The concept of input-to-state stability (ISS) proposed in the late 1980s is one of the central notions in robust nonlinear control. ISS has become indispensable for various branches of nonlinear systems theory, such as robust stabilization of nonlinear systems, design of nonlinear observers, analysis of large-scale networks, etc. The success of the ISS theory of ODEs and the need for robust stability analysis of partial differential equations (PDEs) motivated the development of ISS theory in the infinite-dimensional setting. For instance, the Lyapunov method for analysis of iISS of nonlinear parabolic equations. For an overview of the ISS theory for distributed parameter systems. ISS of control systems with application to robust global stabilization of the chemostat has been studied with the help of vector Lyapunov functions.
This paper studies the finite-time stability and stabilization of linear discrete time-varying stochastic systems with multiplicative noise. Firstly, necessary and sufficient conditions for finite-time stability are presented via state transition matrix approach. Secondly, this paper also develops the Lyapunov function method to study finite-time stability and stabilization of discrete time-varying stochastic systems based on matrix inequalities and linear matrix inequalities (LMIs), so as to Matlab LMI Toolbox can be used. Two numerical examples are given to illustrate the effectiveness of th…
In an infinite-horizon optimal control problem the stability of the adjoint variable implying the vanishing of the adjoint variable at infinity along optimal solution.
This paper discusses stability and robustness properties of a recently proposed observer algorithm for linear time varying systems. The observer is based on the approximation and subsequent modification of the non-negative Lyapunov exponents which yields a (non-uniform) exponentially stable error system. Theoretical insights in the construction of the observer are given and the error system is analyzed with respect to bounded unknown inputs. Therefor, new conditions for bounded input bounded state stability for linear time varying systems are presented. It is shown that for a specific class of…
This paper studies a class of random nonlinear systems with time-varying delay, in which the $r$-order moment ($r\geq1$) of the random disturbance is finite. Firstly, some general conditions are proposed to guarantee the existence and uniqueness of the global solution to random nonlinear time-delay systems. Secondly, some definitions and criteria on noise-to-state stability in the moment sense and in probability sense are given by Lyapunov method respectively. Finally, two regulation controllers are constructed respectively for two corresponding random nonlinear time-delay systems and the effe…
This paper is concerned with exponential stability of a class of infinite dimensional coupled systems. It is proved that under some admissibility conditions, the considered infinite dimensional coupled system is governed by a $C_0$-semigroup. Furthermore, if both the free subsystems are governed by exponentially stable $C_0$-semigroups, then so is the coupled system. The results are applied to simplify the proof of semigroup generation and exponential stability for several coupled systems emerged in control theory literatures.
We analyze the robustness of the exponential stability of infinite-dimensional sampled-data systems with unbounded control operators. The unbounded perturbations we consider are the so-called Desch-Schappacher perturbations, which arise, e.g., from the boundary perturbations of systems described by partial differential equations. As the main result, we show that the exponential stability of the sampled-data system is preserved under all Desch-Schappacher perturbations sufficiently small in a certain sense.
