Input-to-state stability and stabilization of distributed parameter systems

This paper provides a Lyapunov-based small-gain theorem for input-to-state stability (ISS) of networks composed of infinitely many finite-dimensional systems. We model these networks on infinite-dimensional $\ell_{\infty}$-type spaces. A crucial assumption in our results is that the internal Lyapunov gains, modeling the influence of the subsystems on each other, are linear functions. Moreover, the gain operator built from the internal gains is assumed to be subadditive and homogeneous, which covers both max-type and sum-type formulations for the ISS Lyapunov functions of the subsystems. As a consequence, the small-gain condition can be formulated in terms of a generalized spectral radius of the gain operator. Through an example, we show that the small-gain condition can easily be checked if the interconnection topology of the network has some kind of symmetry. While our main result provides an ISS Lyapunov function in implication form for the overall network, an ISS Lyapunov function in a dissipative form is constructed under mild extra assumptions.

We prove new characterisations of exponential stability for positive linear discrete-time systems in ordered Banach spaces, in terms of small-gain conditions. Such conditions have played an important role in the finite-dimensional systems theory, but are relatively unexplored in the infinite-dimensional setting, yet. Our results apply to the large class of ordered Banach spaces that have normal and generating cone. Moreover, we show that our stability criteria can be considerably simplified if the cone has non-empty interior or if the operator under consideration is quasi-compact. To place our results into context we include an overview of known stability criteria for linear (and not necessarily positive) operators and provide full proofs for several folklore characterizations from this domain.

We introduce the concept of non-uniform input-to-state stability for networks, which combines the uniform global stability together with the uniform attractivity of any finite number of modes of the system, but which does not guarantee the uniform convergence of all modes. We show that given an infinite network of ISS subsystems, which do not have a uniform $\mathcal{KL}$-bound on the transient behavior, and if the gain operator satisfies the bounded monotone invertibility property, then the whole network is non-uniformly ISS and its any finite subnetwork is uniformly ISS.

We prove nonlinear small-gain theorems for input-to-state stability of infinite heterogeneous networks, consisting of input-to-stable subsystems of possibly infinite dimension. Furthermore, we prove small-gain results for the uniform global stability of infinite networks. Our results extend available theorems for finite networks of finite- or infinite-dimensional systems. These results are shown either under the so-called monotone limit property or under the monotone bounded invertibility property, which is equivalent to a uniform small-gain condition. We show that for finite networks of nonlinear systems these properties are equivalent to the so-called strong small-gain condition of the gain operator, and for infinite networks with linear gain operator they correspond to the condition that the spectral radius of the gain operator is less than one. We provide efficient criteria for input-to-state stability of infinite networks with linear gains, governed by linear and homogeneous gain operators, respectively.

In this paper we consider countable couplings of finite-dimensional input-to-state stable systems. We consider the whole interconnection as an infinite-dimensional system on the ∞ state space. We develop stability conditions of the small-gain type to guarantee that the whole system remains ISS and highlight the differences between finite and infinite couplings by means of examples. We show that using our methodology it is possible to study uniform global asymptotic stability of nonlinear spatially invariant systems by solving a finite number of nonlinear algebraic inequalities.

This paper presents a small-gain theorem for networks composed of a countably infinite number of finite-dimensional subsystems. Assuming that each subsystem is exponentially input-to-state stable, we show that if the gain operator, collecting all the information about the internal Lyapunov gains, has a spectral radius less than one, the overall infinite network is exponentially input-to-state stable. The effectiveness of our result is illustrated through several examples including nonlinear spatially invariant systems with sector nonlinearities and a road traffic network.

We consider an abstract class of infinite-dimensional dynamical systems with inputs. For this class, the significance of noncoercive Lyapunov functions is analyzed. It is shown that the existence of such Lyapunov functions implies integral-to-integral input-to-state stability. Assuming further regularity it is possible to conclude input-to-state stability. For a particular class of linear systems with unbounded admissible input operators, explicit constructions of noncoercive Lyapunov functions are provided. The theory is applied to a heat equation with Dirichlet boundary conditions.

In a pedagogical but exhaustive manner, this survey reviews the main results on input-to-state stability (ISS) for infinite-dimensional systems. This property allows estimating the impact of inputs and initial conditions on both the intermediate values and the asymptotic bound on the solutions. ISS has unified the input-output and Lyapunov stability theories and is a crucial property in stability theory of control systems as well as for many applications whose dynamics depend on parameters, unknown perturbations, or other inputs. Starting from classic results for nonlinear ordinary differential equations, we motivate the study of ISS property for distributed parameter systems. Then fundamental properties are given, as an ISS superposition theorem and characterizations of (global and local) ISS in terms of Lyapunov functions. We explain in detail the functional-analytic approach to ISS theory of linear systems with unbounded input operators, with a special attention devoted to ISS theory of boundary control systems. Lyapunov method is shown to be very useful for both linear and nonlinear models, including parabolic and hyperbolic partial differential equations. Next, we show the efficiency of ISS framework to study stability of large-scale networks, coupled either via the boundary or via the interior of the spatial domain. ISS methodology allows reducing the stability analysis of complex networks, by considering the stability properties of its components and the interconnection structure between the subsystems. An extra section is devoted to ISS theory of time-delay systems with the emphasis on techniques, which are particularly suited for this class of systems. Finally, numerous applications are considered in this survey, where ISS properties play a crucial role in their study. Furthermore, this survey suggests many open problems throughout the paper.

