1 Introduction↩︎

\(None\) A fundamental tool for characterizing the robustness of control systems in the presence of disturbances is the notion of input-to-state stability (ISS), first introduced by Sontag and his coauthors for finite-dimensional nonlinear systems described by ordinary differential equations in the late 1980s [1], [2]. Early efforts to extend ISS theory to infinite-dimensional settings, particularly systems governed by partial differential equations (PDEs), date back to the late 2000s and early 2010s; see, e.g., [3]–[9].

The notion of ISS captures the sensitivity of a system even in the presence of nonlinear resonances and provides a qualitative characterization of overshoot behavior when external inputs possess finite energy. As such, it offers a robust framework for PDEs, unifying properties such as bounded-input bounded-output stability, small-input small-error behavior under disturbances, and asymptotic stability in the disturbance-free case. Notably, ISS incorporates the influence of the initial state in a manner fully compatible with classical Lyapunov stability theory, while providing a nonlinear generalization of both finite-gain stability and supremum-norm finite-gain stability. This lays the groundwork for a range of analytical and design tools for PDEs, including small-gain theorems and various characterizations of ISS properties.

In the past decade, a considerable effort has been devoted to providing ISS characterizations of abstract infinite-dimensional systems [4], [6], [10]–[19] and establishing various ISS estimates for specific PDEs subject to different types of disturbances [3], [5]–[7], [9], [12], [16]–[31]. For PDEs, external disturbances typically appear either in the domain or on the boundary of the system domain. It is well known that establishing ISS for PDEs with distributed in-domain disturbances can be achieved by directly extending methods developed for finite-dimensional systems, such as the Lyapunov method [32]. In contrast, the investigation of ISS for PDEs with boundary disturbances is considerably more challenging [25], [32]. This is due to the fact that such systems involve unbounded input operators in their abstract formulation, which poses a significant obstacle to ISS analysis; see, e.g., [7], [10], [11], [19], [25], [32]–[37].

To address the ISS of PDEs with boundary disturbances, various approaches have been developed in recent years, including:

(i) the finite-difference scheme and eigenfunction expansion of solutions, which can be used to establish ISS for PDEs governed by Sturm–Liouville operators [33], [34], [38], [39];

(ii) the semigroup and admissibility method, which can be used to establish ISS for PDEs governed by linear differential operators and associated with admissible inputs [13], [14], [23], [24], [35], [40];

(iii) the classical Lyapunov method (CLM) based on variations of Sobolev embedding inequalities, which can be used to establish ISS for PDEs with either Robin or Neumann boundary disturbances [41]–[43]; this approach was first applied to ISS analysis of parabolic PDEs with boundary disturbances;

(iv) the monotonicity method, which can be used to establish ISS for monotone systems such as parabolic PDEs with Dirichlet boundary disturbances [44];

(v) the De Giorgi iteration, which can be used to establish ISS for parabolic PDEs with either Dirichlet or Robin boundary disturbances [42], [45], [46];

(vi) the Riesz-spectral approach, which can be used to establish ISS for PDEs governed by Riesz-spectral operators [47], [48];

(vii) the maximum principle-based approach, which can be used to establish ISS for parabolic PDEs with different types of boundary disturbances [49]–[52];

(viii) the approximative Lyapunov method, which can be used to establish ISS or integral-ISS in different norms, particularly in the \(L^1\)-norm, for parabolic PDEs with various types of boundary disturbances [53], [54].

Note that the methods in ([40ii41]), ([40i41]), and ([40iii41]), which rely on linear operator theory, eigenfunction expansions, or spectral analysis, are particularly well suited for one-dimensional (\(1\)-D) linear PDEs, whereas the other methods can be applied to either linear or nonlinear PDEs defined on higher dimensional domains. The methods presented in ([40vi41]) and ([40vii41]) are effective for establishing global ISS of PDEs in the absence of Dirichlet boundary disturbances, however, they are not readily extendable to PDEs with Dirichlet disturbances. Note also that extending the methods in ([40iv41]), ([40v41]), and ([40viii41]) to PDEs beyond parabolic equations is extremely challenging. Overall, owing to the complexity and diversity of PDEs, each method has its own advantages and limitations. In particular, the applicability of the non-Lyapunov methods in ([40ii41]), ([40i41]), and ([40iv41])-([40viii41]) depends heavily on the structure of the PDEs under consideration. Consequently, developing a unified ISS analysis framework applicable to a broad class of PDEs remains a significant challenge.

Given the dominant role of Lyapunov methods in the stability analysis of control systems, the authors have, over the past few years, been dedicated to developing unified tools for the ISS analysis of PDEs within the Lyapunov framework. This effort led to the introduction of the notion of

1. generalized ISS-Lyapunov functional (GISS-LF) [55]. In particular, it has been successfully applied to establish global ISS for the Burgers’ equation with Dirichlet boundary disturbances [55], thereby providing a positive answer to [32].

Furthermore, using the proposed method, a generalized ISS-Lyapunov theorem was established for abstract control systems, and a GISS-LF was explicitly constructed for a class of parabolic PDEs with Dirichlet boundary disturbances in [56]. A key idea underlying this approach is the construction of a positive-semidefinite Lyapunov candidate that explicitly incorporates external disturbances, thereby offering a new perspective for handling nonhomogeneous Dirichlet boundary terms in the ISS analysis of PDEs.

It is worth noting that an advantage of the application of GISS-LF lies in its capacity to simultaneously accommodate Dirichlet boundary disturbances alongside other types of disturbances, while enabling ISS analysis of PDEs in a manner directly analogous to the CLM. In particular, this generalized Lyapunov method (GLM) can proceed in a systematic manner by involving solely integration by parts (or the Green’s formula for higher dimensional PDEs) and fundamental differential inequalities such as the Agmon’s, Poincaré’s, Jensen’s, Young’s, and Gronwall’s inequalities, thereby circumventing the need to verify admissibility conditions required in ([40i41]) for input operators. Moreover, it obviates the need for spectral analysis or other sophisticated analytical tools typically employed in ([40ii41]) and ([40iii41]) and hence, brings convenience to the ISS analysis of concrete PDEs. Nevertheless, at present, the applicability of the GLM remains largely confined to parabolic PDEs, especially those defined on \(1\)-D domains and with linear boundary conditions (see, e.g., [55], [57]–[59]), and its extension to other types of PDEs has not yet been reported.

This tutorial aims to illustrate how the GLM–although alternative approaches may also be applicable–can be utilized to establish ISS for a wider range of PDEs, in particular, the most representative ones, including higher dimensional parabolic PDEs under general boundary conditions, \(1\)-D first order nonlinear hyperbolic PDEs, and \(1\)-D second order linear hyperbolic PDEs, i.e., the wave equations.

In the rest of the paper, we first introduce some basic notations used in this paper. Section 2 revisits, within the framework of abstract systems, the classical ISS-Lyapunov functionals (CISS-LFs) and the corresponding ISS-Lyapunov theorem, illustrating the effectiveness, as well as limitations, of CISS-LFs in the ISS analysis of PDEs with boundary disturbances. We then recall the concept of GISS-LFs and the associated ISS-Lyapunov theorem, and present a proposition that is particularly well-suited for concrete PDEs. To demonstrate the application of the GLM in the ISS analysis of PDEs, we consider three different types of PDEs:

• in Section 3, for a class of nonlinear parabolic PDEs defined on a higher dimensional domain with mixed nonlinear boundary disturbances, we show how to construct a GISS-LF and establish ISS estimates in arbitrary \(L^q\)-norms for \(q\in[2,\infty]\);

• in Section 4, for a class of \(1\)-D first order hyperbolic equations with boundary disturbances, we show how to construct a GISS-LF and establish both local and global ISS estimates in arbitrary \(L^q\)-norms for \(q\in[2,\infty]\);

• in Section 5, for a class of wave equations with boundary damping and disturbances, we show how to apply the GLM to derive ISS estimates in arbitrary \(L^q\)-norms for \(q\in[2,\infty]\).

In Section 6, we provide some concluding remarks, followed by appendices that provide two lemmas and proofs of several properties of a truncation function used throughout the paper.

Notation

Let \(\mathbb{R}:= (-\infty,+\infty)\), \(\mathbb{R}_{\geq0}:= [0,+\infty)\), \(\mathbb{R}_{>0}:= (0,+\infty)\), and \(\mathbb{R}_{\leq0}:= (-\infty,0]\).

For \(p\in[1,\infty)\) and an open bounded domain \(\Omega \subset\mathbb{R}^N (N\geq 1)\), the norms of a function \(g\in L^p(\Omega )\) and \(L^{\infty}(\Omega)\) are defined by \(\|g\|_{L^p(\Omega)}:=\left(\int_{\Omega}\left|g(y)\right|^p\mathrm{d}y\right)^{\frac{1}{p}}\) and \(\|g\|_{L^{\infty}(\Omega)}:= \mathop{\mathop{\mathrm{ess\,sup}}}\limits_{y\in\Omega}\left|g(y)\right|\), respectively.