In this paper we deal with infinite-dimensional nonlinear forward complete dynamical systems which are subject to external disturbances. We first extend the well-known Datko lemma to the framework of the considered class of systems. Thanks to this generalization, we provide characterizations of the uniform (with respect to disturbances) local, semi-global, and global exponential stability, through the existence of coercive and non-coercive Lyapunov functionals. The importance of the obtained results is underlined through some applications concerning 1) exponential stability of nonlinear retard…
This paper is concerned with the study of both, local and global, uniform asymptotic stability for switched nonlinear time-varying (NLTV) systems through the detectability of output-maps. With this aim the notion of reduced limiting control systems for switched NLTV systems whose switchings verify time/state dependent constraints, and the concept of weakly zero-state detectability for those reduced limiting systems are introduced. Necessary and sufficient conditions for the (global)uniform asymptotic stability of families of trajectories of the switched system are obtained in terms of this det…
This article is devoted to the stability of error bounds (local and global) for semi-infinite convex constraint systems in Banach spaces. We provide primal characterizations of the stability of local and global error bounds when systems are subject to small perturbations. These characterizations are given in terms of the directional derivatives of the functions that enter into the definition of these systems. It is proved that the stability of error bounds is essentially equivalent to verifying that the optimal values of several minimax problems, defined in terms of the directional derivatives…
We consider the stability of Robust Optimization problems with respect to perturbations in their uncertainty sets. We focus on Linear Optimization problems, including those with a possibly infinite number of constraints, also known as Linear Semi-Infinite Optimization (LSIO) problems, and consider uncertainty in both the cost function and constraints. We prove Lipschitz continuity of the optimal value and ε-approximate optimal solution set with respect to the Hausdorff distance between uncertainty sets and with an explicit Lipschitz constant that can be calculated. In addition, we prove closed…
This paper proposes a unified approach for studying global exponential stability of a general class of switched systems described by time-varying nonlinear functional differential equations. Some new delay-independent criteria of global exponential stability are established for this class of systems under arbitrary switching which satisfies some assumptions on the average dwell time. The obtained criteria are shown to cover and improve many previously known results, including, in particular, sufficient conditions for absolute exponential stability of switched time-delay systems with sector non…
For nonlinear discrete time systems satisfying a controllability condition, we present a stability condition for model predictive control without stabilizing terminal constraints or costs. The condition is given in terms of an analytical formula which can be employed in order to determine a prediction horizon length for which asymptotic stability or a performance guarantee is ensured. Based on this formula a sensitivity analysis with respect to the prediction and the possibly time varying control horizon is carried out.
In this paper we provide a set of stability conditions for linear time-varying networked control systems with arbitrary topologies using a piecewise quadratic switching stabilization approach with multiple quadratic Lyapunov functions. We use this set of stability conditions to provide a novel iterative low-complexity algorithm that must be updated and optimized in discrete time for the design of a sparse observer-controller network, for a given plant network with an arbitrary topology. We employ distributed observers by utilizing the output of other subsystems to improve the stability of each…
This paper introduces an advanced Lyapunov stability analysis for an attitude observer that has been developed on the special orthogonal group. In particular, when the attitude observer is constructed based on multiple direction measurements toward known reference points, a local exponential stability has been established by linearization, under the assumption that those reference points are fixed in the inertial frame. Several modifications have been proposed to deal with reference directions changing over time. Here, we present an alternative Lyapunov analysis to show that the attitude obser…
We investigate the stability and stabilization concepts for infinite dimensional time fractional differential linear systems in Hilbert spaces with Caputo derivatives. Firstly, based on a family of operators generated by strongly continuous semigroups and on a probability density function, we provide sufficient and necessary conditions for the exponential stability of the considered class of systems. Then, by assuming that the system dynamics is symmetric and uniformly elliptic and by using the properties of the Mittag-Leffler function, we provide sufficient conditions that ensure strong stabi…
This paper considers $L_2$ and BIBO stability and stabilization issues for systems with time-varying delays which can be of retarded or neutral type. An important role is played by a nominal system with fixed delays which are close to the time-varying ones. Under stability or stabilizability conditions of this nominal system, sufficient conditions are given in order to ensure similar properties for the system with time-varying delays.
Motivated by a paradigm shift towards a hyper-connected world, we develop a computationally tractable small-gain theorem for a network of infinitely many systems, termed as infinite networks. The proposed small-gain theorem addresses exponential input-to-state stability with respect to closed sets, which enables us to analyze diverse stability problems in a unified manner. The small-gain condition, expressed in terms of the spectral radius of a gain operator collecting all the information about the internal Lyapunov gains, can be numerically computed for a large class of systems in an efficien…
In the last decade, copositive formulations have been proposed for a variety of combinatorial optimization problems, for example the stability number (independence number). In this paper, we generalize this approach to infinite graphs and show that the stability number of an infinite graph is the optimal solution of some infinite-dimensional copositive program. For this we develop a duality theory between the primal convex cone of copositive kernels and the dual convex cone of completely positive measures. We determine the extreme rays of the latter cone, and we illustrate this theory with the…
Stability of linear systems with uncertain bounded time-varying delays is studied under assumption that the nominal delay values are not equal to zero. An input-output approach to stability of such systems is known to be based on the bound of the $L_2$-norm of a certain integral operator. There exists a bound on this operator in two cases: in the case where the delay derivative is not greater than 1 and in the case without any constraints on the delay derivative. In the present note we fill the gap between the two cases by deriving an operator bound which is an increasing and continuous functi…
This paper addresses the stability analysis of infinite-dimensional sampled-data systems under unbounded perturbations. We present two classes of unbounded perturbations preserving the exponential stability of sampled-data systems. To this end, we investigate the continuity of strongly continuous semigroups with respect to their generators, considering the uniform operator topology.
