by PDEs, providing conditions for existence and uniqueness of solutions in infinite-dimensional settings.

Contribution

It offers new sufficient conditions for well-posedness of non-autonomous boundary control systems with multiplicative perturbations, applicable to PDE models like wave equations and beams.

Findings

Established criteria for well-posedness of non-autonomous boundary systems

Proved existence and uniqueness of classical and mild solutions

Applicable to PDEs such as wave equations and Timoshenko beams

Abstract

We study a class of non-autonomous boundary control and observation linear systems that are governed by non-autonomous multiplicative perturbations. This class is motivated by different fundamental partial differential equations, such as controlled wave equations and Timoshenko beams. Our main results give sufficient condition for well-posedness, existence and uniqueness of classical and mild solutions.

Equations

$$\frac{\partial }{\partial t}x(t,\zeta )$$

$$x(0,\zeta )$$

$$u(t)$$

$$0$$

$$y(t)$$

$$\tau(x):=\big{(}x(b),x^{\prime}(b),\cdots,x^{N-1}(b),x(a),x^{\prime}(a)\cdots,x^{N-1}(a)\big{)},$$

$$\dot{x}(t)$$

$$\mathfrak{B}M(t)x(t)$$

$$\mathfrak{C}M(t)(t)x(t)$$

$$\mathfrak{A}(t)x=\sum _{k=0}^N{P}_k(t)\frac{{\partial }^k}{\partial {\zeta }^k}x,\mathfrak{B}x:=\left[\begin{array}{c}{W}_{B,1}\\ W_{B,2}\end{array}\right]\tau (x),\mathfrak{C}x:=W_C\tau (x),\text{ and }\mathcal{H}(t,\cdot ):=M(t)$$

$$\|x(\tau)\|^{2}+\int_{0}^{\tau}\|y(s)\|^{2}ds\leq m_{\tau}\big{(}\|x_{0}\|^{2}+\int_{0}^{\tau}\|u(s)\|^{2}ds\big{)}$$

$$2\mathrm{Re}(\mathfrak{A}(t)x|P(t)x{)}_{{}_X}\le (R(t)u|\mathfrak{B}x{)}_{{}_U}-(\mathfrak{C}x|J(t)\mathfrak{C}(x{)}_{{}_Y}$$

$$\dot{x}(t)$$

$$y(t)$$

$$\dot{u}(t)=A(t)u(t)\mathrm{on}[s,\infty ),u(s)=x_s,(s>0).$$

$$W(t,s)=Q^{-1}(t)U(t,s)Q(s)\text{ for }(t,s)\in \Delta .$$

$$D(Q^{-1}AQ)=D(AQ)=\{x\in X|Qx\in D(A)\}=Q^{-1}D(A).$$

$$\Vert Q^{-1}(t_n)U(t_n,s_n)x-Q^{-1}(t)U(t,s)x\Vert$$

$$+\Vert [Q^{-1}(t_n)-Q^{-1}(t)]U(t,s)x\Vert ,$$

$$\widetilde{Y_t}:=\{x\in X|Q(t)x\in Y_t\}.$$

$$\widetilde{Y_t}=Q^{-1}(t)Y_t\subset Q^{-1}(t)D(A(t))=D(A(t)Q(t))=D(A_Q(t))$$

$$\dot{u}(t)=A(t)u(t)\mathrm{on}[s,\infty ),u(s)=Q(s)x_s,(s>0).$$

$$\frac{d}{dt}W(t,s)x_s$$

$$=-Q(t{)}^{-1}\dot{Q}(t)Q(t{)}^{-1}U(t,s)Q(s)x_s+Q(t{)}^{-1}A(t)U(t,s)Q(s)x_s$$

$$\displaystyle=\big{[}Q^{-1}(t)A(t)Q(t)-Q^{-1}(t)\dot{Q}(t)\big{]}W(t,s)x_{s}.$$

$$\dot{u}(t)=A_Q(t)u(t)\mathrm{on}[s,\infty ),u(s)=x_s.$$

$$V(t,s):=Q(t)W(t,s)Q^{-1}(s)=U(t,s),(t,s)\in \Delta$$

$$|\Vert x{\Vert }_t-\Vert x{\Vert }_s|\le c_T|t-s|\Vert x{\Vert }_s,x\in X,t,s\in [0,T]$$

$$x\mapsto \Vert x{\Vert }_t:=\sqrt{(M(t)x|x)}=\Vert M^{1/2}(t)x\Vert$$

$$\dot{x}(t)=Q^{-1}(t)A(t)Q(t)x(t)+F(t)a.e.\mathrm{on}[s,\infty ),x(s)=x_s,s>0,$$

$$x(t)=W(t,s)x_s+{\int }_s^{t}W(t,r)F(r)\mathrm{d}rt\ge s.$$

$$\dot{u}(t)$$

$$u(s)$$

$$u(t)=U(t,s)Q^{-1}(s)x_s+{\int }_s^{t}U(t,r)Q(r)F(r)\mathrm{d}rt\ge s.$$

$$\dot{x}(t)=A(t)M(t)x(t)+P(t)x(t)+F(t)a.e.\mathrm{on}[s,\infty ),x(s)=x_s,s>0.$$

$$R(t)M(t)=M(t)R(t)\text{ for all}t\ge 0.$$

Full text

$1$ University of Wuppertal, School of Mathematics and Natural Science, Gaußstrasse 20, D-42119 Wuppertal, Germany, bjacob@uni-wuppertal.de.

$2$ University of Wuppertal, School of Mathematics and Natural Science, Gaußstrasse 20, D-42119 Wuppertal, Germany, laasri@uni-wuppertal.de. Support by Deutsche Forschungsgemeinschaft (Grant LA 4197/1-1) is gratefully acknowledged.

**Key words: ** infinite-dimensional non-autonomous control system, evolution family, port-Hamiltonian system, well-posedness

MSC: 93C25, 47D06, 93C20.

1. Introduction

We consider the following non-autonomous partial differential equation with boundary input $u$ and boundary output $y$

[TABLE]

Here $\tau$ denotes the trace operator $\tau:H^{N}((a,b);\mathbb{K}^{n})\rightarrow\mathbb{K}^{2Nn}$ defined by

[TABLE]

$P_{k}(t)$ is $n\times n$ matrix for all $t\geq 0$, $k=0,1,\cdots,N$, $\mathcal{H}(t,\zeta)\in\mathbb{K}^{n\times n}$ for all $t\geq 0$ and almost every $\zeta\in[a,b]$, $W_{B,1}$ is a $m\times 2nN$-matrix, $W_{B,2}$ is $(nN-m)\times 2nN$-matrix and $W_{C}$ is a $d\times 2nN$-matrix. Finally, $u(t)\in\mathbb{K}^{m}$ denotes the input and $y(t)\in\mathbb{K}^{d}$ is the output at time $t$.

This partial differential equation is also known as port-Hamiltonian systems, and covers the wave equation, the transport equation, beam equations, coupled beam and wave equations as well as certain networks. Autonomous port-Hamiltonian systems, that is when $\mathcal{H},P_{k}$ are time-independent, have been intensively investigated, see e.g., [15, 16, 3, 2, 17, 22, 36, 41]. The existence of mild/classical solutions with non-increasing energy and well-posedness for autonomous port-Hamiltonian systems can in most cases be tested via a simple matrix condition [22, Theorem 4.1]. Well-posedness of linear systems in general is not easy to prove and a necessary condition is that the state operator generates a strongly continuous semigroup. For the class of autonomous port-Hamiltonian systems of first order i.e., $N=1$, this condition is even sufficient under some weak assumptions on $P_{1}\mathcal{H},$ see [22] or [17, Theorem 13.2.2].

In this paper, we aim to generalize these solvability and well-posedness results to the non-autonomous situation. To our knowledge, in contrast to infinite-dimensional autonomous port-Hamiltonian systems, the non-autonomous counterpart has not been discussed so far. Motivated by this class we start a systematic study of non-autonomous linear boundary control and observation systems, and in particular those of the following form

[TABLE]

which we denote by $\Sigma_{N,M}(\mathfrak{A},\mathfrak{B},\mathfrak{C}).$ Here $\mathfrak{A}(t):\mathfrak{D}\subset X\rightarrow X$ is a linear operator, $u(t)\in U$, $y(t)\in Y$, the boundary operators $\mathfrak{B}:D(\mathfrak{B})\subset X\rightarrow U$ and $\mathfrak{C}:\mathfrak{D}\subset X\rightarrow Y$ are linear such that $\mathfrak{D}\subset D(\mathfrak{B})$, $X$, $U$ and $Y$ are complex Hilbert spaces and $M(t)\in\mathcal{L}(X)$ for all $t\geq 0.$ Setting

[TABLE]

we see that the non-autonomous port-Hamiltonian system is in fact a special class of non-autonomous systems of the form (1)-(3).

A pair $(x,y)$ is a classical solution of (1)-(3) if $x\in C^{1}((0,\infty);X)\cap C([0,\infty);X),y\in C([0,\infty);Y)$ and $x(t)\in D(\mathfrak{A}(t)M(t))$ for all $t\geq 0$ such that $x,y$ satisfy (1)-(2) and (3), respectively. The system $\Sigma_{N,M}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is called well-posed if for each (classical) solution $(x,y)$ and any final time $\tau>0,$ the operator mapping the input functions $u$ and to the initial state $x_{0}$ to $x(\tau)$ and the output functions $y$ is bounded, i.e.

[TABLE]

for some constant $m_{\tau}>0$ independent of $x_{0}$ and $u.$

Our approach for the solvability of $\Sigma_{N,M}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is based on a non-autonomous version of the Fattorini’s trick, the theory of evolution families together with an idea of Schnaubelt and Weiss [29, Section 2].

Evolution families are a generalization of strongly continuous semigroups, and are often used to describe the solution of an abstract non-autonomous Cauchy problem. In Section 2, we therefore review the concept of evolution families and that of $C^{1}$-well posed non-autonomous Cauchy problems. Furthermore, we provide several abstract results which are crucial for the analysis of our non-autonomous boundary control and observation systems.

Fattorini’s trick is well known for autonomous boundary control systems [17, 12, 9]. The basic idea of this approach is to reformulate the state and the control equation into an abstract inhomogeneous Cauchy problem on $X.$ A brief description of the autonomous situation is given in Subsection 3.1. In Subsection 4.1 we provide a generalization to non-autonomous boundary control systems (see Proposition 4.2). This generalization and the results of Section 2 are then used to prove our main classical solvability results: Theorem 4.8 and Theorem 6.5.