Let \(C(\mathbb{R}_{\geq 0}; \mathbb{R}_{\geq 0} )\) (or \(C(\mathbb{R}_{\geq 0}\times \mathbb{R}_{\geq 0}; \mathbb{R}_{\geq 0})\)) be the set of continuous functions from \(\mathbb{R}_{\geq 0}\) (or \(\mathbb{R}_{\geq 0} \times \mathbb{R}_{\geq 0}\)) to \(\mathbb{R}_{\geq 0}\). For a linear normed space \(Y\), let \(PC(\mathbb{R}_{\geq 0};Y)\) be the set of all piecewise-right continuous functions \(f\) from \(\mathbb{R}_{\geq 0}\) to \(Y\) satisfying \(\|f \|_{PC(\mathbb{R}_{\geq 0};Y)}:=\sup_{t\in \mathbb{R}_{\geq 0}}\|f(t)\|_{Y}<\infty\).

Denote by \(f\circ g\) the composition of the functions \(f\) and \(g\), i.e., \(f\circ g (\cdot):=f(g(\cdot))\).

Let \[\begin{align} \mathcal{P}:=&\{\gamma \in C(\mathbb{R}_{\geq 0};\mathbb{R}_{\geq 0})|\gamma(0)=0, \gamma(s)>0,\forall s\in \mathbb{R}_{>0}\},\\ \mathcal{K}:=&\{\gamma \in \mathcal{P}|\gamma\;\text{is\;strictly\;increasing}\},\\ \mathcal{K}_{\infty}:=&\{\gamma \in \mathcal{K}| \lim\limits_{s\to+\infty}\gamma(s)=+\infty\},\\ \mathcal{L}:=&\{\gamma \in C(\mathbb{R}_{\geq 0};\mathbb{R}_{\geq 0})|\gamma\;\text{is\;strictly\;decreasing\;with} \;\lim\limits_{s\to+\infty}\gamma(s)=0\},\\ \mathcal{K}\mathcal{L}:=& \{\beta\in C(\mathbb{R}_{\geq 0}\times \mathbb{R}_{\geq 0};\mathbb{R}_{\geq 0})|\beta (\cdot,t) \in \mathcal{K}, \forall t \in \mathbb{R}_{\geq 0}; \;\beta(s,\cdot)\in \mathcal{L}, \forall s \in {{\mathbb{R}_{>0}}}\}. \end{align}\]

2 Overview on CLM and GLM↩︎

In this section, we provide a review of GISS-LF and the associated ISS-Lyapunov theorem for control systems under an abstract form. To enable a meaningful comparison, we begin with a brief recap of the essential concepts of CLM for control systems as well as the construction of (classical) ISS-Lyapunov functionals (ISS-LFs) for PDEs.

2.1 A brief review of CLM↩︎

In this subsection, we review the concept of control systems, together with the notions of ISS and classical ISS-LFs, and the ISS-Lyapunov theorem.

We begin by defining forward complete control systems evolving on a Banach space \(X\).

The following definition is concerned with the forward complete control systems.

For a real-valued function \(g:\mathbb{R}_{\geq 0}\to \mathbb{R}\) define the right-hand upper Dini derivative at \(t\in \mathbb{R}_{\geq 0}\) by \[\begin{align} D^+g(t):=\limsup_{\tau\to 0^+}\frac{g(t+\tau)-g(t)}{\tau}. \end{align}\]

Let \(x\in X\) and \(W\) be a real-valued function defined in a neighborhood of \(x\). The Lie derivative of \(W\) at \(x\) corresponding to the input \(u\in\mathcal{U}\) along the trajectory of a control system \(\Sigma\) is defined by \[\begin{align} \dot{W}_u(x):=&D^+W(\phi(\cdot,x,u))\Big|_{t=0} = \limsup_{t\to 0^+}\frac{1}{t}(W(\phi(t,x,u))-W(x)). \end{align}\] We also denote \(\dot{W}_u(x)\) by \(\dot{W} (x)\) for notational simplicity.

The concept of ISS for control systems is presented below.

Note that under the conditions

• \(\mathcal{U}:=PC(\mathbb{R}_{\geq 0};U)\),

• \(\Sigma=(X,\mathcal{U},\phi)\) is a forward complete control system with \(x \equiv 0\) being its equilibrium point,

the causality property of a control system provides an equivalent definition of ISS, which is convenient to use for PDEs.

In the following, we present the concept of (classical) ISS-LF for control systems.

For a forward complete control system, the (classical) ISS Lyapunov theorem can be formulated as below.

The ISS Lyapunov theorem characterizes ISS through the existence of an ISS-LF, thereby offering a systematic framework for analyzing how external inputs affect system stability. It establishes a bridge between Lyapunov-based stability analysis and the robustness of nonlinear systems with respect to (w.r.t.) external inputs, serving as a foundational tool for controller design and stability verification in nonlinear control theory.

Since our focus is on demonstrating how to perform ISS analysis within the Lyapunov framework, rather than addressing the existence and regularity of solutions, we assume throughout this paper that

1. all functions involved in PDEs are sufficiently smooth. Consequently, each PDE considered in this paper admits a classical solution, ensuring that all subsequent calculations in deriving ISS estimates are meaningful.

Notably, if weak or generalized solutions are considered without a fine regularity assumption on the functions, the ISS can still be assessed by using a density argument [60].

In the following, we demonstrate the construction of an ISS-LF for the heat equation by means of an example.

Example 1 shows that the Lyapunov functional of the disturbance-free system is sufficient for the ISS analysis in the presence of either in-domain disturbances or Robin boundary disturbances. However, when \(w(0,t)=g(t)\) with \(g(t)\) representing disturbances, namely, a Dirichlet boundary disturbance is present, the boundary term \(w(0,t)w_y(0,t)\) in ?? cannot be handled. Thus, the application of the CLM meets obstacles.

2.2 A brief review of GLM↩︎

To address the limitations of the CLM, the notion of generalized Lyapunov functionals has been introduced recently in [55], [56], which offers more flexibility for constructing Lyapunov candidates in the ISS analysis of PDEs with boundary disturbances.

In the sequel, we always assume that conditions (H1) and (H2) are satisfied.

A generalized ISS Lyapunov theorem was proved in [56].

The following result, which is a direct consequence of Theorem 2, is suited for ISS analysis of specific PDEs involving only first order time derivatives, in particular, parabolic equations and first order hyperbolic equations.

Furthermore, noting that \(\mathcal{U}:=PC(\mathbb{R}_{\geq 0};U)\) (see condition (H1)), we can prove the following result, which is more convenient for conducting ISS analysis of PDEs.

Let \(\widehat{V}(x):= V(x- {z_{u,T}})+V(-x-{z_{u,T}})\). Applying the comparison principle ([62]) to ?? and using condition (i), we deduce that \[\begin{align} \widehat{V}(\phi(T,x,u)) \leq & e^{-\lambda_0 T} \widehat{V}(x)\notag\\ \leq & e^{-\lambda_0 T}\left(\psi_1(\|x-z_{u,T}\|_X)+\psi_1(\|-x-z_{u,T}\|_X)\right) \\ \leq & 2 e^{-\lambda_0 T}\left(\psi_1(\|x \|_X+ \| z_{u,T}\|_X)\right)\\ \leq &2 e^{-\lambda_0 T}\left(\psi_1(2\|x \|_X) + \psi_1(2\| z_{u,T}\|_X)\right)\notag\\ \leq & 2 e^{-\lambda_0 T }\psi_1(2\|x \|_X) +2 \psi_1(2\| z_{u,T}\|_X). \end{align}\] Using condition (ii) and the assumption that \(\|z_{u,T}\|_{X} \leq \rho\left(\sup_{t\in(0,T)}\|u(t)\|_{ {U}}\right)\), we obtain \[\begin{align} \psi_2\left(\| \phi(T,x,u) \|_X\right) \leq& \widehat{V}(\phi(T,x,u)) + \varphi\left( \|z_{u,T} \|_{X} \right) \\ \leq &2 e^{-\lambda_0 T} \psi_1(2\|x \|_X) + 2\psi_1(2\| z_{u,T}\|_X) + \mu\left( \|z_{u,T} \|_{X} \right)\notag\\ \leq & 2 e^{-\lambda_0 T} \psi_1(2\|x \|_X) + 2 \psi_1\left(2\rho\left(\sup_{t\in(0,T)}\|u(t)\|_{ {U}}\right)\right) + \mu\left(\rho\left(\sup_{t\in(0,T)}\|u(t)\|_{ {U}}\right)\right), \end{align}\] which, along with the arbitrariness of \(T\), gives the desired result. \(\blacksquare\)

It is worth noting that, due to the diversity and complexity of PDEs, constructing an explicit GISS-LF within a unified framework is extremely challenging, and such constructions often require a case-by-case treatment. Nevertheless, for the most representative classes of PDEs, in particular, parabolic equations, first order hyperbolic equations, and wave equations, GISS-LFs can be constructed in a relatively unified manner. For these PDEs, a key technique lies in the use of Stampacchia’s truncation to construct truncation functions and disturbance-dependent energy functionals \(V(h_i(x,u))\). The number of such energy functionals is determined by the number of state variables when the PDE is reformulated as an abstract system. Details of constructing GISS-LFs for these three classes of PDEs are provided in Sections 3, 4, and 5, respectively.

3 ISS analysis for \(N\)-D parabolic equations via GLM↩︎

In this section, we construct a GISS-LF for higher dimensional parabolic PDEs with different disturbances under nonlinear boundary conditions.