In this paper, a kind of neural network with time-varying delays is proposed to solve the problems of quadratic programming. The delay term of the neural network changes with time t. The number of neurons in the neural network is n + h, so the structure is more concise. The equilibrium point of the neural network is consistent with the optimal solution of the original optimization problem. The existence and uniqueness of the equilibrium point of the neural network are proved. Application inequality technique proved global exponential stability of the network. Some numerical examples are given …
This paper investigates the robust stability and stabilization analysis of interval fractional-order systems with time-varying delay. The stability problem of such systems is solved first, and then using the proposed results a stabilization theorem is also included, where sufficient conditions are obtained for designing a stabilizing controller with a predetermined order, which can be chosen to be as low as possible. Utilizing efficient lemmas, the stability and stabilization theorems are proposed in the form of LMIs, which is more suitable to check due to various existing efficient convex opt…
The transversality conditions for infinite-horizon optimal control problem with a free right endpoint and the stability of the adjoint variable (in Russian)
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.
Recently, a framework for controller design of sampled-data nonlinear systems via their approximate discrete-time models has been proposed in the literature. In this paper we develop novel tools that can be used within this framework and that are very useful for tracking problems. In particular, results for stability analysis of parameterized time-varying discrete-time cascaded systems are given. This class of models arises naturally when one uses an approximate discrete-time model to design a stabilizing or tracking controller for a sampled-data plant. While some of our results parallel their…
For general time-varying or switched (nonlinear) systems, converse Lyapunov theorems for stability are not available. In these cases, the integral input-to-state stability (iISS) property is not equivalent to the existence of an iISS-Lyapunov function but can still be characterized as the combination of global uniform asymptotic stability under zero input (0-GUAS) and uniformly bounded energy input-bounded state (UBEBS). For impulsive systems, asymptotic stability can be weak (when the asymptotic decay depends only on elapsed time) or strong (when such a decay depends also on the number of imp…
For bilinear infinite-dimensional dynamical systems, we show the equivalence between uniform global asymptotic stability and integral input-to-state stability. We provide two proofs of this fact. One applies to general systems over Banach spaces. The other is restricted to Hilbert spaces, but is more constructive and results in an explicit form of iISS Lyapunov functions.
This paper deals with the stabilization of the linear Biharmonic Schrödinger equation in an $n$-dimensional open bounded domain under Dirichlet-Neumann boundary conditions considering three infinite memory terms as damping mechanisms. We show that depending on the smoothness of initial data and the arbitrary growth at infinity of the kernel function, this class of solution goes to zero with a polynomial decay rate like $t^{-n}$ depending on assumptions about the kernel function associated with the infinite memory terms.