We show by means of counterexamples that many characterizations of input-to-state stability (ISS) known for ODE systems are not valid for general differential equations in Banach spaces. Moreover, these notions or combinations of notions are not equivalent to each other, and can be classified into several groups according to the type and grade of nonuniformity. We introduce the new notion of strong ISS which is equivalent to ISS in the ODE case, but which is strictly weaker than ISS in the infinite-dimensional setting. We characterize strong ISS as a strong asymptotic gain property plus global stability.

In this paper a class of abstract dynamical systems is considered which encompasses a wide range of nonlinear finite-and infinite-dimensional systems. We show that the existence of a non-coercive Lyapunov function without any further requirements on the flow of the forward complete system ensures an integral version of uniform global asymptotic stability. We prove that also the converse statement holds without any further requirements on regularity of the system. Furthermore, we give a characterization of uniform global asymptotic stability in terms of the integral stability properties and analyze which stability properties can be ensured by the existence of a non-coercive Lyapunov function, provided either the flow has a kind of uniform continuity near the equilibrium or the system is robustly forward complete.

We introduce a monotonicity-based method for studying input-to-state stability (ISS) of nonlinear parabolic equations with boundary inputs. We first show that a monotone control system is ISS if and only if it is ISS w.r.t. constant inputs. Then we show by means of classical maximum principles that nonlinear parabolic equations with boundary disturbances are monotone control systems. With these two facts, we establish that ISS of the original nonlinear parabolic PDE with constant boundary disturbances is equivalent to ISS of a closely related nonlinear parabolic PDE with constant distributed disturbances and zero boundary condition. The last problem is conceptually much simpler and can be handled by means of various recently developed techniques.

We prove a small-gain theorem for interconnections of $n$ nonlinear heterogeneous input-to-state stable control systems of a general nature, covering partial, delay and ordinary differential equations. Furthermore, for the same class of control systems we derive small-gain theorems for asymptotic gain, uniform global stability and weak input-to-state stability properties. We show that our technique is applicable for different formulations of ISS property (summation, maximum, semimaximum) and discuss tightness of achieved small-gain theorems. Finally, we introduce variations of asymptotic gain and limit properties, which are particularly useful for small-gain arguments and characterize ISS in terms of these notions.

We introduce a monotonicity-based method for studying input-to-state stability (ISS) of nonlinear parabolic equations with boundary inputs. We first show that a monotone control system is ISS if and only if it is ISS w.r.t. constant inputs. Then we show by means of classical maximum principles that nonlinear parabolic equations with boundary disturbances are monotone control systems. With these two facts, we establish that ISS of the original nonlinear parabolic PDE with constant boundary disturbances is equivalent to ISS of a closely related nonlinear parabolic PDE with constant distributed disturbances and zero boundary condition. The last problem is conceptually much simpler and can be handled by means of various recently developed techniques.

In this paper, a class of abstract dynamical systems is considered which encompasses a wide range of nonlinear finite- and infinite-dimensional systems. We show that the existence of a non-coercive Lyapunov function without any further requirements on the flow of the forward complete system ensures an integral version of uniform global asymptotic stability. We prove that also the converse statement holds without any further requirements on regularity of the system. Furthermore, we give a characterization of uniform global asymptotic stability in terms of the integral stability properties and analyze which stability properties can be ensured by the existence of a non-coercive Lyapunov function, provided either the flow has a kind of uniform continuity near the equilibrium or the system is robustly forward complete.

For a broad class of infinite-dimensional systems, we characterize input-to-state practical stability (ISpS) using the uniform limit property and in terms of input-to-state stability. We specialize our results to the systems with Lipschitz continuous flows and evolution equations in Banach spaces. Even for the special case of ordinary differential equations our results are novel and improve existing criteria for ISpS.

We prove that input-to-state stability (ISS) of nonlinear systems over Banach spaces is equivalent to existence of a coercive Lipschitz continuous ISS Lyapunov function for this system. For linear infinite-dimensional systems, we show that ISS is equivalent to existence of a non-coercive ISS Lyapunov function and provide two simpler constructions of coercive and non-coercive ISS Lyapunov functions for input-to-state stable linear systems.