In the sequel, let \(N\geq 1\), and let \(\Omega\) be an open bounded domain in \(\mathbb{R}^N\) with smooth (when \(N>1\)) boundary denoted by \(\partial \Omega\). At each point \(y\in\partial \Omega\), the unit outward normal vector is denoted by \(\boldsymbol{\nu}(y)\). For a differentiable function \(h\) defined over \(\Omega\), \(\nabla h\) denotes the gradient of \(h\). The notation \(\operatorname{d}\!{S}\) denotes the area element of \(\partial \Omega\). The notation \(\mathscr{L}^N(\Omega)\) (or \(\mathscr{L}^{N-1}(\Gamma)\)) denotes the \(N\)- (or \((N-1)\)-) dimensional Lebesgue measure of \(\Omega\) (or a set of \(\Gamma\in \partial \Omega\)). The notation \(\Delta\) represents the standard Laplace operator. For \(T>0\), let \(Q_T:=\Omega\times(0,T)\) and \(Q_{\infty}:=\Omega\times\mathbb{R}_{>0}\).

We consider the following higher dimensional nonlinear parabolic equation with both distributed in-domain and mixed boundary disturbances: \[\label{parabolicPDE-d} \begin{align} w_t(y,t) =&\operatorname{div}\!\; (a (y,t)\nabla w(y,t) )-c(y,t) \phi(w (y,t)) -{h(y,t,w(y,t) )} \notag\\ & +f(y,t),(y,t)\in Q_{\infty},\\ w(y,t) =&d_1(y,t),(y,t)\in \Gamma_1\times \mathbb{R}_{>0},\\ a(y,t) \frac{\partial w(y,t)}{\partial \boldsymbol{\nu}} =&-\varphi(w(y,t)) +d_2(y,t),(y,t)\in \Gamma_2\times \mathbb{R}_{>0},\\ w(y,0)=&w_0(y),y\in\Omega, \end{align}\tag{1}\] where

1. \(\Gamma_1\) and \(\Gamma_2\) are smooth and disjoint boundary sets (\((N-1)\)-dimensional hypersurfaces) satisfying (see, e.g., Fig. 2) \[\begin{align} \Gamma_1\cup\Gamma_2=& \partial \Omega,\\ 0\leq \mathscr{L}^{N-1}(\Gamma_1)\leq &\mathscr{L}^{N-1}(\partial \Omega); \end{align}\]

2. \(a,c :\Omega\times\mathbb{R}_{> 0}\to {\mathbb{R}_{> 0}}\) are smooth functions satisfying \[\begin{align} a(y,t ) \geq a_0 ,\forall (y,t )\in\Omega\times\mathbb{R}_{> 0},\\ c(y,t ) \geq c_0 ,\forall (y,t )\in\Omega\times\mathbb{R}_{> 0}, \end{align}\] with positive constants \(a_0\) and \(c_0\);

3. \(\phi: \mathbb{R} \to\mathbb{R}\) is a smooth and strictly increasing function satisfying \[\tag{2} \begin{align} \phi(v)v\geq & 0, \forall v\in \mathbb{R} , \\ \phi(-v) \leq &-\phi(v),\forall v\in \mathbb{R}_{\geq 0} ,\tag{3}\\ \phi'(v)\geq& 1,\forall v\in \mathbb{R} ;\tag{4} \end{align}\]

4. \(h:\Omega\times\mathbb{R}_{> 0}\times\mathbb{R}\to\mathbb{R}\) is a smooth function satisfying \[\begin{align} h(y,t,v)v\geq 0,\forall (y,t,v)\in\Omega\times\mathbb{R}_{> 0}\times\mathbb{R} ;\label{condition-h} \end{align}\tag{5}\]

5. \(\varphi: \mathbb{R} \to\mathbb{R}\) is a smooth and strictly increasing function satisfying \[\tag{6} \begin{align} \varphi(v)v\geq &0,\forall v\in \mathbb{R} ,\tag{7}\\ \varphi(-v) \leq &-\varphi(v),\forall v\in \mathbb{R}_{\geq 0} ;\tag{8} \end{align}\]

6. \(f,d_1,d_2:\Omega\times\mathbb{R}_{\geq 0}\to \mathbb{R}\) are smooth functions, representing distributed in-domain and boundary disturbances, respectively;

7. \(w_0:\Omega \to \mathbb{R}\) is a smooth function, representing the initial data.

We apply the technique of Stampacchia’s truncation to construct a GISS-LF. For any \(p\in(1,\infty)\), let \[\tag{9} \begin{align} g(s):=& \left\{\begin{aligned} & s^p,\;s\in \mathbb{R}_{\geq 0},\\ & 0,\;s\in \mathbb{R}_{< 0}, \end{aligned}\right.\tag{10}\\ G(s):=&\int_0^sg(\tau)\operatorname{d}\!{\tau} = \left\{\begin{align} & \frac{1}{p+1}s^{p+1},\;s\in \mathbb{R}_{\geq 0},\\ & 0,\;s\in \mathbb{R}_{< 0}.\end{align}\right. \end{align}\]

Note that \(g\) and \(G\) have the following properties, which will be extensively used in this paper:

1. \(G(s)=g(s)= 0\) for all \(s\in \mathbb{R}_{\leq 0}\);

2. \(G(s)=\frac{1}{p+1}g(s)s\) for all \(s\in \mathbb{R}\);

3. \(g'(s)\geq 0\) for all \(s\in \mathbb{R}\);

4. \(G(s )\leq G(s+\tau)+G(s-\tau)\) for all \(s,\tau\in \mathbb{R}\);

5. \(G(|s|+\tau )\leq G(s+\tau)+G(-s+\tau)\) for all \(s,\tau\in \mathbb{R}\);

6. \(G(s+\tau)\leq 2^p(G(s)+G(\tau))\) for all \(s,\tau\in \mathbb{R}\);

7. \(G(|s|)\leq 2^p (G(s+\tau-M)+G(s-\tau-M) +G(-s+\tau-M)+G(-s-\tau-M)) +2^pG(M)\) for all \(s,\tau,M\in\mathbb{R}\);

8. Young’s inequality: \[\begin{align} g(s)\tau\leq \varepsilon G(s)+\left(\frac{p}{\varepsilon}\right) ^{ p}G(\tau),\forall s,\tau\in \mathbb{R} , \varepsilon\in \mathbb{R}_{>0}. \end{align}\]

The proofs of properties (G1)-(G8) are provided in Appendix 8.

Since \(\phi\) and \(\varphi\) are strictly increasing, their inverses, denoted by \(\phi^{-1}\) and \(\varphi^{-1}\), respectively, exist. Moreover, \(\phi^{-1},\varphi^{-1}\in\mathcal{K}\). For any \(T>0\), let \[\begin{align} M:=&\phi^{-1}\left(\frac{1}{c_0} \|f\|_{\infty,T}\right) + \|d_1\|_{\infty,T} +\varphi^{-1}\left(\|d_2\|_{\infty,T}\right), \end{align}\] where, for notational simplicity, we denote \[\begin{align} \|f\|_{\infty,T}:=&\sup_{(y,t)\in Q_T}|f(y,t)| ,\\ \|d_1\|_{\infty,T}:=& \sup_{(y,t)\in \Gamma_1\times(0,T)}|d_1(y,t)| ,\\ \|d_2\|_{\infty,T}:=& \sup_{(y,t)\in \Gamma_2\times(0,T)}|d_2(y,t)|. \end{align}\] It is clear that \[\begin{align} \|f\|_{\infty,T} \leq& c_0\phi\left(M\right), \\ \|d_1\|_{\infty,T}\leq& M,\\ \|d_2\|_{\infty,T}\leq& \varphi(M). \end{align}\]

Let \[\begin{align} V(w):=& \int_{\Omega}G(w(y,t))\operatorname{d}\!{y}, \end{align}\] and \[\begin{align} \widehat{V}(w):= V (w -M) +V (-w -M).\label{GISS-LF-V} \end{align}\tag{11}\] Note that \(\widehat{V}(w)= 0\) for \(|w|\leq M\). Thus, \(\widehat{V}(w)\) is positive semidefinite.

For \(\widehat{V}(w)\) and system 1 , we have the following result.

We employ Proposition 2 to prove this proposition.

First, it is clear that the conditions (i) and (ii) of Corollary 1 are fulfilled with (see [56]) \[\begin{align} \psi_1(s):=&\frac{1}{p+1} s^{p+1} ,\forall s\in \mathbb{R}_{\geq 0},\\ \psi_2(s):=&\frac{1}{ 2^{p}(p+1)}s^{p+1},\forall s\in \mathbb{R}_{\geq 0},\\ \mu(s):=&2s^{p+1},\forall s\in \mathbb{R}_{\geq 0}. \end{align}\] We shall verify the condition (iii’) of Proposition 2. Indeed, for any \(t\in(0,T)\), using the Green’s formula (see [63]), we deduce that \[\begin{align} \dot{V}(w-M) =& \frac{\diff~}{\operatorname{d}\!{t}} \int_{\Omega} G(w-M) \operatorname{d}\!{y} \notag\\ = & \int_{\Omega} g(w-M)w_t\operatorname{d}\!{y} \notag\\ = & \int_{\Omega} g(w-M)(\operatorname{div}\!\; (a w )-c \phi(w) -{h(y,t,w )}+f )\operatorname{d}\!{y}\notag\\ = & \int_{\Gamma_1} ag(w-M)\frac{\partial w}{\partial \boldsymbol{\nu}}\operatorname{d}\!{S}+ \int_{\Gamma_2} ag(w-M)\frac{\partial w}{\partial \boldsymbol{\nu}}\operatorname{d}\!{S} - \int_{\Omega}a g'(w-M)|\nabla w|^2 \operatorname{d}\!{y} \notag\\ &- \int_{\Omega} g(w-M) (c \phi(w)-f) \operatorname{d}\!{y} - \int_{\Omega} g(w-M) h(y,t,w) \operatorname{d}\!{y}.\label{parabolic-14} \end{align}\tag{12}\]