The second main purpose of this paper is the study of the well-posedness for non-autonomous boundary and observation systems $\Sigma_{N,M}(\mathfrak{A},\mathfrak{B},\mathfrak{C}).$ However, we will restrict ourselves to the case where for every $t\geq 0$ the (unperturbed) autonomous system $\Sigma_{N,\operatorname{id}}(\mathfrak{A}(t),\mathfrak{B},\mathfrak{C})$ is $(R(t),P(t),J(t))$-scattering passive i.e., when

[TABLE]

for all $x$ in an appropriate subspace of $X\times U$ where $P(t),R(t)$ and $J(t)$ are bounded linear operators. A precise definition and a characterization of scattering passive autonomous and non-autonomous systems is the subject of Subsection 3.2 and Subsection 4.2, respectively. Under additional conditions we then prove in Theorem 4.8 that the perturbed system $\Sigma_{N,M}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is well-posed. In particular, we deduce in Theorem 6.5 that well-posedness for a large class of non-autonomous port-Hamiltonian systems can be checked via a simples matrix condition.

In the literature most attention has been devoted to autonomous control systems. However, in view of applications, the interest in non-autonomous systems has been rapidly growing in recent years, see e.g., [13, 26, 7, 29, 19, 6, 18, 28] and the references therein. In particular, a class of scattering passive linear non-autonomous linear systems of the form

[TABLE]

has been studied by R. Schnaubelt and G. Weiss in [29]. Here $(A,D(A))$ generates a strongly continuous semigroup on $X$, $A_{-1}\in\mathcal{L}(X,X_{-1})$ is a bounded extension of $(A,D(A))$, $B\in\mathcal{L}(U,X_{-1}),$ $C\in\mathcal{L}(Z,Y)$ and $D\in\mathcal{L}(U,Y),$ where $X_{-1}$ is the extrapolation space corresponding to $A,$ and $Z:=D(A)+(\alpha-A)^{-1}BU$ for some $\alpha\in\rho(A).$

The control part (1)-(2) of the nonautonomous boundary control system $\Sigma_{N,M}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ can be rewritten in the (standard) abstract formulation (4), however, in the particular case where $\mathfrak{A}(t)=\mathfrak{A}$ is constant which for non-autonomous port-Hamiltonian system correspond to the case where the matrices $P_{k},k=1,\cdots,N,$ are constant with respect to time variable. On the other hand, when $\mathfrak{A}(t)=\mathfrak{A}$ the output part (3) could be also written into (5) using the concept of system nodes. Indeed, well-posed autonomous port-Hamiltonian system fit into the framework of compatible system nodes [37, Theorem 10]. This can be also easily generalized for boundary control and observation systems defined in Definition 3.2. Since we do not follow the approach of [29], this topic will not be discussed in this paper and we refer to [31, 34] for more details on system nodes.

For the general case, that is when $\mathfrak{A}$ is not constant, then $A_{-1},B,C,D$ and $Z$ will be time dependent. Thus, the abstract results in [29] cannot be immediately applied to deduce classical solvability and well-posedness for (1)-(3). We expect that the results in [29] can be generalized to include this general case. However, for the class of boundary control systems defined in Definition 3.2 we deal directly with (1)-(2) in combination with Fattorini’s trick instead of its corresponding system (4)-(5). Our method is indeed much simpler. Moreover, in general it is not clear how the solution of (4)-(5) can be related to that of (1)-(3) even for the special case where $\mathfrak{A}(t)=\mathfrak{A}$ is constant. In the autonomous case this relationship is quite simple as we can see in Section 3. The reason is that $C_{0}$-semigroups can be always extended to the extrapolation space. The situation is more delicate for the non-autonomous setting. Indeed, a general extrapolation theory for evolution families is still missing. Moreover, the extrapolation space may also depend on the time variable. In Section 5 we deal with this question by associating a mild solution to the control part (1)-(2) of the nonautonomous boundary control system $\Sigma_{N,M}(\mathfrak{A},\mathfrak{B},\mathfrak{C}).$

Finally, we apply our abstract results to non-autonomous port-Hamiltonian systems, in particular to the time-dependent vibrating string and the time-dependent Timoschenko beam.

2. Background on evolution families and preliminary results

Throughout this section $(X,\|\cdot\|)$ is a Banach space. Let $\mathcal{A}:=\{A(t)\,|\,t\geq 0\}$ be a family of linear, closed operators with domains $\{D(A(t))\,|\,t\geq 0\}$. Consider the non-autonomous Cauchy problem

[TABLE]

A continuous function $u:[s,\infty)\rightarrow X$ is called a classical solution of (6) if $u(t)\in D(A(t))$ for all $t\geq s,u\in C^{1}((s,\infty),X)$ and $u$ satisfies (6).

Definition 2.1**.**

The non-autonomous Cauchy problem (6) is called $C^{1}$-well posed if there is a family $\{Y_{t}\,|\,t\geq 0\}$ of dense subspaces of $X$ such that:

• $(a)$

$Y_{t}\subseteq D(A(t))$ for all $t\geq 0$.

• $(b)$

For each $s\geq 0$ and $x_{s}\in Y_{s}$ the Cauchy problem (6) has a unique classical solution $u(\cdot,s,x_{s})$ with $u(t,s,x_{s})\in Y_{t}$ for all $t\geq s$.

• $(c)$

The solutions depend continuously on the initial data $s,x_{s}$.

If we want to specify the regularity subspaces $Y_{t}$, $t\geq 0$, we also say (6) is $C^{1}$-well posed on $Y_{t}$.

In the autonomous case, i.e., if $A(t)=A$ is constant, then it is well known that the associated Cauchy problem is well-posed if and only if $A$ generates a $C_{0}$-semigroup $(T(t))_{t\geq 0}$. In this case, for each $x\in D(A)$ the unique classical solution is given by $T(\cdot)x$. The following definition provides a natural generalization of operator semigroups for non-autonomous evolution equations.

Definition 2.2**.**

A family $\mathcal{U}:=\{U(t,s)\,|\,(t,s)\in\Delta\}\subset\mathcal{L}(X)$ where $\Delta:=\{t,s\geq 0\,|\,t\geq s\}$ is called an evolution family if:

• $(i)$

$U(t,t)=I$ and $U(t,s)=U(t,r)U(r,s)$ for every $0\leq s\leq r\leq t$,

• $(ii)$

$U(\cdot,\cdot):\Delta\rightarrow\mathcal{L}(X)$ is strongly continuous.

The evolution family $\mathcal{U}$ is said to be generated by $\mathcal{A}$, if there is a family $\{Y_{t}\,|\,t\geq 0\}$ of dense subspaces of $X$ with $Y_{t}\subset D(A(t))$ and

• $(iii)$

For every $x_{s}\in Y_{s}$, the function $t\rightarrow U(t,s)x_{s}$ is the unique classical solution of (6).

The Cauchy problem (6) is then $C^{1}$-well posed if and only if $A(t)$, $t\geq 0$, generates a unique evolution family, see [10, Proposition 9.3] or [24, Proposition 3.10]. Clearly, if $(T(t))_{t\geq 0}$ is a $C_{0}$-semigroup in $X$ with generator $(A,D(A))$, then $U(t,s):=T(t-s)$ yields an evolution family on $X$ with regularity spaces $Y_{t}=D(A)$.

2.1. Similar evolution families

Let $\mathcal{U}:=\{U(t,s)\,|\,(t,s)\in\Delta\}$ be an evolution family on $X$ and let $\{Q(t)\,|\,t\geq 0\}\subset\mathcal{L}(X)$ be a family of isomorphisms on $X$ such that $Q$ and $Q^{-1}$ are strongly continuous on $[0,\infty)$. Define the two parameters operator family $\mathcal{W}:=\{W(t,s)\,|\,(t,s)\in\Delta\}$ by

[TABLE]

It is well known that if $S$ is a $C_{0}$-semigroup on $X$ with generator $A$ and $Q\in\mathcal{L}(X)$ is an isomorphism, then $T(\cdot):=Q^{-1}S(\cdot)Q$ is again a $C_{0}$-semigroup on $X$, called similar $C_{0}$-semigroup to $S$, and its generator is given by $Q^{-1}AQ$, where

[TABLE]

The purpose of this section is to generalize the concept of similar semigroups to evolution families.

Lemma 2.3**.**

The two parameters family $\mathcal{W}$, defined by (7), defines an evolution family on $X$.

Proof.

Clearly, the evolution law $(i)$ in Definition 2.2 is fulfilled. It remains to prove the strong continuity of $\mathcal{W}$ in $\Delta$. Let $x\in X$ and $T>0$, and set $\Delta_{T}:=\{(t,s)\in[0,T]^{2}\,|\,t\geq s\}$. Let $(t,s)$, $(t_{n},s_{n})\in\Delta_{T}$ for $n\in\mathbb{N}$ such that $(t_{n},s_{n})\rightarrow(t,s)$. Then $\{Q^{-1}(t_{n})\,|\,n\in\mathbb{N}\}$ is bounded by the uniform boundedness theorem. Since

[TABLE]

we deduce that $(t,s)\mapsto Q^{-1}(t)U(t,s)x$ is continuous on $\Delta_{T}$. Thus, using a similar argument for $Q(s)$ and $Q^{-1}(t)U(t,s)$ we obtain that $(t,s)\mapsto W(t,s)x$ is continuous on $\Delta_{T}$. Since $T>0$ is arbitrary, this proves the assertion. ∎

In contrast to semigroups, the evolution law $(i)$ and the strong continuity $(ii)$ do not guarantee that the given evolution family is generated by some family of linear closed operators.

Proposition 2.4**.**

Assume that $Q(\cdot)$ is in addition strongly $C^{1}$-differentiable. Then $\mathcal{U}$ is generated by a family $\mathcal{A}$ with regularity spaces $\{Y_{t}\,|\,t\geq 0\}$ if and only if $\mathcal{W}$ is generated by $\mathcal{A}_{Q}:=\{Q^{-1}(t)A(t)Q(t)-Q^{-1}(t)\dot{Q}(t)\,|\,\ t\geq 0\}$ with regularity spaces $\{\tilde{Y_{t}}\,|\,\ t\geq 0\}$ where

[TABLE]

Proof.

$(i)$ Assume that $\mathcal{U}$ is generated by $\mathcal{A}$ with regularity spaces $\{Y_{t}\,|\,\ t\geq 0\}$. We first remark that $\tilde{Y_{t}}$ is a dense subspace of $X$ and

[TABLE]

for every $t\geq 0$, where $A_{Q}(t):=Q^{-1}(t)A(t)Q(t)-Q^{-1}(t)\dot{Q}(t)$. Next, let $x_{s}\in\tilde{Y_{s}}$. Then $Q(s)x_{s}\in Y_{s}$ and by assumption $U(\cdot,s)Q(s)x_{s}$ is the unique classical solution of

[TABLE]

It follows that $W(t,s)x\in Y_{t}\subset D(A_{Q}(t))$ by (8) and

[TABLE]

Since $Q$ is strongly $C^{1}$-differentiable, it now follows from (10)-(11) that $W(\cdot,s)x_{s}\in C^{1}((s,\infty),X)$ and $W(\cdot,s)x_{s}$ solves the non-autonomous problem

[TABLE]

Clearly, $W(\cdot,s)x_{s}$ is the unique classical solution of (12). We conclude that $\mathcal{W}$ is generated by $\{A_{Q}(t)\,|\,\ t\geq 0\}$ with regularity space $\{\tilde{Y_{t}}\,|\,t\geq 0\}$.

