This paper addresses the tempered dichotomy for dynamical systems, generated by the random linear infinite-dimensional equations over measure-preserving dynamical systems (Ω , ℱ, P, θ ) on the half-line [0,∞ ) . We prove that the solutions of u̇=A(θ _t)u+f, t≥ 0 belong to ℒ^∞ _1 for any f∈ℒ^p_K if and only if the homogenous system u̇=A(θ _t)u has a tempered exponential dichotomy on the half-line ℝ_+ .
The main purpose of this paper is to provide an explicit description of the invariant control sets for a class of control systems induced on the unit quaternion sphere $$S^3$$ S 3 by the action of the Lorentz group $$\textrm{SO}(1,4)$$ SO ( 1 , 4 ) and then generalize it to the sphere $$S^{n-1}$$ S n - 1 . These control sets are the maximal subsets of approximate controllability for the control systems. Describing them in detail is generally challenging due to the complexity of the geometry and topology of the underlying differentiable manifold and the behavior of the vector fields defining the control system. In this work, the Lie theory and the quaternions play a fundamental role in achieving our main results.
We compensate for the scalability issues in controller synthesis by developing a hierarchical control scheme within the framework of ( $$\upgamma ,\updelta $$ γ , δ )-similarity, which measures to what extent a potentially non-deterministic system satisfies specifications expressed as solution trajectories of a dynamical ‘specification’ system. This scheme synthesizes a controller for a non-deterministic ‘concrete’ system in three hierarchical steps. First, an ‘abstract’ system, which is a low-dimensional model of the concrete system, is obtained. Then, a controller is designed for the abstract system. At last, the abstract controller is refined into the concrete controller through an ‘interface’. To enable this, we introduce and characterize the notion of ( $$\upgamma ,\updelta $$ γ , δ )-abstraction that utilizes an $$\mathcal {L}_2$$ L 2 approximation metric to measure the behavioral similarity of the concrete system and its abstraction in the presence of the interface. We utilize this characterization to propose a step-by-step procedure to construct the interface. We then synthesize the abstract controller and refine it into a concrete one.
We study optimal control problems for mechanical systems with control-dependent inertia evolving on Lie groups, with application to the attitude dynamics of foldable multirotor unmanned aerial vehicles (UAVs). In this setting, internal controls modify the inertia tensor of the system, producing geometric effects that are fundamentally different from those of external force inputs. We derive the necessary conditions for optimality using a presymplectic formulation on an extended phase space, which intrinsically accounts for the interplay between internal and external controls. Since the configuration space is the Lie group SO(3), we construct variational integrators that preserve the geometric structure of the optimal control flow, including the manifold structure and symplecticity, by discretizing the variational principle rather than the equations of motion directly. We prove that the resulting discrete scheme satisfies a regularity condition analogous to that of the continuous-time problem. Numerical simulations for a foldable quadrotor validate the proposed framework and illustrate the structure-preserving properties of the integrator in comparison with standard numerical methods.
This paper studies properties of homogeneous approximations in a geometric, coordinate-free setting. Specifically, we develop the concept of homogeneity in the 0-limit and ∞ -limit (i.e., homogeneity in the bi-limit) with respect to general dilations induced by semi-Euler vector fields. A key contribution of the paper is a result relating the regularity properties of homogeneous in the bi-limit functions and vector fields to their degree of homogeneity and the local behavior of the associated dilation flow near the equilibrium set and infinity. Building on these concepts, we establish new finite time and fixed time semistability results for a class of dynamical systems possessing a continuum of equilibria that are homogeneous in the bi-limit with respect to a semi-Euler vector field. Finally, we use these results to develop thermodynamically inspired distributed consensus control protocols for multiagent network systems for achieving coordination tasks in fixed time.
We study the adaptive infinite-horizon discounted control problem for Piecewise Deterministic Markov Processes (PDMPs) using a Nonstationary Value Iteration (NVI) scheme. PDMPs, as introduced by Davis (Markov models and optimization, monographs on statistics and applied probability, Chapman and Hall, London, 1993) evolve deterministically between random jumps whose jump rate λ , transition measure Q, and cost C depend on an unknown parameter β ^* . The proposed NVI algorithm recursively updates the value function using current parameter estimates, enabling online implementation. We show that, for any sequence of strongly consistent estimators {β ^*_n} converging almost surely to β ^* , the resulting policy is asymptotically optimal under the discounted criterion.
We study data-driven optimal control of continuous-time linear systems over finite- and infinite-time horizons. Our approach builds on our continuous-time version of Willems et al.’s fundamental lemma and on the use of orthogonal basis functions to approximate system trajectories. We show that the solution to an optimal control problem can be approximated by a finite linear combination of basis functions and we establish error bounds for such approximations. Moreover, we approximately solve the algebraic Riccati equation and the associated optimal controller gain directly from data, opening up the possibility of optimal controller design directly from data analogue devices.
We consider the stability analysis of feedback systems with rectified linear unit (ReLU) activations, and model this problem with polynomial optimization. Stability can be certified by means of copositive multipliers in the framework of integral quadratic constraints. Based on a duality argument, we show how to certify instability by considering a complete hierarchy of linear matrix inequalities. This hierarchy is obtained by leveraging the specific equality constraints arising from the ReLU encoding. We illustrate the effectiveness of the proposed approach through several numerical examples.