Using the boundary conditions, the definition of \(M\), 10 , and the monotonicity of \(\varphi\), we have \[\begin{align} g(w-M)=&0 \; \text{on}\; \Gamma_1\times (0,T) \end{align}\] and \[\begin{align} g(w-M)\frac{\partial w}{\partial \boldsymbol{\nu}} =&-g(w-M)(\varphi (w) -d_2)\notag\\ \leq& -g(w-M)\left( \varphi (w) -\|d_2\|_{\infty,T}\right)\notag\\ \leq& -g(w-M)\left( \varphi (w) -\varphi (M)\right)\notag\\ \leq& 0 \; \text{on}\; \Gamma_2\times (0,T). \end{align}\] The condition 5 ensures that \[\begin{align} h(y,t,v) \geq &0,\forall (y,t,v)\in\Omega\times(0,T)\times\mathbb{R}_{\geq 0} , \end{align}\] which, along with the definition \(g\), ensures that \[\begin{align} - g(w-M) h(y,t,w) \leq &0,\forall (y,t,w)\in\Omega\times(0,T)\times\mathbb{R} . \end{align}\] In addition, we deduce that \[\begin{align} -\int_{\Omega} g(w-M)(c\phi(w)- f)\operatorname{d}\!{y} \leq& -\int_{\Omega} g(w-M)\left(c\phi(w)- \|f\|_{\infty,T}\right)\operatorname{d}\!{y}\notag\\ \leq& -\int_{\Omega} g(w-M)\left(c\phi(w)- c_0\phi\left(M\right)\right)\operatorname{d}\!{y}\notag\\ \leq& -c_0\int_{\Omega} g(w-M)\left( \phi(w)- \phi\left(M\right)\right)\operatorname{d}\!{y}\notag\\ =& -c_0\int_{\Omega} g(w-M)\phi'(v)\left( w- M\right)\operatorname{d}\!{y}\notag\\ \leq & -c_0\int_{\Omega} g(w-M)(w- M)\operatorname{d}\!{y}\quad (\text{by}\;\eqref{condition-phi})\notag\\ =&-c_0(p+1)\int_{\Omega} G(w-M) \operatorname{d}\!{y}, \quad (\text{by}\;\boldsymbol{(G2)}) \end{align}\] where \(v\) is a function between \(w\) and \(M\).

Then, in view of (G3), for any \(t\in(0,T)\), we get \[\begin{align} \dot{V}(w-M) \leq & -c_0(p+1) \int_{\Omega} G(w-M) \operatorname{d}\!{y}\notag \\ =& -c_0(p+1) V(w-M).\label{parabolic-16} \end{align}\tag{13}\]

Now let \(\widetilde{w}:=-w\), which satisfies \[\label{parabolicPDE-d39} \begin{align} \widetilde{w}_t(y,t) =&\operatorname{div}\!\; (a (y,t)\nabla \widetilde{w}(y,t) ) +c(y,t) \phi(-\widetilde{w} (y,t)) +{h(y,t,-\widetilde{w}(y,t) )}\notag\\ & -f(y,t),(y,t)\in Q_{\infty},\\ \widetilde{w}(y,t) =&-d_1(y,t),(y,t)\in \Gamma_1\times \mathbb{R}_{>0},\\ a(y,t) \frac{\partial \widetilde{w}(y,t)}{\partial \boldsymbol{\nu}} =&\varphi(-\widetilde{w}(y,t)) -d_2(y,t),(y,t)\in \Gamma_2\times \mathbb{R}_{>0},\\ \widetilde{w}(y,0)=&-w_0(y),y\in\Omega, \end{align}\tag{14}\]

Note that \[\begin{align} g(\widetilde{w}-M)=&0 \; \text{on}\; \Gamma_1\times (0,T) \end{align}\] and \[\begin{align} g(\widetilde{w}-M)\frac{\partial \widetilde{w}}{\partial \boldsymbol{\nu}} =& g(\widetilde{w}-M)(\varphi (-\widetilde{w}) -d_2)\notag\\ \leq& g(\widetilde{w}-M)\left( \varphi (-\widetilde{w}) +\|d_2\|_{\infty,T}\right)\notag\\ \leq& g(\widetilde{w}-M)\left( \varphi (-\widetilde{w}) +\varphi (M)\right)\; \text{on}\; \Gamma_2\times (0,T). \end{align}\] Furthermore, in view of the nonnegativity of \(g\) and the degeneracy property (G1), the condition of \(\varphi\) (see 8 ) and the increasing monotonicity of \(\varphi\) imply that \[\begin{align} g(\widetilde{w}-M)\frac{\partial \widetilde{w}}{\partial \boldsymbol{\nu}} \leq& g(\widetilde{w}-M)\left( \varphi (-\widetilde{w}) +\varphi (M)\right) \notag\\ \leq & g(\widetilde{w}-M)\left( -\varphi (\widetilde{w}) +\varphi (M)\right)\\ \leq& 0 \; \text{on}\; \Gamma_2\times (0,T). \end{align}\]

Similarly, it holds that \[\begin{align} \int_{\Omega} g(\widetilde{w}-M)(c\phi(-\widetilde{w})+ f)\operatorname{d}\!{y} \leq& \int_{\Omega} g(\widetilde{w}-M)\left(-c\phi(\widetilde{w})+ c_0\phi\left(M\right)\right)\operatorname{d}\!{y}\notag\\ \leq& -c_0\int_{\Omega} g(\widetilde{w}-M)\left( \phi(\widetilde{w})- \phi\left(M\right)\right)\operatorname{d}\!{y}\notag\\ =& -c_0\int_{\Omega} g(\widetilde{w}-M)\phi'(v)\left( \widetilde{w}- M\right)\operatorname{d}\!{y}\notag\\ \leq &-c_0(p+1)\int_{\Omega} G(\widetilde{w}-M) \operatorname{d}\!{y},\forall t\in (0,T), \end{align}\] where \(v\) is a function between \(\widetilde{w}\) and \(M\).

In addition, due to the fact that \(h(y,t,v) \geq 0\) for all \((y,t,v)\in\Omega\times(0,T)\times\mathbb{R}_{\leq 0}\), we have \[\begin{align} g(\widetilde{w}-M) h(y,t,-\widetilde{w}) \leq &0,\forall (y,t,w)\in\Omega\times(0,T)\times\mathbb{R} . \end{align}\] Analogous to 13 , for any \(t\in(0,T)\), we have \[\begin{align} \dot{V}(\widetilde{w}-M) \leq & -c_0(p+1) \int_{\Omega} G(\widetilde{w}-M) \operatorname{d}\!{y} \notag\\ =& -c_0(p+1) V(\widetilde{w}-M), \label{parabolic-17} \end{align}\tag{15}\] which, along with 13 , indicates that the condition (iii’) of Proposition 2 is fulfilled with \(x:=w\), \(u:=(f,d_0,d_1)\), \(z_{u,T}:=M\), and \(\rho\left(\sup_{t\in(0,T)}\|u(t)\|_{ {U}}\right):=M\).

Applying Proposition 2 and Remark 6, we deduce that the functional \(\widehat{V} (w)\) defined by 11 is a GISS-LF for system 1 and hence, system 1 is ISS in the \(L^{p+1}\)-norm with arbitrary \(p\in (1,\infty)\).

Furthermore, we deduce by ?? that \[\begin{align} \psi_2\left( \| w(\cdot,T) \|_{L^{p+1}(\Omega)}\right)\leq &2 e^{-c_0(p+1)T}\psi_1\left(2\| w_0 \|_{L^{p+1}(\Omega)}\right) + 2\psi_1(2M)+\mu(M), \end{align}\] which implies that \[\begin{align} \| w(\cdot,T) \|_{L^{p+1}(\Omega)} \leq & \Big( 4 \| w_0 \|_{L^{p+1}(\Omega)} e^{-c_0 T} + 2M+ \left( p+1 \right)^{\frac{1}{p+1}}2M\Big)\notag\\ \leq& 4 \| w_0 \|_{L^{p+1}(\Omega)} e^{-c_0 T}+8M. \end{align}\] Thus, system 1 is ISS in the \(L^{p+1}\)-norm.

Furthermore, due to the arbitrariness of \(p\in(1,\infty)\), we conclude that system 1 is ISS in the \(L^{q}\)-norm for any \(q\in[2,\infty]\), having the estimate ?? . \(\blacksquare\)

4 ISS analysis for \(1\)-D first order hyperbolic equations via GLM↩︎

In this section, we employ the GLM to establish ISS estimates for the following \(1\)-D hyperbolic equation: \[\label{first32hyperbolic32PDE} \begin{align} \rho_t(y,t)+\left(\lambda(W(t))\rho(y,t)\right)_y=&0, (y,t)\in (0,1)\times\mathbb{R}_{>0},\\ \rho(0,t)-k\rho(1,t)=&d(t),t\in \mathbb{R}_{>0},\\ \rho(y,0)=&\rho_0(y),y\in(0,1), \end{align}\tag{16}\] where \(\lambda\in C^1(\mathbb{R};\mathbb{R}_{>0})\), \(k\in(-1,1)\) is a constant, \(d,\rho_0\) are smooth functions, and \(W(t):=\int_0^1\rho(y,t)\operatorname{d}\!{y}\).