$(ii)$ Conversely, assume that $\mathcal{A}_{Q}$ generates the evolution family $\mathcal{W}$ with some regularity spaces $\{\tilde{Y}_{t}\,|\,\ t\geq 0\}.$ Since $Q^{-1}$ is $C^{1}$-strongly continuous we obtain by $(i)$ that the family $(\mathcal{A}_{Q})_{Q^{-1}}=\mathcal{A}$ generates the evolution $\mathcal{V}$ defined by

[TABLE]

with regularity space $Y_{t}=Q(t)\tilde{Y}_{t}.$ This completes the proof. ∎

If $A:D(A)\subset X\rightarrow X$ is the generator of a $C_{0}$-semigroup and $B\in\mathcal{L}(X)$, then the perturbed operator $\tilde{A}:=A+B$ is again the generator of a $C_{0}$-semigroup, see e.g., [10, Section 1.2] or [35]. This perturbation results fails to be true in general for non-autonomous evolution equations [10, Example 9.2]. Thus one cannot conclude from Proposition 2.4 that the family $\{Q^{-1}(t)A(t)Q(t)\,|\,t\geq 0\}$ generates an evolution family. Nevertheless, inspired by an idea of Schnaubelt and Weiss [29], using Proposition 2.4 we show that a positive answer can be given under some additional regularity assumptions.

For this we first need to introduce the following definition.

Definition 2.5**.**

(Kato’s class)

1. (1)

A family $\mathcal{A}$ is said to be Kato-stable if for each $t\geq 0$ there exists a norm $\|\cdot\|_{t}$ on $X$ equivalent to the original one such that for each $T\geq 0$ there exists a constant $c_{T}\geq 0$ with

[TABLE]

and $A(t)$ generates a contractive $C_{0}$-semigroup on $X_{t}:=(X,\|\cdot\|_{t})$ for all $t\geq 0$. 2. (2)

A family $\mathcal{A}$ is said to belong to Kato’s class if it is Kato-stable and the operators $A(t)$, $t\geq 0$, have a common domain $D$ such that $A(\cdot):[0,\infty)\rightarrow\mathcal{L}(D,X)$ is strongly $C^{1}$-differentiable.

It is known that Kato-stability is a sufficient condition for $C^{1}$-well posedness of (hyperbolic) non-autonomous evolution equations [21, 35, 32]. In particular, each non-autonomous evolution equation that is governed by a Kato-class family is $C^{1}$-well posed.

Obviously, $\mathcal{A}$ is Kato-stable if each operator $A(t)$ generates a contractive $C_{0}$-semigroup, as one can simply choose $\|\cdot\|_{t}=\|\cdot\|$, $t\geq 0$. In this case we say that $\mathcal{A}$ belongs to the elementary Kato class. Starting from this simple case many less trivial Kato-stable families can be constructed.

Example 2.6**.**

Assume that $(H,\|\cdot\|_{H})$ is a Hilbert space. Let $M:[0,\infty)\rightarrow\mathcal{L}(H)$ be self-adjoint and uniformly coercive, i.e., $M(t)^{*}=M(t)$ and $(M(t)x|x)_{H}\geq\beta\|x\|^{2}_{H}$ for some constant $\beta>0$ and all $t\geq 0$. If $M$ is strongly $C^{1}$-continuous and $M^{-1}$ is strongly continuous, then for each $t\in[0,\infty)$ the function

[TABLE]

defines a norm on $H$ which is equivalent to the norm $\|\cdot\|_{H}$ and satisfies (13). Moreover, if $\mathcal{A}$ has a common domain $D$ and for each $t\geq 0$ the operator $(A(t),D)$ generates a contraction $C_{0}$-semigroup in $H$, then $(A(t)M(t),D(A(t)M(t))$ and $(M(t)A,D(A(t)))$ generate contractive $C_{0}$-semigroups on $H_{t}$, and thus both families $\{A(t)M(t)\,|\,t\geq 0\}$ and $\{M(t)A(t)\,|\,t\geq 0\}$ are Kato-stable. We refer to [17, Lemma 7.2.3] and to the proof of [29, Proposition 2.3] for precise details. Finally, if $P:[0,\infty)\rightarrow\mathcal{L}(X)$ is a locally uniformly bounded function, then $\{M(t)A(t)+P(t)\,|\,t\geq 0\}$ and $\{A(t)M(t)+P(t)\,|\,t\geq 0\}$ are Kato-stable [32, Propositions 4.3.2 and 4.3.3].

Proposition 2.7**.**

Let $\mathcal{A}$ belong to the Kato-class and let $D$ denote the common domain of $A(t)$, $t\geq 0$. Assume that $Q(\cdot)$ is strongly $C^{2}$-continuous. Then $\{Q^{-1}(t)A(t)Q(t)\,|\,\ t\geq 0\}$ generates a unique evolution family $\mathcal{W}$ with regularity spaces $Y_{t}=Q{{-1}}(t)D$, $t\geq 0$. Moreover, for each $F\in C^{1}([0,\infty);X)$ and $x_{s}\in Q{{}^{-1}}(s)D$ the inhomogeneous non-autonomous Cauchy problem

[TABLE]

has a unique classical solution given by

[TABLE]

Proof.

It is not difficult to verify that (13) implies that $\|x\|_{t}\leq e^{c_{T}|t-s|}\|x\|_{s}$ for all $x\in X$, $t,s\in[0,T]$ and $T>0$. Using [32, Propositions 4.3.2 and 4.3.3] and [32, Corollary of Theorem 4.4.2] we obtain that $\{A(t)+\dot{Q}(t)Q^{-1}(t)\,|\,t\geq 0\}$ generates a unique evolution family $\mathcal{U}$ on $X$. Thus the first assertion follows from Proposition 2.4. Next, let $F\in C^{1}([0,\infty);X)$ and $x_{s}\in Q(s)D$. By [32, Theorem 4.5.3] the inhomogeneous Cauchy problem

[TABLE]

has a unique classical solution $x$ given by

[TABLE]

On the other hand, arguing as in the proof of Proposition 2.4 we see that $x:=Q^{-1}(\cdot)u$ is a classical solution of (15). The uniqueness of classical solutions of (15) follows from the uniqueness of classical solutions of (17). Finally, (16) follows from (19) and (7). ∎

Using Example 2.6 and Proposition 2.7 one can formulate the following two corollaries.

Corollary 2.8**.**

Assume that $X$ is a Hilbert space. Assume that $\mathcal{A}$ belongs to the elementary Kato class and denote by $D$ the common of $A(t)$, $t\geq 0$. Let $M:[0,\infty)\rightarrow\mathcal{L}(X)$ and $P:[0,\infty)\rightarrow\mathcal{L}(X)$ be self-adjoint and uniformly coercive such that $M$ is strongly $C^{2}$-continuous while $P$ is strongly $C^{1}$-differentiable. Then $\{A(t)M(t)+P(t)\,|\,\ t\geq 0\}$ generates a unique evolution family $\mathcal{W}$ with regularity spaces $Y_{t}={M^{-1}}(t)D$, $t\geq 0$. Moreover, for each $F\in C^{1}([0,\infty);X)$ and $x_{s}\in{M^{-1}}(s)D$ the inhomogeneous non-autonomous Cauchy problem

[TABLE]

has a unique classical solution given by (16).

Proof.

For the proof we just have to apply Proposition 2.7 for $M(t)A(t)+M(t)P(t)M^{-1}(t)$ instead of $A(t)$ and $M(t)$ instead of $Q(t)$. ∎

Corollary 2.9**.**

Let $X$ be a Hilbert space and let $(A,D(A))$ be generator of a contractive $C_{0}$-semigroup on $X$. Let $M:[0,\infty)\rightarrow\mathcal{L}(X)$ and $P:[0,\infty)\rightarrow\mathcal{L}(X)$ be as in Corollary 2.8. Further, let $R:[0,\infty)\rightarrow\mathcal{L}(X)$ be self-adjoint and uniformly coercive such that $R$ is strongly $C^{1}$-continuous and commute with $M$ i.e.

[TABLE]

Then the family $\{R(t)AM(t)+P(t)\,|\,\ t\geq 0\}$ generates a unique evolution family $\mathcal{W}$ with regularity spaces $Y_{t}={M^{-1}}(t)D(A)$, $t\geq 0.$ Moreover, for each $F\in C^{1}([0,\infty);X)$ and $x_{s}\in{M^{-1}}(s)D(A)$ the inhomogeneous non-autonomous Cauchy problem

[TABLE]

has a unique classical solution given by (16).

Proof.

From Example 2.6 we deduce that the family $\{M(t)R(t)A\,|\,t\geq 0\},$ and therefore $\{M(t)R(t)A+M(t)P(t)M^{-1}(t)\,|\,t\geq 0\},$ belongs to Kato’s class. In fact, using (22) we see that $M(\cdot)R(\cdot):[0,\infty)\rightarrow\mathcal{L}(X)$ is selfadjoint and uniformly coercive. Now, applying Proposition 2.7 for $M(t)R(t)A+M(t)P(t)M^{-1}(t)$ instead of $A(t)$ and $M(t)$ instead of $Q(t)$ concludes the proof. ∎

Remark 2.10**.**

Corollary 2.8 has been proved in [29, Proposition 2.8-(a)] using a slightly different method for $A(t)=A$ and $F=P=0$.

2.2. Backward evolution families

Let $X$ be a Hilbert space over $\mathbb{K}=\mathbb{C}$ or $\mathbb{R}$.

Definition 2.11**.**

A family $\mathcal{V}:=\{V(t,s)\,|\,\ (t,s)\in\Delta\}\subset\mathcal{L}(X)$ is called a backward evolution family if

• $(i)$

$V(t,t)=I$ and $V(r,s)V(t,r)=V(t,s)$ for every $0\leq s\leq r\leq t$,

• $(ii)$

$V(\cdot,\cdot):\Delta\rightarrow\mathcal{L}(X)$ is strongly continuous.