For large classes of infinite-dimensional time-varying control systems, the equivalence between integral input-to-state stability (iISS) and the combination of global uniform asymptotic stability under zero input (0-GUAS) and uniformly bounded-energy input/bounded state (UBEBS) is established under a reasonable assumption of continuity of the trajectories with respect to the input, at the zero input. By particularizing to specific instances of infinite-dimensional systems, such as time-delay, or semilinear over Banach spaces, sufficient conditions are given in terms of the functions defining t…
We develop tools for investigation of input-to-state stability (ISS) of infinite-dimensional control systems. We show that for certain classes of admissible inputs the existence of an ISS-Lyapunov function implies the input-to-state stability of a system. Then for the case of systems described by abstract equations in Banach spaces we develop two methods of construction of local and global ISS-Lyapunov functions. We prove a linearization principle that allows a construction of a local ISS-Lyapunov function for a system which linear approximation is ISS. In order to study interconnections of no…
In this work, the null controllability problem for a linear system in $\ell^2$ is considered, where the matrix of a linear operator describing the system is an infinite matrix with $λ\in \mathbb R$ on the main diagonal and 1s above it. We show that the system is asymptotically stable if and only if $λ\le-1$, which shows the fine difference between the finite and the infinite-dimensional systems. When $λ\le-1$ we also show that the system is null controllable in large. We also show a dependence of the stability on the norm i.e. the same system considered in $\ell^\infty$ is not asymptotically s…
We provide novel sufficient conditions for stability of nonlinear and time-varying impulsive systems. These conditions generalize, extend, and strengthen many existing results. Different types of input-to-state stability (ISS), as well as zero-input global uniform asymptotic stability (0-GUAS), are covered by employing a two-measure framework and considering stability of both weak (decay depends only on elapsed time) and strong (decay depends on elapsed time and the number of impulses) flavors. By contrast to many existing results, the stability state bounds imposed are uniform with respect to…
For time-invariant (nonimpulsive) systems, it is already well-known that the input-to-state stability (ISS) property is strictly stronger than integral input-to-state stability (iISS). Very recently, we have shown that under suitable uniform boundedness and continuity assumptions on the function defining system dynamics, ISS implies iISS also for time-varying systems. In this paper, we show that this implication remains true for impulsive systems, provided that asymptotic stability is understood in a sense stronger than usual for impulsive systems.
We introduce the notions of semi-uniform input-to-state stability and its subclass, polynomial input-to-state stability, for infinite-dimensional systems. We establish a characterization of semi-uniform input-to-state stability based on attractivity properties as in the uniform case. Sufficient conditions for linear systems to be polynomially input-to-state stable are provided, which restrict the range of the input operator depending on the rate of polynomial decay of the product of the semigroup and the resolvent of its generator. We also show that a class of bilinear systems are polynomially…
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 norm-to-integral input-to-state stability. This property in turn is equivalent to input-to-state stability, if the system satisfies certain mild regularity assumptions. 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…
We consider infinite heterogeneous networks, consisting of input-to-state stable subsystems of possibly infinite dimension. We show that the network is input-to-state stable, provided that the gain operator satisfies a certain small-gain condition. We show that for finite networks of nonlinear systems this condition is equivalent to the so-called strong small-gain condition of the gain operator (and thus our results extend available results for finite networks), and for infinite networks with a linear gain operator they correspond to the condition that the spectral radius of the gain operator …
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 not…
On the stability of gradient descent with second order dynamics for time-varying cost functions
Massachusetts Institute of Technology（麻省理工学院）；Brigham and Women’s Hospital（布莱根妇女医院）；Harvard Medical School（哈佛医学院）；Broad Institute of MIT and Harvard（麻省理工学院与哈佛大学博德研究所）；Stanford University（斯坦福大学）
This paper analyses the interplay between dissipativity and stability properties in continuous-time infinite-horizon Optimal Control Problems (OCPs). We establish several relations between these properties, which culminate in a set of equivalence conditions. Moreover, we investigate convergence and stability of the infinite-horizon optimal adjoint trajectories. The workhorse for our investigations is a notion of strict dissipativity in OCPs, which has been coined in the context of economic model predictive control. With respect to the link between stability and dissipativity, the present paper…
In this paper, we analyze the dynamics of spreading processes taking place over time-varying networks. A common approach to model time-varying networks is via Markovian random graph processes. This modeling approach presents the following limitation: Markovian random graphs can only replicate switching patterns with exponential inter-switching times, while in real applications these times are usually far from exponential. In this paper, we introduce a flexible and tractable extended family of processes able to replicate, with arbitrary accuracy, any distribution of inter-switching times. We th…
We prove characterizations of input-to-state stability (ISS) for a large class of infinite-dimensional control systems, including some classes of evolution equations over Banach spaces, time-delay systems, ordinary differential equations (ODE), switched systems. These characterizations generalize well-known criteria of ISS, proved by Sontag and Wang for ODE systems. For the special case of differential equations in Banach spaces we prove even broader criteria for ISS and apply these results to show that (under some mild restrictions) the existence of a non-coercive ISS Lyapunov functions impli…