Equation 16 is commonly used to model semiconductor manufacturing systems; see, e.g., [64]–[66]. The key feature of this PDE is characterized by the velocity function \(\lambda\) depending on the total mass \(W(t)\).

We propose either one of the following assumptions on \(\lambda\):

1. \(\lambda(s)\geq \lambda_0\) for all \(s\in \mathbb{R}\), where \(\lambda_0\) is a positive constant;

2. \(\lambda(s)\geq \lambda(|s|)\) for all \(s\in \mathbb{R}\) and \(\lambda(s)\) is decreasing in \(s\in \mathbb{R}_{\geq 0}\).

Typical forms of \(\lambda(s)\) satisfying (A2) include \[\begin{align} \lambda(s):=Ae^{-s^2}+B,\forall s\in \mathbb{R}_{\geq 0} , \end{align}\] and \[\begin{align} \lambda(s):=\frac{A}{B+s},\forall s\in \mathbb{R}_{\geq 0} , \end{align}\] where \(A\) and \(B\) are positive constants.

For any \(p\in(1,\infty)\), let \(g\) and \(G\) be defined by 9 . For any \(T>0\), let \[\begin{align} M:= \frac{1}{1-|k|}\sup_{t\in (0,T)}|d (t)| = \left\{\begin{aligned} & \frac{1}{1-k}\sup_{t\in (0,T)}|d (t)|,\;k\in[0,1),\\ & \frac{1}{1+k}\sup_{t\in (0,T)}|d (t)|,\;k\in(-1,0). \end{aligned}\right. \end{align}\]

Let \[\begin{align} V(\rho):= \int_0^1e^{-r y}G(\rho(y,t))\operatorname{d}\!{y}, \end{align}\] where \(r\) is a positive constant satisfying \[\begin{align} 0< r\leq (p+1)\ln \frac{1}{|k|}.\label{condition-sigma} \end{align}\tag{17}\] Let \[\begin{align} \widehat{V}(\rho):= V (\rho -M) +V (-\rho -M).\label{GISS-LF-V-2} \end{align}\tag{18}\]

The first result for system 16 is regarded as a global ISS property.

Using the GLM, we may establish local ISS estimates for system 16 with a general \(\lambda\).

Now, we employ the GLM to prove Proposition 4 and Proposition 5, respectively.

We proceed the proof with three steps.

Step 1: We show that the functional \(\widehat{V} (\rho)\) defined by 18 is a GISS-LF for system 16 .

First, analogous to [56], it is direct to verify that the conditions (i) and (ii) of Corollary 1 are fulfilled. Next, for any \(t\in(0,T)\), we claim that \[\begin{align} \dot{\widehat{V}}(\rho) \leq& -r \lambda(W)\widehat{V}(\rho) \tag{19}\\ \leq& -r \lambda_0\widehat{V}(\rho) \forall t\in (0,T), \tag{20} \end{align}\] which, along with the conditions (i) and (ii) of Corollary 1, ensures that \(\widehat{V} (\rho)\) is a GISS-LF.

Now we prove 20 . Indeed, for any \(t\in(0,T)\), using integrating by parts, we obtain \[\begin{align} \dot{V}(\rho-M) =&\int_0^1e^{-r y}g(\rho-M)\rho_t\operatorname{d}\!{y}\notag\\ = &-\int_0^1e^{-r y}g(\rho-M) \left(\lambda(W)\rho(y,t)\right)_y \operatorname{d}\!{y}\notag\\ =&-\lambda(W)\int_0^1e^{-r y}\left(G(\rho-M) \right)_y \operatorname{d}\!{y}\notag\\ =&-\lambda(W)\left[e^{-r y}G(\rho-M)\right]_{y=0}^{y=1} -r \lambda(W)\int_0^1e^{-r y} G(\rho-M) \operatorname{d}\!{y}\notag\\ =&-\lambda(W)\left(e^{-r}G(\rho(1,t)-M)- G(\rho(0,t)-M)\right) -r \lambda(W)V(\rho-M)\notag\\ =&- \lambda(W) e^{-r}G(\rho(1,t)-M) +\lambda(W)G(k\rho(1,t)-M+d(t)) -r \lambda(W)V(\rho-M).\label{PDE2-V1} \end{align}\tag{21}\]

Analogously, we have \[\begin{align} \dot{V}(-\rho-M) =&- \lambda(W) e^{-r}G(-\rho(1,t)-M) +\lambda(W)G(-k\rho(1,t)-M-d(t)) \notag\\ &-r \lambda(W)V(-\rho-M), \forall t\in (0,T).\label{PDE2-V2} \end{align}\tag{22}\] Combining 21 and 22 , we have \[\begin{align} \dot{\widehat{V}}(\rho) \leq & \lambda(W) G(k\rho(1,t)-M+d(t)) +\lambda(W)G(-k\rho(1,t)-M-d(t)) - \lambda(W) e^{-r} G(\rho(1,t)-M) \notag\\ &- \lambda(W) e^{-r} G(-\rho(1,t)-M) -r \lambda(W)\widehat{V}(\rho),\forall t\in (0,T).\label{PDE2-hatV} \end{align}\tag{23}\]

Now we prove 20 in two cases: \(k\in[0,1)\) and \(k\in(-1,0)\).

Case 1: \(k\in[0,1)\). Note that \(G\) has the following property: \[\begin{align} \label{G-m} G(ms) = m^{p+1}G(s),\;\forall m\in\mathbb{R}_{\geq 0}, s\in\mathbb{R}. \end{align}\tag{24}\] Using the definition of \(M\), the monotonicity of \(G\), and 24 , we have \[\begin{align} G(k\rho(1,t)-M+d(t)) \leq & G(k\rho(1,t)-M+|d(t)|)\notag\\ \leq & G(k\rho(1,t)-M+(1-k)M) \notag\\ = &G( k(\rho(1,t)- M) )\notag\\ =&k^{p+1}G (\rho(1,t)- M),\forall t\in (0,T),\label{PDE2-34} \end{align}\tag{25}\] and \[\begin{align} G(-k\rho(1,t)-M-d(t)) \leq & G(-k\rho(1,t)-M+|d(t)|)\notag\\ \leq & G(-k\rho(1,t)-M+(1-k)M) \notag\\ = & G(k(-\rho(1,t) - M)) \notag\\ = &k^{p+1} G(-\rho(1,t) - M) ,\forall t\in (0,T).\label{PDE2-35} \end{align}\tag{26}\]

Putting 25 and 26 into 23 , we get \[\begin{align} \dot{\widehat{V}}(\rho) \leq & \lambda(W) \left(k^{p+1}-e^{-r}\right)G( \rho(1,t)-M) +\lambda(W) \left(k^{p+1}-e^{-r}\right)G( -\rho(1,t)-M) \notag\\ & -r \lambda(W)\widehat{V}(\rho),\forall t\in (0,T), \forall t\in (0,T).\label{PDE2-hatV39} \end{align}\tag{27}\] Note that the choice of \(r\) (see 17 ) implies that \[\begin{align} k^{p+1}- e^{-r}\leq 0 . \end{align}\] Then, 27 leads to 20 .

Case 2: \(k\in(-1,0)\). Analogous in Case 1, using the definition of \(M\), the monotonicity of \(G\), and 24 , we have \[\begin{align} G(k\rho(1,t)-M+d(t)) \leq & G(k\rho(1,t)-M+(1-|k|)M) \notag\\ = &G( k(\rho(1,t)+ M) )\notag\\ =&(-k)^{p+1}G (-\rho(1,t)- M),\forall t\in (0,T),\label{PDE2-3439} \end{align}\tag{28}\] and \[\begin{align} G(-k\rho(1,t)-M-d(t)) \leq & G(-k\rho(1,t)-M+(1-|k|)M) \notag\\ = & G(k(-\rho(1,t) + M)) \notag\\ = &(-k)^{p+1} G( \rho(1,t) - M) ,\forall t\in (0,T).\label{PDE2-3539} \end{align}\tag{29}\]

Putting 28 and 29 into 23 , we get \[\begin{align} \dot{\widehat{V}}(\rho) \leq & \lambda(W) \left((-k)^{p+1}-e^{-r}\right)G( \rho(1,t)-M) +\lambda(W) \left((-k)^{p+1}-e^{-r}\right)G( -\rho(1,t)-M) \notag\\ & -r \lambda(W)\widehat{V}(\rho),\forall t\in (0,T), \forall t\in (0,T).\label{PDE2-hatV393939} \end{align}\tag{30}\] Note that the choice of \(r\) (see 17 ) implies that \[\begin{align} (-k)^{p+1}=|k|^{p+1}\leq e^{-r} . \end{align}\] Then, 30 leads to 20 . We conclude that 20 holds true for all \(k\in(-,1,1)\).