A family $A(t):D(A(t))\subset X\rightarrow X$, $t\geq 0$, of linear operators generates a backward evolution equation $\mathcal{V}$ if there is a family $\{Y_{t}\,|\,t\geq 0\}$ of dense subspaces of $X$ with $Y_{t}\subset D(A(t))$ and

[TABLE]

$V(t,\cdot)x_{t}\in C^{1}([0,t],X)$ for every $x_{t}\in Y_{t}$ and $V(t,\cdot)x_{t}$ solves uniquely the backward non-autonomous problem

[TABLE]

Lemma 2.12**.**

1. (1)

Assume that $\mathcal{A}=\{A(t)\,|\,t\geq 0\}$ belongs to the elementary Kato-class. Then $\mathcal{A}$ generates a backward evolution family. 2. (2)

Assume that $\mathcal{A}$ generates an evolution family $\mathcal{U}.$ If the adjoint operators $\mathcal{A}^{*}:=\{A^{*}(t)\,|\,t\geq 0\}$ generate a backward evolution family $\mathcal{U}_{*}:=\{U_{*}(t,s)\,|\,\ (t,s)\in\Delta\},$ then for $(t,s)\in\Delta$ we have

[TABLE]

Proof.

$(i)$ Let $T>0$ be fixed and set $\mathcal{A}_{T}:=\{A(T-t)\,|\,t\in[0,T]\}$. Then, obviously $\mathcal{A}_{T}$ belongs to the Kato-class and thus generates an evolution family $\mathcal{U}_{T}:=\{U_{T}(t,s)\,|\,0\leq s\leq t\leq T\}$ [35, Theorem 4.8] such that for each $x\in D$ and $0\leq s\leq t\leq T$

[TABLE]

It is easy to see that $S(t,s):=U_{T}(T-s,T-t)$ for each $0\leq s\leq t\leq T$ defines a backward evolution family with generator $\{A(t)\,|\,t\in[0,T]\}$. This completes the proof since $T$ is arbitrary.

$(ii)$ Denote by $Y_{t}$ and $Y_{t,*},t\geq 0$ the regularity spaces corresponding to $\mathcal{A}$ and $\mathcal{A}^{*}$, repectively. Let $t>s\geq 0$ and let $x_{s}\in Y_{s}$ and $y_{t}\in Y_{t,*}$. Then for $s\geq r\geq t$ we have

[TABLE]

Integrating over $[s,t]$ and using that $Y_{s}$ and $Y_{t,*}$ are dense in $X$ yield the desired identity. ∎

3. Review on Autonomous boundary control and observation systems

Many systems governed by linear partial differential equations with inhomogeneous boundary conditions are described by an abstract boundary system of the form

[TABLE]

Here $\mathfrak{A}:D(\mathfrak{A})\subset X\rightarrow X$ is a linear operator, $u(t)\in U$, $y(t)\in Y$, the boundary operators $\mathfrak{B}:D(\mathfrak{B})\subset X\rightarrow U$ and $\mathfrak{C}:D(\mathfrak{A})\subset X\rightarrow Y$ are linear such that $D(\mathfrak{A})\subset D(\mathfrak{B})$, and $X$, $U$ and $Y$ are complex Hilbert spaces. We shall call $X$ the state space, $U$ the input space and $Y$ the output space of the system.

In this section, we recall some well-known results on well-posedness of these system which are needed throughout this paper.

Definition 3.1**.**

Let $x_{0}\in X$ and $u:[0,\infty)\rightarrow U$ be given.

• $(i)$

$x$ is called a classical solution of (28)-(29), if $x\in C^{1}([0,\infty),X)$, $x(t)\in D(\mathfrak{A})$ for all $t\geq 0$ and $x$ satisfies (28)-(29).

• $(ii)$

A pair $(x,y)$ is called a classical solution of (28)-(30), if $x$ is a classical solution of (28)-(29), $y\in C([0,\infty);Y)$ and $y$ satisfies (30).

• $(iii)$

The system $\Sigma(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is called well-posed, if for any final time $\tau>0$ there exists $m_{\tau}>0$ such that for all classical solution of (28)-(30) we have

[TABLE]

Remark that, if $\mathfrak{C}\in\mathcal{L}(D(\mathfrak{A}),Y)$, then $(x,y)$ is a classical solution of (28)-(30) if and only if $x$ is a classical solution of (28)-(29).

3.1. Existence of classical solutions

In order to study existence of classical solutions it is often useful to write the boundary control system (28)-(29) as a $C^{1}$-well posed (inhomogeneous) autonomous Cauchy problem. We introduce the following definition which is based on Curtain and Zwart [9, Definition 3.3.2].

Definition 3.2**.**

The linear (autonomous) system (28)-(30) is called a boundary control and observation autonomous system, and we write $\Sigma(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is a BCO-system, if the following assertions hold:

• $(i)$

The operator $A:D(A)\subset X\rightarrow X$, called the main operator, defined by

[TABLE]

generates a strongly continuous semigroup on $X$.

• $(ii)$

There exists a linear operator $\tilde{B}\in\mathcal{L}(U,X)$ such that for all $u\in U$ we have

[TABLE]

• $(iii)$

$\mathfrak{C}:D(\mathfrak{A})\subset X\rightarrow Y$ is a linear bounded operator, where $D(\mathfrak{A})$ is equipped with the graph norm.

In the following $\Sigma(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is assumed to be a BCO-system. The following remark will be very useful for non-autonomous boundary control systems.

Remark 3.3**.**

Let $\Sigma(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ be a BCO-system. Then for each $(x,u)\in X\times U$ we have

[TABLE]

This is an easily consequence of Definition 3.2.

We denote by $X_{-1}$ the extrapolation space associated to $A$, i.e., the completion of $X$ with respect to the norm $x\mapsto\|(\beta I-A)^{-1}x\|$ for some arbitrary $\beta\in\rho(A)$. Let $A_{-1}$ be the extension of $A$ to $X_{-1}$. It is well known that $A_{-1}$ with domain $X$ generates a $C_{0}$-semigroup $(T_{-1}(t))_{t\geq 0}$ on $X_{-1}$ and for all $t\geq 0$ the operator $T_{-1}(t)$ is the unique continuous extension of $T(t)$ to $X_{-1}$. We associate with $\Sigma(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ the linear operator $B\in\mathcal{L}(U,X_{-1})$ called control operator defined by

[TABLE]

It turns out, that for sufficiently smooth initial data and inputs the two Cauchy problems

[TABLE]

and the BCO-system (28)-(30) are equivalent. More precisely, we have

Proposition 3.4**.**

Let $(x_{0},u)\in D(\mathfrak{A})\times W^{2,2}([0,\infty);U)$ such that $\mathfrak{B}x_{0}=u(0)$. Then (28)-(29) has a unique classical solution $x$ given by

[TABLE]

Therefore, $x$ is the unique classical solution of (34) and $w:=x-\tilde{u}$ is the unique classical solution of (33) with initial value $w_{0}=x_{0}-\tilde{B}u(0)$.

Proof.

The proof follows from a combination of [17, Theorem 11.1.2] (see also [9, Theorem 3.3.3]) and [17, Corollary 10.1.4] taking Remark 3.3 into account. ∎

3.2. Scattering passive BCO-systems

Let $\Sigma(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ be a BCO-system on $(X,U,Y)$ and let $P=P^{*}\in\mathcal{L}(X)$, $R=R^{*}\in\mathcal{L}(U)$ and $J=J^{*}\in\mathcal{L}(Y)$. The admissible space $\mathcal{V}\subset X\times U$ is defined by

[TABLE]

Definition 3.5**.**

We say that $\Sigma(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is $(P,R,J)$-scattering passive if

[TABLE]

and all classical solutions $(x,y)$ of (28)-(30). Further, $\Sigma(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is called (R,P,J)-scattering energy preserving if equality holds in (38). If $P=I,R=I$ and $J=I$, then we simply say that $\Sigma(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is scattering passive (or dissipative).

Each $(P,R,J)$-scattering passive boundary system $\Sigma(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is well-posed if $P$ and $J$ are invertible. This can be seen by using Gronwall’s Lemma (see the proof of Lemma 4.5). The following lemma characterizes $(P,R,J)$-scattering passive BCO-systems. A comparable results has been proved in [23, Theorem 3.2, Proposition 5.2] for systems nodes.

Lemma 3.6**.**

The BCO-system $\Sigma(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is $(P,R,J)$-scattering passive if and only if for each $(x_{0},u_{0})\in\mathcal{V}$ we have

[TABLE]

or equivalently,

[TABLE]

Then the BCO-system $\Sigma(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is $(P,R,J)$-energy preserving if and only if equality holds in (39), or equivalently in (40).

Proof.

Obviously, the inequalities (39) and (40) are equivalent since $\mathfrak{B}x_{0}=u_{0}$ for each $(x_{0},u_{0})\in\mathcal{V}$. Assume that $\Sigma(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is $(R,P,J)$-scattering passive. Let $(x_{0},u_{0})\in\mathcal{V}$ and $u:[0,\infty)\rightarrow U$ such that $u(0)=u_{0}$. Assume that $(x,y)$ is a classical solution of (28)-(30) corresponding to $(x_{0},u)$. Then $(x(t),u(t))\in\mathcal{V}$ and

[TABLE]

for all $t\geq 0$. Inserting this into (38) yields

[TABLE]

for all $t\geq 0$. The previous inequality implies (39) by taking $t=0$. The converse implication and the last assertion can be proved similarly. ∎

4. Non-autonomous boundary and observation systems

In this section, our aim is to extend the results of Section 3 to the more general case where $\mathfrak{A},\mathfrak{B}$, and $\mathfrak{C}$ are time dependent. Let $X,U$ and $Y$ be Hilbert spaces over $\mathbb{K}=\mathbb{C}$ or $\mathbb{R}$. For each $t\geq 0$ we consider the linear operators $\mathfrak{A}(t):D(\mathfrak{A}(t))\subset X\rightarrow X$, $\mathfrak{B}(t):D(\mathfrak{B}(t))\subset X\rightarrow U$ and $\mathfrak{C}(t):D(\mathfrak{A}(t))\subset X\rightarrow Y$ such that $D(\mathfrak{A}(t))\subset D(\mathfrak{B}(t))$ for each $t\geq 0$.

We consider the following abstract non-autonomous boundary system

[TABLE]

which we denote by $\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$.

Definition 4.1**.**

Let $s\geq 0,x_{s}\in X$ and $u:[0,\infty)\rightarrow U$ be given.

• $(i)$

A function $x:[s,\infty)\rightarrow X$ is called a classical solution of (42)-(43), if $x\in C^{1}([s,\infty),X)$, $x(t)\in D(\mathfrak{A}(t))$ for all $t\geq s$ and $x$ satisfies (42)-(43).

• $(ii)$

A pair $(x,y)$ is a classical solution of (42)-(44), if $x$ is classical solution of (42)-(43), $y\in C([s,\infty);Y)$ and $(x,y)$ satisfies (42)-(44).