This paper deals with strong versions of input-to-state stability and integral input-to-state stability of infinite-dimensional linear systems with an unbounded input operator. We show that infinite-time admissibility with respect to inputs in an Orlicz space is a sufficient condition for a system to be strongly integral input-to-state stable but, unlike in the case of exponentially stable systems, not a necessary one.

We introduce a monotonicity-based method for studying input-to-state stability (ISS) of nonlinear parabolic equations with boundary inputs. We first show that a monotone control system is ISS if and only if it is ISS w.r.t. constant inputs. Then we show by means of classical maximum principles that nonlinear parabolic equations with boundary disturbances are monotone control systems. With these two facts, we establish that ISS of the original nonlinear parabolic PDE with constant \textit{boundary disturbances} is equivalent to ISS of a closely related nonlinear parabolic PDE with constant \textit{distributed disturbances} and zero boundary condition. The last problem is conceptually much simpler and can be handled by means of various recently developed techniques. As an application of our results, we show that the PDE backstepping controller which stabilizes linear reaction-diffusion equations from the boundary is robust with respect to additive actuator disturbances.

We show that practical uniform global asymptotic stability (pUGAS) is equivalent to existence of a bounded uniformly globally weakly attractive set. This result is valid for a wide class of robustly forward complete distributed parameter systems, including time-delay systems, switched systems, many classes of PDEs and evolution differential equations in Banach spaces. We apply this criterion to show that existence of a non-coercive Lyapunov function ensures pUGAS of robustly forward complete systems. For ordinary differential equations with uniformly bounded disturbances the concept of uniform weak attractivity is equivalent to the usual weak attractivity. It is however essentially stronger than weak attractivity for infinite-dimensional systems, even for linear ones.

In this work, the relation between input-to-state stability and integral input-to-state stability is studied for linear infinite-dimensional systems with an unbounded control operator. Although a special focus is laid on the case $L^{\infty}$, general function spaces are considered for the inputs. We show that integral input-to-state stability can be characterized in terms of input-to-state stability with respect to Orlicz spaces. Since we consider linear systems, the results can also be formulated in terms of admissibility. For parabolic diagonal systems with scalar inputs, both stability notions with respect to $L^\infty$ are equivalent.

This work contributes to the recently intensified study of input-to-state stability for infinite-dimensional systems. The focus is laid on the relation between input-to-state stability and integral input-to-state stability for linear systems with a possibly unbounded control operator. The main result is that for parabolic diagonal systems both notions coincide, even in the setting of inputs in L ∞ , and a simple criterion is derived.

We prove characterizations of ISS for a large class of infinite-dimensional control systems, including differential equations in Banach spaces, time-delay systems, ordinary differential equations, switched systems. These characterizations generalize well-known criteria of ISS, proved by Sontag and Wang in \cite{SoW96} for ODE systems. For the special case of differential equations in Banach spaces we prove even broader criteria for ISS property. Using an important technical result from \cite{SoW96} we show that the characterizations obtained in \cite{SoW96} are a special case of our results. We introduce the new notion of strong ISS which is equivalent to ISS in the ODE case, but which is strictly weaker than ISS in the infinite-dimensional setting and prove several criteria for sISS property. At the same time we show by means of counterexamples, that many characterizations, which are valid in the ODE case, are not true for general infinite-dimensional systems.

We show that existence of a non-coercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinite-dimensional systems with external disturbances provided an additional mild assumption is fulfilled. For UGAS infinite-dimensional systems with external disturbances we derive a novel ‘integral’ construction of non-coercive Lipschitz continuous Lyapunov functions. Finally, converse Lyapunov theorems are used in order to prove Lyapunov characterizations of input-to-state stability of infinite-dimensional systems.

We prove that uniform global asymptotic stability of bilinear infinite-dimensional control systems is equivalent to their integral input-to-state stability. Next we present a method for construction of iISS Lyapunov functions for such systems if the state space is a Hilbert space. Unique issues arising due to infinite-dimensionality are highlighted.

We show that the existence of a non-coercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinite-dimensional systems with external disturbances provided the speed of decay is measured in terms of the norm of the state and an additional mild assumption is satisfied. For evolution equations in Banach spaces with Lipschitz continuous nonlinearities these additional assumptions become especially simple. The results encompass some recent results on linear switched systems on Banach spaces. Some examples show the necessity of the assumptions which are made.

We show that existence of a non-coercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinite-dimensional systems with external disturbances provided an additional mild assumption is fulfilled. For UGAS infinite-dimensional systems with external disturbances we derive a novel 'integral' construction of non-coercive Lipschitz continuous Lyapunov functions. Finally, converse Lyapunov theorems are used in order to prove Lyapunov characterizations of input-to-state stability of infinite-dimensional systems.