Step 2: We prove the ISS estimate ?? . Indeed, applying the Gronwall’s inequality to 20 , we obtain \[\begin{align} {\widehat{V}}( \rho(y,T) ) \leq& e^{-r \lambda_0 T}\widehat{V}(\rho_0(y)).\notag \end{align}\] Therefore, using (G6) and (G5), we have \[\begin{align} \frac{1}{p+1}\int_0^1|\rho(y,T)|^{p+1}\operatorname{d}\!{y} = & \int_0^1G\left(|\rho(y,T)|\right)\operatorname{d}\!{y}\notag\\ \leq &2^{p}\int_0^1G\left(|\rho(y,T)|-M \right)\operatorname{d}\!{y}+2^{p}G(M)\notag\\ = &2^{p}\int_0^1\left(G\left( \rho(y,T)-M \right)+G\left(- \rho(y,T)-M \right)\right)\operatorname{d}\!{y} +2^{p}G(M)\notag\\ \leq &2^{p}e^{r}\left(V\left( \rho(y,T)-M \right)+ V\left( -\rho(y,T)-M \right)\right)+2^{p}G(M)\notag\\ =&2^{p}e^{r}{\widehat{V}}( \rho(y,T) )+2^{p}G(M)\notag\\ \leq &2^{p}e^{r}e^{-r \lambda_0 T}\widehat{V}(\rho_0(y))+2^{p}G(M)\notag\\ =&2^{p} e^{r} e^{-r \lambda_0 T}\int_0^1e^{-r y} \left(G(\rho_0(y)-M)+G(-\rho_0(y)-M) \right) \operatorname{d}\!{y} +2^{p}G(M)\notag\\ = &2^{p}e^{r}e^{-r \lambda_0 T} \int_0^1e^{-r y} G(|\rho_0(y)|-M) \operatorname{d}\!{y} +2^{p}G(M)\notag\\ \leq &2^{p}e^{r}e^{-r \lambda_0 T} \int_0^1 G(|\rho_0(y)| ) \operatorname{d}\!{y} +2^{p}G(M)\notag\\ =& \frac{2^{p}e^{r}}{p+1}e^{-r \lambda_0 T} \int_0^1|\rho_0(y)|^{p+1}\operatorname{d}\!{y} +\frac{2^{p}}{p+1}M^{p+1}, \label{PDE2-26} \end{align}\tag{31}\] which implies that \[\begin{align} \|\rho(\cdot,T)\|_{L^{p+1}(0,1)} \leq & 2^{\frac{p}{p+1}}e^{\frac{r}{p+1}} e^{-\frac{r \lambda_0}{p+1} T}\|\rho_0 \|_{L^{p+1}(0,1)} + 2^{\frac{p}{p+1}} M,\forall T\in \mathbb{R}_{>0}. \label{PDE2-27} \end{align}\tag{32}\] Therefore, the ISS estimate ?? holds true.

Step 3: We prove the ISS estimate ?? . Now let \(r\) satisfy \[\begin{align} 0<-\frac{1}{2}(p+1)\ln |k|\leq r\leq -(p+1)\ln |k|,\label{condition-sigma39} \end{align}\tag{33}\] which ensures that \[\begin{align} e^{\frac{r}{p+1}} \leq \frac{1}{|k|} \;\;\text{and}\;\; e^{-\frac{r}{p+1}} \leq |k|^{\frac{1}{2}} . \end{align}\] For any \(T\in \mathbb{R}_{>0}\), it follows from 32 that \[\begin{align} \|\rho(\cdot,T)\|_{L^{p+1}(0,1)} \leq & \frac{ 2 }{|k|} |k|^{\frac{\lambda_0 T}{2}} \|\rho_0 \|_{L^{p+1}(0,1)} + 2 M .\label{PDE2-28} \end{align}\tag{34}\] Therefore, the ISS estimate ?? holds true for any \(q:=p+1 \in (2,\infty)\).

letting \(p\to 1\) and \(p\to \infty\) in 34 yields that the ISS estimate ?? holds for \(q=2\) and \(q=\infty\), respectively. \(\blacksquare\)

For any \(T\in \mathbb{R}_{>0}\), applying the Gronwall’s inequality to 19 , we obtain \[\begin{align} {\widehat{V}}( \rho(y,T) ) \leq& e^{-r \int_0^T\lambda(W(t))\operatorname{d}\!{t}}\widehat{V}(\rho_0(y)).\label{hatV-W40t41} \end{align}\tag{35}\] It follows that \[\begin{align} {\widehat{V}}( \rho(y,T) ) \leq& \widehat{V}(\rho_0(y)), \end{align}\] which, along with similar derivations for 31 , 32 , and 34 (by setting \(\lambda_0=0\) in 31 , 32 , and 34 ), leads to \[\begin{align} \|\rho(\cdot,T)\|_{L^{p+1}(0,1)} \leq & \frac{ 2 }{|k|} \|\rho_0 \|_{L^{p+1}(0,1)} + 2 M . \end{align}\] It follows that \[\begin{align} |W(T)| \leq & \|\rho(\cdot,T)\|_{L^{p+1}(0,1)}\notag\\ \leq & \frac{ 2 }{|k|} \|\rho_0 \|_{L^{p+1}(0,1)} + \frac{2}{1-|k|}\sup_{t\in (0,T)}|d (t)|. \end{align}\] Thus, for \(\rho_0\) and \(d\) satisfying ?? , i.e., \[\begin{align} \|\rho_0 \|_{L^{p+1}(0,1)} + \sup_{t\in \mathbb{R}_{\geq 0}}|d (t)|\leq R_0 \end{align}\] with some positive constant \(R_0\), it holds that \[\begin{align} |W(T)| \leq & 2R_0\max\left\{\frac{ 1 }{|k|}, \frac{ 1 }{1-|k|} \right\} . \end{align}\] According to the assumption (A2), we have \[\begin{align} \lambda(W(T))\geq &\lambda(|W(T)|)\notag\\ \geq & \Lambda_0:= \lambda\left( 2R_0\max\left\{\frac{ 1 }{|k|}, \frac{ 1 }{1-|k|} \right\}\right). \end{align}\] Then, 35 becomes \[\begin{align} {\widehat{V}}( \rho(y,T) ) \leq& e^{-r \int_0^T\lambda(W(t))\operatorname{d}\!{t}}\widehat{V}(\rho_0(y)) \\ \leq & e^{-r \int_0^T\Lambda_0 \operatorname{d}\!{t}}\widehat{V}(\rho_0(y))\notag \\ =& e^{-r \Lambda_0 T}\widehat{V}(\rho_0(y)).\notag \end{align}\] The remaining part of the proof is exactly the same as the proof of Proposition 4. \(\blacksquare\)

5 ISS analysis for \(1\)-D wave equations via GLM↩︎

In this section, we employ the GLM to establish ISS estimates for the following \(1\)-D wave equation with boundary velocity damping: \[\label{second32hyperbolic32PDE} \begin{align} w_{tt}(y,t)=&c^2w_{yy}(y,t) +f(y,t), (y,t)\in (0,1)\times\mathbb{R}_{>0},\\ w(0,t)=&0,t\in \mathbb{R}_{>0},\\ w_y(1,t)=&-kw_t(1,t)+d(t),t\in \mathbb{R}_{>0},\\ w(y,0)=&w_0(y), y\in(0,1),\\ w_t(y,0)=&\phi_0(y), y\in(0,1), \end{align}\tag{36}\] where \(c\) and \(k\) are positive constants satisfying \[\begin{align} ck=1, \end{align}\] \(f,d:\mathbb{R}_{>0}\to \mathbb{R}\) are smooth functions, representing distributed in-domain and boundary disturbances, respectively, and \(w_0\) and \(\phi_0\) are smooth functions, representing the initial data.

It is worth mentioning that the CLM has been employed to establish the ISS in the norm of Hilbert spaces for the \(1\)-D wave equation with different dampings; see [67], [68]. In this paper, we show how to use the GLM to establish the ISS in the norm of Banach spaces for the wave equation with boundary velocity damping.

For any \(p\in(1,\infty)\), let \(g\) and \(G\) be defined by 9 . For any \(T>0\), let \[\begin{align} M:= \frac{1}{c}\sup_{t\in (0,T)}|d (t)|. \end{align}\]

For \(v\in C(\mathbb{R}_{>0};L^{p+1}(0,1))\), define \[\begin{align} V_1 (v):= &\int_0^1e^{r y}G(v(y,t))\operatorname{d}\!{y},\\ V_2(v):=&\int_0^1e^{-r y}G(v(y,t))\operatorname{d}\!{y}, \end{align}\] where \(r\) is a positive constant. Let \[\begin{align} \widehat{V}(w):= &V_1 (w_t+cw_y -M) +V_1 (-w_t-cw_y -M) +V_2 (w_t-cw_y -M) +V_2 (-w_t+cw_y -M). \end{align}\]

We proceed the proof in four steps.

Step 1: Calculate the derivative of \(\widehat{V}\).