• $(iii)$

$\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is a non-autonomous boundary control and observation system, and we write NBCO-systems, if for each $t\geq 0$ the autonomous system $\Sigma(\mathfrak{A}(t),\mathfrak{B}(t),\mathfrak{C}(t))$ is a BCO-system such that the family $\{A(t)\,|\,t\geq 0\}$ of main operators generates an evolution family.

• $(iv)$

The non-autonomous system $\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is called well-posed if for any final time $\tau>0$ there exists a constant $m_{\tau}>0$ such that for all classical solution of (42)-(44) we have

[TABLE]

4.1. Existence of classical solutions

Let $\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ be a NBCO-system. In this subsection, we study existence and uniqueness of classical solutions of $\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ without output, i.e., classical solution of (42)-(43). In the previous section we have seen in the autonomous case that (42)-(43) can be equivalently written as a $C^{1}$-well-posed inhomogeneous Cauchy problem (in $X$) for sufficiently smooth initial data and inputs. This idea can be extended to the non-autonomous setting.

For each $t\geq 0$, we denote by $A(t):D(A(t)\subset X\rightarrow X$ the main operator of $\Sigma(\mathfrak{A}(t),\mathfrak{B}(t),\mathfrak{C}(t))$, and by $\mathcal{U}$ the evolution family generated by $\{A(t)\,|\,t\geq 0\}$. Further, according to Definition 4.1-$(iii)$ there exists $\{\tilde{B}(t)\,|\,t\geq 0\}\subset\mathcal{L}(U,X)$ such that for all $t\geq 0$ we have

[TABLE]

We also consider the time-dependent admissible spaces $\mathcal{V}(t),t\geq 0$, i.e,

[TABLE]

Since $\{A(t)\,|\,t\geq 0\}$ generates an evolution family $\mathcal{U}$ on $X$, for a given $f\in L_{Loc}^{1}([0,\infty);X)$ the inhomogeneous non-autonomous Cauchy problem

[TABLE]

has at most one classical solution given by

[TABLE]

see e.g., [35, Section 5.5.1]. Thus the following proposition provides a generalization of [9, Theorem 3.3.3] (see also Proposition 3.6).

Proposition 4.2**.**

Assume that $u\in C^{1}([0,\infty);U)$, $\tilde{B}(\cdot)u_{0}\in C^{1}([0,\infty);X)$ and $\mathfrak{A}(\cdot)\tilde{B}(\cdot)u_{0}\in L^{1}([0,\infty);X)$ for each $u_{0}\in U$. Let $x_{s}\in X$ such that $(x_{s},u_{s})\in\mathcal{V}(s)$. Then $x$ is a classical solution of (42)-(43) if and only if $v:=x-\tilde{B}u$ is a classical solution of (46)-(47) with inhomogeneity

[TABLE]

and initial data $v_{s}=x_{s}-\tilde{B}(s)u(s)$. Therefore, (42)-(43) has at most one classical solution $x$ given by

[TABLE]

for each $t\geq s$.

Proof.

Let $s\geq 0.$ Clearly $x\in C^{1}([s,\infty);X)$ if and only if $v\in C^{1}([s,\infty);X)$. Assume now that $x$ is a classical solution of (42)-(43). Then $v(t)\in\mathcal{V}_{t}\subset D(A(t))$ for every $t\geq s$ by Remark 3.3 and

[TABLE]

Thus $v$ is a classical solution of (46) with $f$ given by (48). The converse implication can be proved similarly. Finally, (49) follows by the above the remark. ∎

4.2. Scattering passive NBCO-systems

Let $R:[0,\infty)\rightarrow\mathcal{L}(U)$, $P:[0,\infty)\rightarrow\mathcal{L}(X)$ and $J:[0,\infty)\rightarrow\mathcal{L}(Y)$ be continuous functions such that $P$ is strongly differentiable and $R(t)^{*}=R(t)$, $P(t)^{*}=P(t)$, $J(t)^{*}=J(t)$ for all $t\geq 0$.

Definition 4.3**.**

Let $(x,y)$ be classical solution of (42)-(44). Then $\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is called (R,P,J)-scattering passive if for all $t\geq s$

[TABLE]

Further, $\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is called (R,P,J)-scattering energy preserving if equality holds in (50). If $P=I,R=I$ and $J=I$ then $\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is called scattering passive, and scattering energy preserving if we have equality in (50).

We have seen in Section 3.2 that for autonomous BCO-systems $\Sigma(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ the $(R,P,J)$-scattering passivity can be characterized in terms of $\mathfrak{A}$, $\mathfrak{B}$ and $\mathfrak{C}$ and $(R,P,J)$-scattering passivity is a sufficient condition for well-posedness, if additionally $P$ and $J$ are invertible. Proposition 4.4 and Lemma 4.5 generalize this facts for non-autonomous boundary control and observation systems.

Proposition 4.4**.**

The following assertion are equivalent.

• $(i)$

$\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$* is (R,P,J)-scattering passive.*

• $(ii)$

For each $t\geq 0$ and all $(x,u)\in\mathcal{V}(t)$ we have

[TABLE]

• $(iii)$

For each $t\geq 0$, the autonomous BCO-system $\Sigma(\mathfrak{A}(t),\mathfrak{B}(t),\mathfrak{C}(t))$ is $(R(t),P(t),J(t))$-scattering passive.

Proof.

The equivalence of $(ii)$ and $(iii)$ has been proved in Proposition 3.6. It remains to prove the equivalence of $(i)$ and $(ii)$. Assume that $(i)$ holds and let $s\geq 0$ and let $(x_{s},u_{s})\in\mathcal{V}(s)$. Let $u:[s,\infty)\rightarrow U$ such that $u(s)=u_{s}$. If $(x,y)$ is a classical solution of (42)-(44) corresponding to $(x_{s},u)$ then $(x(t),u(t))\in\mathcal{V}(t),\ y(t)=\mathfrak{C}(t)x(t)$ and

[TABLE]

for all $t\geq s.$ Inserting this into (50) yields

[TABLE]

for all $t\geq s$. The last inequality $(ii)$ by taking $t=s$. Conversely, assume that $(ii)$ holds and let $(x,y)$ be a classical solution of (42)-(44). Then $(x(t),u(t))\in\mathcal{V}(t)$ and (52)-(53) holds for all $t\geq s.$ This together with (51) imply (50), which completes the proof. ∎

Lemma 4.5**.**

Let $\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ be $(R,P,J)$-scattering passive such that $J\geq 0.$ Assume that $P$ is strongly $C^{1}$-continuous and uniformly coercive with

[TABLE]

for some constant $\beta>0.$ Then each classical solution of (42)-(44) satisfies the following inequality

[TABLE]

where $c_{t,s}=\max\{1,\underset{r\in[s,t]}{\max}\|P(r)\|\}$. Therefore, $\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is well-posed provided that $J$ is uniformly coercive and $R\in L^{\infty}_{Loc}([0,\infty);\mathcal{L}(U))$.

Proof.

For the proof we follow a similar argument as in fourth steps of the proof of [29, Theorem 4.1]. Assume that $\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is $(R,P,J)$-scattering passive. Clearly (50) holds if and only if

[TABLE]

for all $t\geq s\geq 0$. Thus using (54) and that $J\geq 0$ we obtain

[TABLE]

Applying Gronwall’s Lemma yields

[TABLE]

which implies (55). This completes the proof. ∎

4.3. Multiplicative perturbed of NBCO-systems

We will adopt the same notations of the previous sections. The main purpose of this section is the study of some classes of NBCO-systems which are governed by a time-dependent multiplicative perturbation. More precisely, let $\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ be a NBCO-system such that the boundary operators are constant, that is $\mathfrak{C}(t)=\mathfrak{C}$ and $\mathfrak{B}(t)=\mathfrak{B}$ for all $t\geq 0$. Thus the domain $\mathfrak{A}(t)$ should also be constant and we set $\mathcal{D}(\mathfrak{A}(t))=\mathfrak{D}$ for all $t\geq 0$.

Further, throughout this section we assume that the following assumption holds:

Assumption 4.6**.**

1. (1)

$M:[0,\infty)\rightarrow\mathcal{L}(X)$and $R:[0,\infty)\rightarrow\mathcal{L}(X)$ be two self-adjoint and uniformly coercive functions. 2. (2)

$M(\cdot)x\in C^{2}([0,\infty);X)$ and $M^{-1}(\cdot)x\in C([0,\infty);X)$ for each $x\in X$. 3. (3)

$L(\cdot)x\in C^{1}([0,\infty);X)$ for each $x\in X$ such that $L$ and $M$ commute.

For each $t\geq 0$ we set

[TABLE]

We consider the following perturbed system

[TABLE]

which we denote by $\Sigma_{N,M}(\mathfrak{A},\mathfrak{B},\mathfrak{C})=\Sigma_{N}(\mathfrak{A}M,\mathfrak{B}M,\mathfrak{C}M)$. Let $\tilde{B}(t)$ be operators associated with $\Sigma_{M}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ provided by Definition 3.2-$(ii).$ Then $\tilde{B}_{M}(t):=M^{-1}(t)\tilde{B}(t)$ satisfies for each $t\geq 0$ all properties listed in Definition 3.2-$(ii)$. Moreover, the main operators associated with $\Sigma_{M,N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ are given by $\{A(t)M(t)\,|\,t\geq 0\}$, where $D(A(t)M(t))=M^{-1}(t)(\mathfrak{D}\cap\ker(\mathfrak{B}))$ for each $t\geq 0$.

Lemma 4.7**.**

The perturbed system $\Sigma_{N,M}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is $(R,P,J)$-scattering passive if and only if $\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is $(R,PM^{-1},J)$-scattering passive.

Proof.

For each $t\geq 0$ we set

[TABLE]

Then, $(x,u)\in\mathcal{V}_{M}(t)$ if and only if $(M(t)x,u)\in\mathcal{V}(t)$ for all $t\geq 0$. Assume now that $\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is $(R,M^{-1}P,J)$-scattering passive and let $(x,u)\in\mathcal{V}_{M}(t)$. Using Proposition 4.4 we obtain

[TABLE]

This implies, again by Proposition 4.4, that $\Sigma_{N,M}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is $(R,P,J)$-scattering passive.

Conversely, assume that $\Sigma_{N,M}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is $(R,P,J)$-scattering passive. This means that $\Sigma_{N}(\mathfrak{A}M,\mathfrak{B}M,\mathfrak{C}M)$ is $(R,PM^{-1}M,J)$-scattering passive. Applying the first part of the proof yields that

[TABLE]

is $(R,PM^{-1},J)$-scattering passive. This completes the proof. ∎

In particular, the system $\Sigma_{N,M}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is $(R,M,J)$-scattering passive if and only if the unperturbed system $\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is $(R,I,J)$-scattering passive. According to the above assumptions, we remark that $(x,y)$ is a classical solution of (58)-(60) if and only if $x$ is a classical solution of (58)-(59).

