In this paper, we introduce a weak maximum principle-based approach to input-to-state stability (ISS) analysis for certain nonlinear partial differential equations (PDEs) with certain boundary disturbances. Based on the weak maximum principle, a classical result on the maximum estimate of solutions to linear parabolic PDEs has been extended, which enables the ISS analysis for certain nonlinear parabolic

Model order reduction (MOR) techniques are often used to reduce the order of spatially discretized (stochastic) partial differential equations and hence reduce computational complexity. A particular class of MOR techniques is balancing related methods which rely on simultaneously diagonalizing the system Gramians. This has been extensively studied for deterministic linear systems. The balancing procedure

We establish the local input-to-state stability of a large class of disturbed nonlinear reaction–diffusion equations w.r.t. the global attractor of the respective undisturbed system.

Classical discrete-time adaptive controllers provide asymptotic stabilization and tracking; neither exponential stabilization nor a bounded noise gain is typically proven. In our recent work, it is shown, in both the pole placement stability setting and the first-order one-step-ahead tracking setting, that if the original, ideal, projection algorithm is used (subject to the common assumption that the

This article presents a new method for computing sharp bounds on the solutions of nonlinear dynamic systems subject to uncertain initial conditions, parameters, and time-varying inputs. Such bounds are widely used in algorithms for uncertainty propagation, robust state estimation, system verification, global dynamic optimization, and more. Recently, it has been shown that bounds computed via differential

This article considers the problem of optimally recovering stable linear time-invariant systems observed via linear measurements made on their transfer functions. A common modeling assumption is replaced here by the related assumption that the transfer functions belong to a model set described by approximation capabilities. Capitalizing on recent optimal recovery results relative to such approximability

We study the sample-data control problem of output tracking and disturbance rejection for unstable well-posed linear infinite-dimensional systems with constant reference and disturbance signals. We obtain a sufficient condition for the existence of finite-dimensional sampled-data controllers that are solutions of this control problem. To this end, we study the problem of output tracking and disturbance

The purpose of this work is to obtain restrictions on the asymptotic structure of two-dimensional hybrid dynamical systems. Previous results have been achieved by the authors concerning hybrid dynamical systems with a single impact surface and a single state space. Here, this work is extended to hybrid dynamical systems defined on a directed graph; each vertex corresponds to a state space and each

This paper shows that within the space of all LTI systems, equipped with the Zariski topology, the set of impulse controllable systems contains an open dense set of systems; in other words, impulse controllable systems are generic. This genericity persists for many closed subsets of LTI systems of interest, such as the class of singular descriptor systems.

This work revisits the constant stepsize stochastic approximation algorithm for tracking a slowly moving target and obtains a bound for the tracking error that is valid for the entire time axis, using the Alekseev nonlinear variation of constants formula. It is the first non-asymptotic bound for the entire time axis in the sense that it is not based on the vanishing stepsize limit and associated limit

We establish characterizations of weak input-to-state stability for abstract dynamical systems with inputs, which are similar to characterizations of uniform and of strong input-to-state stability established in a recent paper by A. Mironchenko and F. Wirth. We also investigate the relation of weak input-to-state stability to other common stability concepts, thus contributing to a better theoretical

Symmetries, e.g. rotational and translational invariances for the class of mechanical systems, allow to characterize solution trajectories of nonlinear dynamical systems. Thus, the restriction to symmetry-induced dynamics, e.g. by using the concept of motion primitives, may be considered as a quantization of the system. Symmetry exploitation is well established in both motion planning and control.

In a recent paper by Bourdin and Trélat, a version of the Pontryagin maximum principle (in short, PMP) has been stated for general nonlinear finite-dimensional optimal sampled-data control problems. Unfortunately, their result is only concerned with fixed sampling times, and thus, it does not take into account the possibility of free sampling times. The present paper aims to fill this gap in the literature

We provided a detailed study of the Schur–Cohn stability algorithm for Schur stable polynomials of one complex variable. Firstly, a real analytic principal \(\mathbb {C}\times \mathbb {S}^1\)-bundle structure in the family of Schur stable polynomials of degree n is constructed. Secondly, we consider holomorphic \(\mathbb {C}\)-actions \(\mathscr {A}\) on the space of polynomials of degree n. For each

This paper deals with the analysis of the internal control of a parabolic PDE with nonlinear diffusion, nonlocal in space. In our main result, we prove the local exact controllability to the trajectories with distributed controls, locally supported in space. The main ingredients of the proof are a compactness–uniqueness argument and Kakutani’s fixed-point theorem in a suitable functional setting. Some

In this paper, we investigate connectedness and convexity properties of the subspace \(\mathbf {L}_{n,m}^c(\mathbb {R})\) of controllable input pairs \((A,B)\in \mathbf {L}_{n,m}(\mathbb {R}):= \mathbb {R}^{n\times n}\times \mathbb {R}^{n\times m}\). We introduce three restricted convexity properties (“dense”, “almost sure” and “generic” convexity). In order to prove that the space \(\mathbf {L}_{n

We study a sequence of many-agent exit time stochastic control problems, parameterized by the number of agents, with risk-sensitive cost structure. We identify a fully characterizing assumption, under which each such control problem corresponds to a risk-neutral stochastic control problem with additive cost, and sequentially to a risk-neutral stochastic control problem on the simplex that retains only

In this paper, the problem of robust stability for linear time-varying implicit dynamic equations is generally studied. We consider the effect of uncertain structured perturbations on all coefficient matrices of equations. A formula of stability radius with respect to dynamic structured perturbations acting on the right-hand side coefficients is obtained. In case where structured perturbations affect

The Grassmann manifold\(Gr_m({\mathbb {R}}^n)\) of all m-dimensional subspaces of the n-dimensional space \({\mathbb {R}}^n\)\((m

This paper deals with some reachability issues for piecewise linear switched systems with time-dependent coefficients and multiplicative noise. Namely, it aims at characterizing data that are almost reachable at some fixed time \(T>0\) (belong to the closure of the reachable set in a suitable \({\mathbb {L}}^2\)-sense). From a mathematical point of view, this provides the missing link between approximate

Adaptive control deals with systems that have unknown and/or time-varying parameters. Most techniques are proven for the case in which any time variation is slow, with results for systems with fast time variations limited to those for which the time variation is of a known form or for which the plant has stable zero dynamics. In this paper, a new adaptive controller design methodology is proposed in

Along the ideas of Curtain and Glover (in: Bart, Gohberg, Kaashoek (eds) Operator theory and systems, Birkhäuser, Boston, 1986), we extend the balanced truncation method for (infinite-dimensional) linear systems to arbitrary-dimensional bilinear and stochastic systems. In particular, we apply Hilbert space techniques used in many-body quantum mechanics to establish new fully explicit error bounds for

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