Indeed, for any \(t\in(0,T)\), using integrating by parts, we obtain \[\begin{align} \dot{V}_1(w_t+cw_y -M) =&\int_0^1e^{ r y}g(w_t+cw_y -M)(w_{tt}+cw_{yt})\operatorname{d}\!{y}\notag\\ =&\int_0^1e^{ r y}g(w_t+cw_y -M)( c^2w_{yy}+f+cw_{yt})\operatorname{d}\!{y}\notag\\ =&c\int_0^1e^{ r y}g(w_t+cw_y -M)( w_{t}+c w_{y })_y\operatorname{d}\!{y}+\int_0^1e^{ r y}g(w_t+cw_y -M) f \operatorname{d}\!{y}\notag\\ =&c\int_0^1e^{ r y}\left(G(w_t+cw_y -M)\right)_y\operatorname{d}\!{y} +\int_0^1e^{ r y}g(w_t+cw_y -M) f \operatorname{d}\!{y}\notag\\ =&c \left[e^{ r y}G(w_t+cw_y -M)\right]_{y=0}^{y=1} -cr{V}_1(w_t+cw_y -M)\notag\\ & +\int_0^1e^{ r y}g(w_t+cw_y -M) f \operatorname{d}\!{y}.\label{PDE3-53} \end{align}\tag{37}\]

Using the Young’s inequality (see (G8)), we have \[\begin{align} \int_0^1e^{ r y}g(w_t+cw_y -M) f \operatorname{d}\!{y} \leq &\varepsilon \int_0^1e^{ r y}G(w_t+cw_y -M) \operatorname{d}\!{y}\notag\\ & +\left(\frac{p}{\varepsilon}\right) ^{ p} \int_0^1e^{ r y}G(|f|) \operatorname{d}\!{y},\forall \varepsilon\in \mathbb{R}_{>0}.\label{PDE3-54} \end{align}\tag{38}\] We infer from 37 and 38 that \[\begin{align} \dot{V}_1(w_t+cw_y -M) \leq & -(cr-\varepsilon) {V}_1(w_t+cw_y -M)+\left(\frac{p}{\varepsilon}\right) ^{ p} V_1(|f|) \notag\\ & + c \left[e^{ r y}G(w_t+cw_y -M)\right]_{y=0}^{y=1} ,\forall t\in(0,T). \label{PDE3-55} \end{align}\tag{39}\] Analogously, we deduce that \[\begin{align} \dot{V}_1(-w_t-cw_y -M) \leq & -(cr-\varepsilon) {V}_1(-w_t-cw_y -M)+\left(\frac{p}{\varepsilon}\right) ^{ p} V_1(|f|)\notag\\ & + c \left[e^{ r y}G(-w_t-cw_y -M)\right]_{y=0}^{y=1} ,\forall t\in(0,T), \label{PDE3-56} \end{align}\tag{40}\] \[\begin{align} \dot{V}_2(w_t-cw_y -M) \leq & -(cr-\varepsilon) {V}_2(w_t-cw_y -M)+\left(\frac{p}{\varepsilon}\right) ^{ p} V_2(|f|) \notag\\ &- c \left[e^{ -r y}G(w_t-cw_y -M)\right]_{y=0}^{y=1} ,\forall t\in(0,T), \label{PDE3-57} \end{align}\tag{41}\] and \[\begin{align} \dot{V}_2(-w_t+cw_y -M) \leq & -(cr-\varepsilon) {V}_2(-w_t+cw_y -M)+\left(\frac{p}{\varepsilon}\right) ^{ p} V_2(|f|)\notag\\ & - c \left[e^{ -r y}G(-w_t+cw_y -M)\right]_{y=0}^{y=1} ,\forall t\in(0,T), \label{PDE3-58} \end{align}\tag{42}\] where \(\varepsilon\) keeps the same value and satisfies \[\begin{align} cr-\varepsilon>0. \end{align}\]

Combining 39 , 40 , 41 , and 42 , we obtain \[\begin{align} \dot{\widehat{V}}(w) \leq & -(cr-\varepsilon) \widehat{V} (w)+2\left(\frac{p}{\varepsilon}\right) ^{ p} \left( V_1(|f|)+V_2(|f|)\right) +cB(t) \notag\\ \leq & -(cr-\varepsilon) \widehat{V} (w)+4\left(\frac{p}{\varepsilon}\right) ^{ p} V_1(|f|) +cB(t), \forall t\in(0,T), \label{PDE3-59} \end{align}\tag{43}\] where \[\begin{align} B(t):=& \left[e^{ r y}G(w_t+cw_y -M)\right]_{y=0}^{y=1} + \left[e^{ r y}G(-w_t-cw_y -M)\right]_{y=0}^{y=1} - \left[e^{ -r y}G(w_t-cw_y -M)\right]_{y=0}^{y=1} \notag\\ &- \left[e^{ -r y}G(-w_t+cw_y -M)\right]_{y=0}^{y=1} . \end{align}\]

Step 2: Handle the boundary term \(B(t)\).

Using the boundary conditions, the fact that \(ck=1\), and the monotonicity of \(G\), we have \[\begin{align} B(t) =& e^{ r }G(w_t(1,t)+cw_y(1,t) -M) - G(w_t(0,t)+cw_y(0,t) -M) \notag\\ &+e^{ r }G(-w_t(1,t)-cw_y(1,t) -M) - G(-w_t(0,t)-cw_y(0,t) -M)\notag\\ & - e^{ -r }G(w_t(1,t)-cw_y(1,t) -M) + G(w_t(0,t)-cw_y(0,t) -M) \notag\\ &- e^{ -r }G(-w_t(1,t)+cw_y(1,t) -M) +G(-w_t(0,t)+cw_y(0,t) -M)\notag\\ =& e^{ r }G(w_t(1,t)+c(-kw_t(1,t)+d(t)) -M) - G(w_t(0,t)+cw_y(0,t) -M)\notag\\ & +e^{ r }G(-w_t(1,t)-c(-kw_t(1,t)+d(t)) -M) - G(-cw_y(0,t) -M)\notag\\ &- e^{ -r }G(w_t(1,t)-c(-kw_t(1,t)+d(t)) -M) + G(-cw_y(0,t) -M) \notag\\ &- e^{ -r }G(-w_t(1,t)+c(-kw_t(1,t)+d(t)) -M) +G(cw_y(0,t) -M)\notag\\ =& e^{ r }G( c d(t) -M) - G(cw_y(0,t) -M) +e^{ r }G(-c d(t) -M) - G(-cw_y(0,t) -M) \notag\\ &+ G(-cw_y(0,t) -M) - e^{ -r }G(-2w_t(1,t)+c d(t) -M) +G(cw_y(0,t) -M)\notag\\ =& e^{ r }G( c d(t) -M) +e^{ r }G(-c d(t) -M) - e^{ -r }G(2w_t(1,t)-c d(t) -M) \notag\\ & - e^{ -r }G(-2w_t(1,t)+c d(t) -M) \notag\\ \leq & e^{ r }G( c d(t) -M) +e^{ r }G(-c d(t) -M) \notag\\ \leq & e^{ r }G( c |d(t)| -M) +e^{ r }G(|-c d(t)| -M) \notag\\ \leq & e^{ r }G( M -M) +e^{ r }G(M -M) \notag\\ =& 0,\forall t\in (0,T). \end{align}\]

Step 3: Establish the ISS estimate ?? .

Since \(B(t)\leq 0\) for all \(t\in (0,T)\), we deduce from 43 that \[\begin{align} \dot{\widehat{V}}(w) \leq & -(cr-\varepsilon) \widehat{V} (w) +4\left(\frac{p}{\varepsilon}\right) ^{ p} V_1(|f|) \notag\\ \leq & -(cr-\varepsilon) \widehat{V} (w) +4e^{ r} \left(\frac{p}{\varepsilon}\right) ^{ p}\frac{1}{p+1}\|f\|_{\infty,T}^{p+1} , \forall t\in(0,T), \label{PDE3-61} \end{align}\tag{44}\] where, and in the sequel, \[\begin{align} \|f\|_{\infty,T}:=\sup_{(y,t)\in (0,1)\times (0,T)}|f(y,t)|. \end{align}\]