Now we can formulate the first main result of this section.

Theorem 4.8**.**

Assume that the following additional assumptions holds

• $(a)$

$\mathfrak{A}:[0,\infty)\rightarrow\mathcal{L}(\mathfrak{D},X)$* is strongly $C^{1}$-continuous.*

• $(b)$

The main operators $A(t):\mathfrak{D}\cap\ker(\mathfrak{B})\rightarrow X$, $t\geq 0$ generate contraction $C_{0}$-semigroups.

• $(c)$

$\tilde{B}(\cdot)u\in C^{2}([0,\infty);U)$* for each $u\in U$.*

Then the perturbed system $\Sigma_{N,M}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is a NBCO-system on $(X,U,Y)$. Furthermore, if we denote by $\mathcal{W}$ the associated evolution family, then for each $s\geq 0$ and $(x_{s},u)\in X\times C^{2}([0,\infty);U)$ with $(M(s)x_{s},u(s))\in\mathcal{V}(s)$ the system (58)-(60) has a unique classical solution $(x,y)$ given by

[TABLE]

The system $\Sigma_{N,M}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is well-posed if in addition

[TABLE]

for all $t\geq 0$ and $(x_{0},u_{0})\in\mathcal{V}(t)$ where $R=R^{*}\in L^{\infty}_{Loc}([0,\infty);\mathcal{L}(U))$ and $J=J^{*}$ is uniformly coercive, where

[TABLE]

Proof.

The first and the second assertion follow from Proposition 4.2 and Corollary 2.8, whereas the last assertion follows from Lemma 4.7, Proposition 4.4 and Lemma 4.5. ∎

Next we consider the case where $\mathfrak{A}(t)=L(t)\mathfrak{A}$ with $L(t)$ is as in Assumption 4.6 and such that $(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is an autonomous BCO-system. This implies that $(L(t)\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is again an autonomous BCO-system for each $t\geq 0$ such that the associated operator $\tilde{B}$ is time-independent. In fact, if $\tilde{B}$ denotes the operator associated with the autonomous BCO-system $\Sigma(\mathfrak{A},\mathfrak{B},\mathfrak{C})$, then it is easy to see that $\tilde{B}$ satisfies all properties listed in Definition 3.2-$(ii)$ corresponding to $(L(t)\mathfrak{A},\mathfrak{B},\mathfrak{C}).$ We consider the following perturbed system

[TABLE]

which we denote by $\Sigma_{N,M,L}(\mathfrak{A},\mathfrak{B},\mathfrak{C})=\Sigma_{N}(L\mathfrak{A}M,\mathfrak{B}M,\mathfrak{C}M)$.

Clearly, the main operators associated with $\Sigma_{M,N,L}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ are given by $\{L(t)AM(t)\,|\,t\geq 0\}.$

Theorem 4.9**.**

Assume that the main operators $A:\mathfrak{D}\cap\ker(\mathfrak{B})\rightarrow X$ generate a contraction $C_{0}$-semigroup on $X.$ Then the perturbed system $\Sigma_{N,M,L}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is a NBCO-system on $(X,U,Y)$. Furthermore, if we denote by $\mathcal{W}$ the associated evolution family, then for each $s\geq 0$ and $(x_{s},u)\in X\times C^{2}([0,\infty);U)$ with $(M(s)x_{s},u(s))\in\mathcal{V}(s)$ the system (62)-(64) has a unique classical solution $(x,y)$ given by

[TABLE]

The system $\Sigma_{N,M,L}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is well-posed if in addition

[TABLE]

for all $t\geq 0$ and $(x_{0},u_{0})\in\mathcal{V}$ where $R=R^{*}\in L^{\infty}_{Loc}([0,\infty);\mathcal{L}(U))$ and $J=J^{*}$ is uniformly coercive.

Proof.

The first and the second assertion follow from Proposition 4.2 and Corollary 2.9, whereas the last assertion follows from Lemma 4.7, Proposition 4.4 and Lemma 4.5. ∎

Remark 4.10**.**

Theorem 4.9 is not a special case of Theorem 4.8 since we do not assume that $P(t)A$ generates a contractive $C_{0}$-semigroup on $X.$

5. Mild solutions for NBC-systems

As mentioned in Section 3, for an autonomous BCO-system $\Sigma(\mathfrak{A},\mathfrak{B},\mathfrak{C})$, for smooth input $u$ and initial data $x_{0}$, the classical solution of the corresponding boundary control system can be formulated as

[TABLE]

We recall that $B\in\mathcal{L}(U,X_{-1})$ is given by (32). If $x_{s}\in X$ and $u\in L^{2}([0,\infty);U)$, then the above formula makes sense and it is called the mild solution in $X_{-1}$ of (28)-(30). Moreover, it is well known that the mild solution belongs to $C([0,\infty);X)$ if $B$ is admissible for the semigroup $(T(t))_{t\geq 0},$ i.e., if for some $\tau>0$ one has

[TABLE]

see, e.g., [33, Proposition 4.2.4].

The main purpose of this section is to extend the conceps of mild solutions to non-autonomous boundary control and observation systems $\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$. In contrast to the autonomous case, this is more delicate. In fact, firstly we remark that the extrapolation spaces $X_{-1,t}$ associated with the family $\{A(t)\,|\,t\geq 0\}$ of the main operators are in general time-dependent. Secondly, in contrast to semigroups, it is not clear whether the evolution family $\mathcal{U}$ generated by $\{A(t)\,|\,t\geq 0\}$ can be extended to the extrapolation space even if the spaces $X_{-1,t}$ are constant. However, if the latter condition holds, then we can still use the adjoint problem, i.e, $A^{*}(t)$, $t\geq 0$, and the associated backward evolution family to extend $\mathcal{U}$ to $\mathcal{L}(X_{-1})$. The idea to use a duality argument can be found in [7, 25, 29] to study some classes of non-autonomous systems.

We will adopt here the notations of the previous sections. Let $\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ be a NBCO-system. Then the main operators $\{A(t)\,|\,t\geq 0\}$ generate, by definition, an evolution family $\mathcal{U}=\{U(t,s)\,|\,(s,t)\in\Delta\}$ with regularity space $Y_{t}$, $t\geq 0$. We restrict ourselves to case where $\{A(t)\,|\,t\geq 0\}$ have a common extrapolation space $X_{-1}$, i.e.,

[TABLE]

According to [33, Proposition 2.10.2], (67) holds if and only if $D(A^{*}(t))=D(A^{*}(s))$ for $t,s\in[0,\infty)$ and the corresponding graph norms are locally uniformly equivalent. In fact, $X_{-1}(t)$ is the dual space of $D(A^{*}(t))$ with respect to the pivot space $X$. This condition holds, if for instance $A(t)=AM(t)$ or $A(t)=A+M(t)$ and $M\in C^{1}([0,\infty);\mathcal{L}(X))$.

In the following we denote $\mathfrak{D}_{*}:=D(A^{*}(0))$ equipped with the graph norm and by $\langle\cdot,\cdot\rangle$ the duality between $X_{-1}$ and $\mathfrak{D}_{*}$. Recall from (32)

[TABLE]

Proposition 5.1**.**

Assume that $\mathcal{A}^{*}:=\left\{A^{*}(t)\,|\,t\geq 0\right\}$ generates a backward evolution family $\mathcal{U}_{*}$. Then $U(t,s)$ has a unique extension $U_{-1}(t,s)\in\mathcal{L}(X_{-1})$ for each $(t,s)\in\Delta$ and for each $T>0$ there is $c_{T}>0$ such that

[TABLE]

Moreover, if the assumptions of Proposition 4.2 hold, then each classical solution $x$ of the boundary control system (42)-(43) satisfies

[TABLE]

Proof.

By [33, Proposition 2.9.3-(b)] we obtain that for each $(t,s)\in\Delta$ the operator $U(t,s)$ has a unique extension $U_{-1}(t,s)\in\mathcal{L}(X_{-1})$ since $[U(t,s)]^{*}\mathfrak{D}_{*}=U_{*}(t,s)\mathfrak{D}_{*}\subset\mathfrak{D}_{*}$. Next, similar to the proof of [29, Proposition 2.7-(c)] we show the uniform boundedness of $\mathcal{U}_{-1}$ on compact intervals. Next, we claim that for each $y\in\mathfrak{D}_{*},x\in X_{-1}$ we have

[TABLE]

In fact, this equality holds for $x\in X$ by Lemma 2.12-$(ii)$ since

[TABLE]

Remark that $U_{*}(t,s)y\in\mathfrak{D}_{*}$, thus the claim follows since $X$ is dense in $X_{-1}$.

Using again Lemma 2.12 and (71), we obtain for each $y\in\mathfrak{D}_{*}$

[TABLE]

Integrating over $[s,t]$, we obtain

[TABLE]

Inserting this equality in (49), we obtain that a classical solution $x$ of (42)-(43) satisfies (70). ∎

If the assumptions of Proposition 5.1 hold, then for $x_{s}\in X$ and $u\in L^{2}_{Loc}([0,\infty);U)$ we see that (70) is well defined with value in $X_{-1}$ provided $B(\cdot)u(\cdot)\in L^{1}_{Loc}([0,\infty);X_{-1})$. In fact, (69) guaranties that the integral term on the right hand side of (73) is well defined. Thus the following definition makes sense.

Definition 5.2**.**

Let $\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ be a NBCO-system and let $\mathcal{U}$ and $\{B(t)\,|\,t\geq 0\}$ the associated evolution family and control operators, respectively. Let $(x_{s},u)\in X\times L^{2}_{Loc}([0,\infty);U)$. If $U(t,s)$ has a unique extension $U_{-1}(t,s)\in\mathcal{L}(X_{-1})$ for each $(t,s)\in\Delta$ such that $U_{-1}(t,\cdot)B(\cdot)u(\cdot)\in L^{1}_{Loc}([0,\infty);X_{-1})$, then the function

[TABLE]

is called the mild solution of (42)-(44) in $X_{-1}$. Further, (73) is called a mild solution of (42)-(44) (in $X$), if in addition

[TABLE]

and $x\in C([s,\infty);X)$.

This definition is related to the notion of admissibility for non-autonomous linear systems. More precisely, recall that a family $\{B(t)\,|\,t\geq 0\}\subset\mathcal{L}(U,X_{-1})$ is $L^{2}$-admissible for a given evolution family $\mathcal{U}$ that admit an extension to $\mathcal{L}(X_{-1})$ if $U_{-1}(t,\cdot)B(\cdot)u(\cdot)\in L^{1}_{Loc}([0,\infty);X_{-1})$, (74) holds and for each $T>0$ there exists $c_{T}>0$ such that

[TABLE]

for each $u\in L^{2}_{Loc}([0,\infty);U)$ and all $0\leq s\leq t\leq T$ [28, Definition 3.3]. For $L^{2}$-admissible control operators we have that $(t,s)\mapsto\Phi_{t,s}u$ is continuous on $\Delta$ with value in $X$ [28, Proposition 3.5-(2)].