Applying the Gronwall’s inequality to 44 , we obtain \[\begin{align} \widehat{V} (w(y,T)) \leq & e^{-(cr-\varepsilon)T} \widehat{V} (w(y,0)) + \frac{4e^{ r}}{p+1}\left(\frac{p}{\varepsilon}\right) ^{ p}\|f\|_{\infty,T}^{p+1}\int_0^Te^{-(cr-\varepsilon)(T-t)} \operatorname{d}\!{t}\notag\\ \leq &e^{-(cr-\varepsilon)T} \widehat{V} (w(y,0)) +\frac{4e^{ r}}{p+1} \left(\frac{p}{\varepsilon}\right) ^{ p} \frac{1}{cr-\varepsilon} \|f\|_{\infty,T}^{p+1}. \label{PDE3-62} \end{align}\tag{45}\] Using the definition of \(\widehat{V}\), (G7), and (G6), we infer from 45 that \[\begin{align} &\frac{1}{p+1}\left(\|w_t(\cdot,T)\|^{p+1}_{L^{p+1}(0,1)}+\|w_y(\cdot,T)\|^{p+1}_{L^{p+1}(0,1)}\right) \notag\\ =& \int_0^1G(|w_t(y,T)|)\operatorname{d}\!{y}+\int_0^1G(|w_y(y,T)|)\operatorname{d}\!{y}\notag\\ \leq & e^r\widehat{V} (w(y,T))+2^{p}G(M)+e^r\widehat{V} (w(y,T))+2^{p}G(M)\notag\\ \leq &2 e^{r-(cr-\varepsilon)T} \widehat{V} (w(y,0)) +\frac{8e^{ 2r}}{p+1} \left(\frac{p}{\varepsilon}\right) ^{ p} \frac{1}{cr-\varepsilon} \|f\|_{\infty,T}^{p+1} +2^{p+1}G(M)\notag\\ = &2 e^{r-(cr-\varepsilon)T} V_1(|\phi_0(y)+cw_{0y}(y)|-M) +2 e^{r-(cr-\varepsilon)T} V_2(|\phi_0(y)+cw_{0y}(y)|-M) \notag\\ &+\frac{8e^{ 2r}}{p+1} \left(\frac{p}{\varepsilon}\right) ^{ p} \frac{1}{cr-\varepsilon} \|f\|_{\infty,T}^{p+1} +2^{p+1}G(M)\notag\\ \leq & 2 e^{r-(cr-\varepsilon)T} V_1(|\phi_0(y)+cw_{0y}(y)| ) +2 e^{r-(cr-\varepsilon)T} V_2(|\phi_0(y)+cw_{0y}(y)| ) \notag\\ &+\frac{8e^{ 2r}}{p+1} \left(\frac{p}{\varepsilon}\right) ^{ p} \frac{1}{cr-\varepsilon} \|f\|_{\infty,T}^{p+1} +2^{p+1}G(M)\notag\\ \leq & 2 e^{r-(cr-\varepsilon)T} 2^{p} ( V_1(|\phi_0(y)|)+V_1(|cw_{0y}(y)| ) ) +2 e^{r-(cr-\varepsilon)T} 2^{p} ( V_2(|\phi_0(y)|)+V_2(|cw_{0y}(y)| ) ) \notag\\ &+\frac{8e^{ 2r}}{p+1} \left(\frac{p}{\varepsilon}\right) ^{ p} \frac{1}{cr-\varepsilon} \|f\|_{\infty,T}^{p+1} +2^{p+1}G(M)\notag\\ \leq & 2^{p+2}e^{2r-(cr-\varepsilon)T} \int_0^1 G(|\phi_0(y)| )\operatorname{d}\!{y} + 2^{p+2}c^{p+1}e^{2r-(cr-\varepsilon)T}\int_0^1 G(|w_{0y}(y)| )\operatorname{d}\!{y} \notag\\ &+\frac{8e^{ 2r}}{p+1} \left(\frac{p}{\varepsilon}\right) ^{ p} \frac{1}{cr-\varepsilon} \|f\|_{\infty,T}^{p+1} +2^{p+1}G(M)\notag\\ =& \frac{2^{p+2}}{p+1}e^{2r-(cr-\varepsilon)T}\left(\|\phi_0\|^{p+1}_{L^{p+1}(0,1)}+\|w_{0y}\|^{p+1}_{L^{p+1}(0,1)}\right) +\frac{8e^{ 2r}}{p+1} \left(\frac{p}{\varepsilon}\right) ^{ p} \frac{1}{cr-\varepsilon} \|f\|_{\infty,T}^{p+1} \notag\\ &+\frac{2^{p+1}}{p+1}M^{p+1}, \end{align}\] which implies that system 36 is ISS in the \(L^{p+1}\)-norm, having the estimate \[\begin{align} \|w_t(\cdot,T)\|_{L^{p+1}(0,1)}+\|w_y(\cdot,T)\|_{L^{p+1}(0,1)} \leq& 2^{\frac{p}{p+1}} \bigg[ 2^{\frac{p+2}{p+1} } e^{\frac{2r-(cr-\varepsilon)T}{p+1}}\left(\|\phi_0\|_{L^{p+1}(0,1)}+\|w_{0y}\|_{L^{p+1}(0,1)}\right)\notag\\ &+ \left(\frac{p}{\varepsilon}\right) ^{ \frac{p}{p+1}} \left(\frac{8e^{2r}}{cr-\varepsilon}\right)^{\frac{1}{p+1}} \|f\|_{\infty,T} + 2M \bigg],\forall T\in \mathbb{R}_{>0}.\label{PDE3-ISS-139} \end{align}\tag{46}\]

In addition, due to the arbitrariness of \(p\), we conclude that ?? holds true for all \(q\in [2,\infty)\).

Step 4: Establish the ISS estimate ?? .

For any given \(m>0\), letting \(r=\frac{2m}{c}p\) and \(\varepsilon=mp\) in 46 yields \[\begin{align} &\|w_t(\cdot,T)\|_{L^{p+1}(0,1)}+\|w_y(\cdot,T)\|_{L^{p+1}(0,1)}\\ \leq& 2^{\frac{p}{p+1}} \bigg[ 2^{\frac{p+2}{p+1} } e^{\frac{4m}{c}\frac{p}{p+1}}e^{-\frac{p}{p+1}mT}\left(\|\phi_0\|_{L^{p+1}(0,1)}\right. \left. +\|w_{0y}\|_{L^{p+1}(0,1)}\right)\notag\\ &+ \left(\frac{1}{m}\right)^{\frac{p}{p+1}}\left( \frac{8}{mp}\right)^{\frac{1}{p+1}} e^{\frac{4m}{c}\frac{p}{p+1}} \|f\|_{\infty,T} + 2M \bigg] \notag\\ \leq & 2 \bigg[ 4 e^{\frac{4m}{c} }e^{-\frac{1}{2}mT}\left(\|\phi_0\|_{L^{p+1}(0,1)} +\|w_{0y}\|_{L^{p+1}(0,1)}\right) + \left(\frac{1}{m}\right)^{\frac{p}{p+1}}\left( \frac{8}{m}\right)^{\frac{1}{p+1}} e^{\frac{4m}{c} } \|f\|_{\infty,T} + 2M \bigg] \notag\\ = &8 e^{\frac{4m}{c} }e^{-\frac{1}{2}mT}\left(\|\phi_0\|_{L^{p+1}(0,1)} +\|w_{0y}\|_{L^{p+1}(0,1)}\right)+ \frac{16}{m} e^{\frac{4m}{c} } \|f\|_{\infty,T} + 4M ,\forall T\in \mathbb{R}_{>0}, \end{align}\] which, along with the arbitrariness of \(p>1\), implies ?? . \(\blacksquare\)

6 Conclusion↩︎

In this paper, within the framework of abstract control systems, we have revisited the concept of (classical) ISS-LF, the (classical) ISS-Lyapunov theorem, as well as the notion of GISS-LF and the GISS-Lyapunov theorem. A distinctive feature of the GISS-LF is its dependence on the external inputs, which generalizes the classical ISS-LF and provides greater flexibility for the ISS analysis of concrete PDEs. In particular, in the presence of Dirichlet boundary disturbances, the GISS-LF demonstrates superior efficiency than the classical ISS-LF. We refer to this approach of ISS analysis via the construction of GISS-LFs as the generalized Lyapunov method (GLM). To illustrate the general applicability of GLM, we have shown in detail how to construct GLFs for three representative examples, namely, an \(N\)-dimensional parabolic PDE, a first order hyperbolic PDE, and a wave equation with boundary damping, and based on which, established the corresponding ISS estimates in general \(L^q\) spaces. These examples cover the main features of parabolic, first order hyperbolic, and second order hyperbolic equations, providing a validation of the applicability of the GLM to different classes of PDEs. It is worth noting that, due to the diversity of PDEs, constructing a universally applicable GISS-LF in an explicit form for general PDEs remains an open problem. In future work, we will continue to focus on developing GISS-LFs tailored to a wider range of specific PDEs, explore the intrinsic connection between GISS-LF and controller design, particularly in the context of observer design and output feedback control.

7 Technical lemmas↩︎

8 Proofs of properties (G1)-(G8)****↩︎

Indeed, properties (G1)-(G3) can be directly verified via the definition of \(g\) and \(G\).

Using the increasing monotonicity of \(G\), i.e., (G3), we get \(G(s)\leq G(s+\tau)\) for \(\tau\geq 0\) and \(G(s)\leq G(s-\tau)\) for \(\tau\leq 0\), respectively. Thus, (G4) holds true.

Since \(G(|s|+\tau)= G(s+\tau)\) for \(s\geq 0\) and \(G(|s|+\tau)= G(-s+\tau)\) for \(s\leq 0\), we obtain (G5).

For (G6), it suffices to note that \((s+\tau)^{p+1}\leq 2^{p}(s^{p+1}+\tau^{p+1})\) for \(s,\tau\in \mathbb{R}_{\geq 0}\).

By subsequently using (G6), (G5), and (G4), we have \[\begin{align} G(|s|)\leq &2^pG(|s|-M)+2^pG(M)\notag\\ = &2^p(G(s-M)+G(-s-M))+2^pG(M)\notag\\ \leq & 2^p (G(s+\tau-M)+G(s-\tau-M) +G(-s+\tau-M)+G(-s-\tau-M)) \notag\\ &+2^pG(M) ,\forall s,\tau,M\in\mathbb{R}. \end{align}\] Therefore, (G7) holds true.

For (G8), it suffices to prove the result for positive \(s,\tau\). Indeed, for \(s,\tau\in\mathbb{R}_{>0}\), setting \(a:=g(s)=s^{p}\), \(b:=\tau\), \(r:=\frac{p+1}{p}\), \(q:=p+1\), \(\epsilon :=\frac{\varepsilon}{p+1}\) in Lemma 1 and using (G2), we obtain \[\begin{align} g(s)\tau\leq &\frac{1}{p+1}\varepsilon (s^{p})^{\frac{p+1}{p}}+C(\epsilon)\tau^{p+1}\notag\\ =& \frac{\varepsilon}{p+1}s^{p+1} +\left(\frac{\varepsilon}{p+1} \frac{p+1}{p}\right)^{-\frac{p+1}{\;\frac{p+1}{p}\;}} \frac{1}{p+1}\tau^{p+1}\notag\\ =& \varepsilon G(s)+\left(\frac{p}{\varepsilon}\right) ^{ p}G(\tau) . \end{align}\] Therefore, (G8) holds true.

References↩︎