Proposition 5.3**.**

Assume that $\mathcal{A}^{*}:=\left\{A^{*}(t)\,|\,t\geq 0\right\}$ belongs to the Kato-class and $\{B(t)\,|\,t\geq 0\}$ is $L^{2}$-admissible. Then for each $(x_{s},u)\in X\times L^{2}_{Loc}([0,\infty);U)$ with $B(\cdot)u(\cdot)\in L^{1}_{Loc}([0,\infty);X_{-1})$ the system (42)-(44) has a unique mild solution in $X$.

Proof.

The proof follows from Lemma 2.12-$(i)$ and Proposition 5.1. ∎

If $\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is a well-posed NBCO-system and the classical solutions is given by (70), then the corresponding family $\{B(t)\,|\,t\geq 0\}$ is $L^{2}$-admissible provided

[TABLE]

Thus the following corollary follows from Proposition 5.3, Lemma 4.5 and (69).

Corollary 5.4**.**

Assume $\Sigma_{N}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is $(R,P,J)$-scattering passive such that $R\in L^{\infty}_{Loc}([0,\infty);\mathcal{L}(U))$ and $J,P$ are uniformly coercive. In addition, we assume that $\mathcal{A}^{*}:=\left\{A^{*}(t)\,|\,t\geq 0\right\}$ belongs to the Kato-class. Then for each $(x_{s},u)\in X\times L^{2}_{Loc}([0,\infty);U)$ with $B(\cdot)u(\cdot)\in L^{1}_{Loc}([0,\infty);X_{-1})$ the system (42)-(44) has a unique mild solution in $X$.

Finally, if Assumption 4.6 holds such that $A(t)$ generates contractive $C_{0}$-semigroup on $X$ for each $t\geq 0$ then we can follow [29, Section 2, page 8] to deduce that the extrapolation spaces corresponding to $A(t)M(t)$, $t\geq 0$ can be all identified with $X_{-1}$ and that $[A(t)M(t)]_{-1}=A_{-1}(t)M(t)$ for every $t\geq 0$.

Corollary 5.5**.**

Assume that Assumption 4.6 holds such that the adjoint operators $\left\{A(t)^{*}\,|\,t\geq 0\right\}$ have a common domain. Then the perturbed system (58)-(59) has a unique mild solution in $X$ if the unperturbed system $\Sigma_{N,\operatorname{id}}(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is $(R,I,J)$-scattering passive with $R\in L^{\infty}_{Loc}([0,\infty);\mathcal{L}(U))$ and $J,P$ are uniformly coercive.

Proof.

The proof is an easy consequence of Corollary 5.4 and Lemma 4.7. ∎

6. Application to non-autonomous Port-Hamiltonian systems

Let $N,n\in\mathbb{N}$ be fixed and let $X:=L^{2}([a,b];\mathbb{K}^{n})$ where $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. In this section we investigate the well-posedness of the linear non-autonomous port-Hamiltonian systems of order $N\in\mathbb{N}$, given by the boundary control and observation system

[TABLE]

Here $\tau$ denotes the trace operator $\tau:H^{N}((a,b);\mathbb{K}^{n})\rightarrow\mathbb{K}^{2Nn}$ defined by

[TABLE]

$P_{k}(t)$ is $n\times n$ matrix for all $t\geq 0$, $k=0,1,\cdots,N$, $\mathcal{H}(t,\zeta)\in\mathbb{K}^{n\times n}$ for all $t\geq 0$ and almost every $\zeta\in[a,b]$, $W_{B,1}$ is a $m\times 2nN$-matrix, $W_{B,2}$ is $(nN-m)\times 2nN$-matrix and $W_{C}$ is a $d\times 2nN$-matrix. Finally, $u(t)\in U:=\mathbb{K}^{m}$ denotes the input and $y(t)\in Y:=\mathbb{K}^{d}$ is the output at time $t$.

Set $W_{B}:=\left[\begin{smallmatrix}W_{B,1}\\ W_{B,2}\end{smallmatrix}\right]$, $\Sigma:=\left[\begin{smallmatrix}0&I\\ I&0\end{smallmatrix}\right]$ and for each $t\geq 0$ we set

[TABLE]

$R_{ext}(t):=\left[\begin{smallmatrix}Q(t)&-Q(t)\\ \mathrm{I}&\mathrm{I}\end{smallmatrix}\right]$ and $W_{B}(t):=W_{B}R_{ext}^{-1}(t),W_{C}(t):=W_{B}R_{ext}^{-1}(t)$.

In this section we assume the following assumptions:

Assumption 6.1**.**

• •

$W_{B}$ has full rank and $W_{B}(t)\Sigma W_{B}^{*}(t)\geq 0$ for all $t\geq 0$.

• •

$P_{N}(t)$ is invertible and $P^{*}_{k}(t)=(-1)^{k-1}P_{k}(t)$ for all $k\geq 1$, $t\geq 0$,

• •

$P_{k}\in C^{1}([0,\infty);L^{\infty}(a,b;\mathbb{C}^{n\times n}))$ for all $t\geq 0$ and $k=0,1,\cdots N$.

• •

$\mathcal{H}\in C^{2}([0,\infty);L^{\infty}([a,b];\mathbb{C}^{n\times n}))$ and there exist $m,M\geq 0$ such that

[TABLE]

Under these assumptions, the port-Hamiltonian system (76)-(80) can be written as a non-autonomous boundary control and observation system in the sense of Definition 4.1-$(ii)$. In fact, on the Hilbert space $X$ we consider the (maximal) port-Hamiltonian operators

[TABLE]

Then $(\mathfrak{A}(t),D(\mathfrak{A}(t)))$ is a closed and densely defined operator and its graph norm $\|\cdot\|_{D(\mathfrak{A}(t))}$ is equivalent to the Sobolev norm $\|\cdot\|_{H^{N}((a,b);\mathbb{K}^{n})}$ as $P_{N}(t)$ is invertible. Moreover, for each $t\geq 0$ the operator $A(t):D(A(t))\subset X\rightarrow X$ defined by

[TABLE]

generates a contractive $C_{0}$-semigroup on $X$. Further, we define the input operator $\mathfrak{B}$ and output operator $\mathfrak{C}$ a follows

[TABLE]

and

[TABLE]

The operator $\mathfrak{C}$ is a linear and bounded operator from $D(\mathfrak{A}(t))$ to $Y$, since the trace operator $\tau$ is bounded and the norm graph norm of $D(\mathfrak{A})$ is equivalent to the $H^{N}((a,b);\mathbb{K}^{n})$-norm. Moreover, Lemma 6.2 below shows that there exists an operator $\tilde{B}\in\mathcal{L}(U,X)$ which is independent of $t\geq 0$ satisfying the assumption $(ii)$ of Definition 3.2. The proof of this fact follows by a minor modification of the proof of [17, Theorem 11.3.2] and that of [2, Lemma 3.2.19] (see also the second step of the proof of [22, Theorem 4.2]).

Lemma 6.2**.**

There exists a linear operator $\tilde{B}\in\mathcal{L}(\mathbb{K}^{m},X)$ such that $\tilde{B}\mathbb{K}^{m}\subset D(\mathfrak{A}(t)),$ $\mathfrak{A}(t)\tilde{B}\in\mathcal{L}(\mathbb{K}^{m},X)$ for each $t\geq 0$ and $\mathfrak{B}\tilde{B}=\mathrm{I}_{\mathbb{K}^{m}}=I_{U}.$

Proof.

Since the $nN\times 2nN$-matrix $W_{B}$ has full rank $nN$ there exists a $2nN\times nN$-matrix $S$ such that

[TABLE]

In fact, one can choose $S$ as follows

[TABLE]

Let us write $S=\left[\begin{smallmatrix}S_{11}&S_{12}\\ S_{21}&S_{22}\\ \cdot&\cdot\\ \cdot&\cdot\\ \cdot&\cdot\\ S_{(2nN)1}&S_{(2nN)2}\end{smallmatrix}\right]=:\left[\begin{smallmatrix}\tilde{S}_{1}&\tilde{S}_{2}\end{smallmatrix}\right],$ where $S_{j1},j=1,\cdots,2nN,$ are $1\times m$ matrices.

Next, let $\{e_{j}\}_{j=1}^{2Nn}$ be the standard orthogonal basis in $\mathbb{K}^{2nN}.$ For each $j=1,2,\cdots 2nN$ we take $f_{j}\in H^{N}(a,b;\mathbb{K}^{n})$ such that $\tau(f_{j})=e_{j}$ [37, Lemma A.3], and we define the operator $\tilde{B}\in\mathcal{L}(\mathbb{K}^{m},X)$ by

[TABLE]

Thus $B\in\mathcal{L}(\mathbb{K}^{m},H^{N}((a,b);\mathbb{K}^{n})).$ Furthermore, (84) implies that $W_{B,2}\tilde{S}_{1}=0$ and thus

[TABLE]

for every $u\in\mathbb{K}^{m}.$ We deduce that $\tilde{B}\mathbb{K}^{m}\subset D(\mathfrak{A}(t))$ for all $t\geq 0.$ It follows that $\Sigma(\mathfrak{A}(t),\mathfrak{B},\mathfrak{C})$ is for each $t\geq 0$ a BCO-system on $(L^{2}([a,b];\mathbb{K}^{n}),\mathbb{K}^{m},\mathbb{K}^{d})$. Using (84) once more, we obtain that

[TABLE]

for all $u\in\mathbb{K}^{m}.$ This completes the proof. ∎

Moreover, if in addition the following assumption holds

Assumption 6.3**.**

• •

$nN=m=d$ (and thus $W_{B}=W_{B,1}$ or equivalently $W_{B,2}=0$),

• •

$R=\{R(t)\,|\,t\geq 0\}$ and $J=\{J(t)\,|\,t\geq 0\}$ are bounded and self adjoint operators on $\mathbb{K}^{n}$,

• •

$\operatorname{Re}P_{0}(t,\zeta)\leq 0$ for all $t\geq 0$ and a.e. $\zeta\in[a,b]$,

• •

the matrix $W_{C}$ has full rank,

then we obtain:

Lemma 6.4**.**

Under assumptions 6.3 and 6.1 for each $t\geq 0$ the autonomous port-Hamiltonian system $\Sigma(\mathfrak{A}(t),\mathfrak{B},\mathfrak{C})$ is $(R(t),\mathrm{I},J(t))$-scattering passive if

[TABLE]

Proof.

Using [2, Lemma 3.2.13] we obtain

[TABLE]

Inserting

[TABLE]

into (87) we obtain that

[TABLE]

holds for every $x\in H^{N}([a,b];\mathbb{K}^{n})$, since $ReP_{0}(t,\zeta)\leq 0$. Now the claim follows by Lemma 3.6. ∎

Finally, the assumption on $\mathcal{H}$ ensures that the family of operators $M(t):=\mathcal{H}(t)(\cdot):=\mathcal{H}(t,\cdot)$ as matrix multiplication operators on $L^{2}(a,b;\mathbb{K}^{n})$ satisfies all assumptions of Section 4.3.

Our abstract results in the previous sections hence yield the following main result.

Theorem 6.5**.**

If Assumption 6.1 holds, then the port-Hamiltonian system (76)-(80) is a non-autonomous boundary control and observation system. Furthermore, there exists a unique evolution family $\mathcal{W}$ in $L^{2}([a,b];\mathbb{K}^{n})$ such that for each $x_{0}\in H^{N}((a,b);\mathbb{K}^{n})$ and $u\in C^{2}([a,b];\mathbb{K}^{m})$ with, $W_{B,1}\tau(x_{0})=u(0)$ and $W_{B,2}\tau({x_{0}})=0$ we have

[TABLE]

[TABLE]

is the unique classical solution of (76)-(80). If in addition Assumption 6.3 and (86) hold, then (76)-(80) is $(R,\mathcal{H},J)$-scattering passive and the classical solution $(x,y)$ satisfies the balance inequality

[TABLE]

where $c_{t,s}=\max\{1,\underset{r\in[s,t]}{\max}\|\mathcal{H}(r)\|\}$. Moreover, (76)-(80) is well posed if in addition $J$ is uniformly coercive and $R\in L_{Loc}^{\infty}([0,\infty);\mathcal{L}(\mathbb{K}^{n}))$.

Finally, we give a result on the existence of mild solution of the non-autonomous port-Hamiltonian system. For that we assume that $nN=m=d$. Then it is known [22, Lemma A1] (see also [17, Section 7.3]) that there exist a matrix $V\in\mathbb{K}^{nN\times nN}$ and an invertible matrix $S\in\mathbb{K}^{nN\times nN}$ such that

[TABLE]

with $VV^{*}\geq\mathrm{I}$. Further, we have $\ker W_{B}=\operatorname{Ran}\left[\begin{smallmatrix}I-V\\ -I-V\end{smallmatrix}\right]$ . For each $t\geq 0$, the adjoint operator $A^{*}(t):D(A^{*}(t))\rightarrow X$ of (82)-(83) is given by

[TABLE]

see e.g., [36, Theorem 2.24], [2, Proposition 3.4.3]. We deduce that the domain of $A^{*}(t)$ are time-independent if for instance all matrices $P_{k},k=1,2,\cdots N$ are constant. Thus using Corollary 5.5 we obtain the followin proposition.

Proposition 6.6**.**

Assume that Assumption 6.1 and Assumption 6.3 hold with $P_{k},k=1,2,\cdots N$ are constant and $J$ is uniformly coercive and $R\in L_{Loc}^{\infty}([0,\infty);\mathcal{L}(\mathbb{K}^{n}))$. If (86) holds, then the non-autonomous system (76)-(79) has a unique mild solution.

We closed this section by some examples of physical systems which can be modelled as a non-autonomous port-Hamiltonian system. Then the existence of classical and mild solutions as well as well-posedness can be checked by a simple application of the abstracts results presented in this section. Here we will present just two relevant examples, however various other control systems fit into the framework of port-Hamiltonian system and into the general class of NBCO-systems.

6.1. Vibrating string

Let us consider the model of vibrating string on the compact interval $[a,b]$. The string is fixed at the left end point $a$ and at the right end point $b$ a damper is attached. The Young’s modulus and the mass density of the string are assumed to be time- and spatial dependent. Let us denote by $\omega(t,\zeta)$ the vertical position of the string at position $\zeta\in[a,b]$ and time $t\geq 0$. Then the evolution of the controlled vibrating string can be modelled by a non-autonomous wave equation of the form

[TABLE]

We assume that $k\geq 0$ and $T,\rho\in C^{2}([0,\infty);L^{\infty}(a,b))\cap C_{b}([0,\infty);L^{\infty}(a,b))$ such that for some $m>0$, for a.e  $\zeta\in[a,b]$ and all $t\geq 0$ we have $m^{-1}\leq\rho(t,\zeta),T(t,\zeta)\leq m$, moreover, $\alpha\in C^{1}([0,\infty))$ is strictly positive. We take as state variable the momentum-strain couple $x:=(\alpha\rho\frac{\partial w}{\partial t},\frac{\partial w}{\partial\zeta})$. Then the first equation can be equivalently written as follows

[TABLE]

where

[TABLE]

Indeed, we have

[TABLE]

Moreover, the boundary conditions (94)-(93) with $u=(u_{1},u_{2})=0$ can be equivalently written as follows

[TABLE]

The $2\times 4$ matrix $W_{B}$ has full rank. Next,

[TABLE]

and $W_{B}(t)\Sigma W_{B}^{*}(t)=\begin{bmatrix}4k\alpha(t)&0\\ 0&0\end{bmatrix}\geq 0$. The corresponding matrices $W_{B,1},W_{B,2}$ and the corresponding boundary operator $\mathfrak{B}$ can be defined as follows:

Case $u_{2}=0:$ $W_{B,2}=\begin{bmatrix}0&0&1&0\end{bmatrix}$ and

[TABLE]

Case $u_{2}\neq 0:$ $W_{B,2}=0$ and

[TABLE]

For each $S,V\in\mathbb{K}^{2\times 2}$ such that $S$ is invertible and $VV^{*}\geq I$ we can we take

[TABLE]

as an output of (94)-(93). Thus, we are in the position to apply Theorem 6.5. However, Proposition 6.6 concerning mild solutions can be applied only if $\alpha(t)\equiv\alpha>0$ is constant.

Proposition 6.7**.**

Under the conditions on the physical parameters $T,\alpha,\rho,k$ listed above we have:

1. (1)

The abstract linear system associated with the controlled vibrating string (92), (94) with output (96) yields a non-autonomous boundary control and observation system on $(L^{2}([a,b];\mathbb{K}^{2}),\mathbb{K}^{1},\mathbb{K}^{2})$ if $u_{2}=0$, i.e., when the string is clamped at the end point $a,$ and in $(L^{2}([a,b];\mathbb{K}^{2}),\mathbb{K}^{2},\mathbb{K}^{2})$ if $u_{2}\neq 0.$ 2. (2)

Let $\omega_{0},\omega_{1}\in H^{1}(a,b;\mathbb{K})$ be such that $k\omega_{0}(b)+\omega_{1}(b)=u_{1}(0)$ and $\omega_{0}(a)=u_{2}(0)$. Then (92)-(94) with output equation (96) and initial conditions

[TABLE]

has a unique solution $(\omega,y)$ such that $y\in C([0,\infty);\mathbb{K}^{2})$ and

[TABLE] 3. (3)

Let $u_{2}\neq 0.$ Let $R(t),J(t)$ be self adjoint $2\times 2$-matrices such that $R\in L_{Loc}^{\infty}([0,\infty);\mathcal{L}(\mathbb{K}^{2}))$ and $c_{0}^{-1}\leq J(t)\leq c_{0}$ for all $t\geq 0$ and some constant $c_{0}>0.$ Choose $V,S$ in (96) such that (86) holds for all $t\geq 0.$ Then the linear system associated with the non-autonomous controlled vibrating string (92)-(94) and (96) is a well-posed non-autonomous boundary control and observation system. 4. (4)

Assume that $\alpha(t)\equiv\alpha>0$ is constant such that the assumptions in $(3)$ hold. Let $\omega_{0},\omega_{1}\in L^{2}(a,b;\mathbb{C}).$ Then (92)-(94) with initial conditions

[TABLE]

has a unique (mild) solution $\omega$ such that

[TABLE]

6.2. Timoschenko beam

Consider the following model of the Timoshenko beam with time-dependent coefficient and time dependent boundary control

[TABLE]

[TABLE]

for some positive constants $\alpha_{1},\alpha_{2}\geq 0$. Here $\zeta\in(a,b)$, $t\geq 0$, $w(t,\zeta)$ is the transverse displacement of the beam and $\phi(t,\zeta)$ is the rotation angle of the filament of the beam. We assume that $K$, $\rho$, $EI$, $I_{\rho}\in C^{2}([0,\infty);L^{\infty}(a,b))\cap C_{b}([0,\infty);L^{\infty}(a,b))$ and there exists $m>0$ such that for a.e  $\zeta\in[a,b]$ and all $t\geq 0$ we have

[TABLE]

where $\rho(t,\zeta)$ and $I_{\rho}$ are strictly positive. Moreover, $\tilde{\rho},\tilde{I}_{\rho}\in C^{1}([0,\infty))$ are strictly positive.

Indeed, taking as state variable $x:=(\frac{\partial w}{\partial\zeta}-\phi,\tilde{\rho}\rho\frac{\partial w}{\partial t},\frac{\partial\phi}{\partial\zeta},\tilde{I}_{\rho}I_{\rho}\frac{\partial\phi}{\partial t})$ one can easily see that (97)-(98) can be written as a system of the form (76)-with

$P_{1}=\begin{bmatrix}0&\tilde{\rho}^{-1}&0&0\\ \tilde{\rho}^{-1}&0&0&0\\ 0&0&0&\tilde{I}_{\rho}^{-1}\\ 0&0&\tilde{I}_{\rho}^{-1}&0\\ \end{bmatrix}$, $P_{0}=\begin{bmatrix}0&0&0&-\tilde{I}_{\rho}^{-1}\\ 0&0&0&0\\ 0&0&0&0\\ \tilde{I}_{\rho}^{-1}&0&0&0\\ \end{bmatrix}$ and $\mathcal{H}=\begin{bmatrix}K&0&0&0\\ 0&\rho^{-1}&0&0\\ 0&0&EI&0\\ 0&0&0&I_{\rho}^{-1}\\ \end{bmatrix}$.

The boundary condition can be formulated as follows

[TABLE]

Thus $W_{B}$ has full rank and the corresponding $4\times 8$ matrix $W_{B}(t)$ is given by

[TABLE]

Thus $W_{B}(t)\Sigma W_{B}^{*}(t)=\left[\begin{smallmatrix}0&0&0&0\\ 0&0&0&0\\ 0&0&4\alpha_{1}\tilde{\rho}(t)&0\\ 0&0&0&\alpha_{2}\tilde{I}_{\rho}(t)\end{smallmatrix}\right]\geq 0$. As in Example 6.1, the output equation can be choosing similarly as (96). Thus the above Timoshenko beam fit into the framework of port-Hamiltonian system and thus one obtain a similar results to that presented in Proposition 6.7.