We compensate for the scalability issues in controller synthesis by developing a hierarchical control scheme within the framework of ( γ , δ )-similarity, which measures to what extent a potentially non-deterministic system satisfies specifications expressed as solution trajectories of a dynamical 'specification' system. This scheme synthesizes a controller for a non-deterministic 'concrete' system in three hierarchical steps. First, an 'abstract' system, which is a low-dimensional model of the concrete system, is obtained. Then, a controller is designed for the abstract system. At last, the abstract controller is refined into the concrete controller through an 'interface'. To enable this, we introduce and characterize the notion of ( γ , δ )-abstraction that utilizes an L 2 approximation metric to measure the behavioral similarity of the concrete system and its abstraction in the presence of the interface. We utilize this characterization to propose a step-by-step procedure to construct the interface. We then synthesize the abstract controller and refine it into a concrete one.
In this paper, we revisit a fundamental technical issue within the theory of stochastic approximation (SA) in a Markovian framework, first proposed in the book by Derevitskii and Fradkov (Applied theory of discrete adaptive control systems, Nauka, 1981), and further developed in much detail in the book by Benveniste, Métivier, and Priouret (Adaptive algorithms and stochastic approximations, Springer, Berlin, 1990). This theory is instrumental in many application areas such as the statistical analysis of Hidden Markov Models arising in telecommunication, quantized linear stochastic systems, and reinforcement learning. The problem at hand is the verification of the existence, uniqueness, and Lipschitz continuity of the solution of a parameter-dependent Poisson equation, in an appropriate weighted sup-norm, associated with a collection of Markov chains on general state spaces. Verification of the above facts is vital in the analysis of SA processes presented in the cited book via the ODE (ordinary differential equations) method, requiring substantial technical effort. The motivation and focus of the paper is to address this technical issue, by presenting a simple set of conditions, under which the above properties of the Poisson equation at hand can be conveniently established. A distinctive feature of our work is that it is based on a remarkable result of Hairer and Mattingly (2011), proving that by tilting standard conditions of mainstream stability theory for Markov chains, the transition kernels prove to be contractions in the space of differences of probability measures in a suitable metric. To demonstrate the applicability of our results, the proposed conditions are verified for a class of queuing system with open-loop control.
We investigate the uniform stabilization of two elastic strings in series, coupled with a dynamic mass at an interior node, under three damping schemes: classical boundary damping, lower-order nodal (tip-velocity) feedback, and a novel higher-order nodal (strain-velocity) feedback. It is shown that when higher-order nodal damping is paired with boundary damping the full system is unconditionally exponentially stable; by contrast, boundary damping alone, or boundary plus lower-order nodal feedback, admits at best the sharp t^-1 decay first established by [Littman-Taylor’02] and found in the strong stabilization result of [Hansen-Zuazua’95]. Remarkably, even in the absence of any boundary dissipation, higher-order nodal feedback alone enforces exponential decay provided the wave-speed ratio satisfies an explicit arithmetic condition, whereas lower-order nodal feedback remains confined to the t^-1 rate, refining and completing earlier partial results of [Chen-Coleman-West’87] and [Lee-You’89]. These findings are illustrated by finite-difference simulations of solution profiles, eigenvalue spectra, and energy-decay curves across varying damping configurations, speed ratios, and mesh resolutions, which confirm the decisive role of the arithmetic condition in distinguishing exponential, polynomial, or no decay.
We compensate for the scalability issues in controller synthesis by developing a hierarchical control scheme within the framework of ( , )-similarity, which measures to what extent a potentially non-deterministic system satisfies specifications expressed as solution trajectories of a dynamical ‘specification’ system. This scheme synthesizes a controller for a non-deterministic ‘concrete’ system in three hierarchical steps. First, an ‘abstract’ system, which is a low-dimensional model of the concrete system, is obtained. Then, a controller is designed for the abstract system. At last, the abstract controller is refined into the concrete controller through an ‘interface’. To enable this, we introduce and characterize the notion of ( , )-abstraction that utilizes an ℒ_2 approximation metric to measure the behavioral similarity of the concrete system and its abstraction in the presence of the interface. We utilize this characterization to propose a step-by-step procedure to construct the interface. We then synthesize the abstract controller and refine it into a concrete one.
This paper provides two results that are useful in the study of the existence and the stability properties of a periodic solution for a given dynamical system. The first result deals with scalar time-periodic systems and establishes the equivalence of the existence of a periodic solution and the existence of a bounded solution. The second result provides sufficient conditions for the existence and the stability of a periodic solution for a time-periodic dynamical system. Both results are applied to extremum seeking problems for a static output map with no plant dynamics, and novel non-local results are provided without invoking averaging theorems and singular perturbation arguments.
Landmark manifolds consist of finite collections of distinct points in an underlying space, referred to as landmark configurations. The dynamics of these configurations can be used to represent flows, such as solutions to ODEs or shape deformations. In this work, we consider landmark configurations in Euclidean space and study how they can be connected via flows of vector fields. For dimensions greater than or equal to two, we explicitly construct two vector fields whose flows can connect any pair of landmark configurations with the same cardinality. This property is known as exact universal interpolation. In dimension one, we show that the same result holds for pairs of landmark configurations that share the same relative ordering. In all dimensions, controllability is achieved using one constant vector field and one polynomial vector field of degree three.
The main purpose of this paper is to provide an explicit description of the invariant control sets for a class of control systems induced on the unit quaternion sphere S^3 by the action of the Lorentz group SO(1,4) and then generalize it to the sphere S^n-1 . These control sets are the maximal subsets of approximate controllability for the control systems. Describing them in detail is generally challenging due to the complexity of the geometry and topology of the underlying differentiable manifold and the behavior of the vector fields defining the control system. In this work, the Lie theory and the quaternions play a fundamental role in achieving our main results.
Numerical methods for developing port-Hamiltonian representations of general linear time-invariant systems are studied. The approach extends previous port-Hamiltonian characterizations to include the general non-minimal case and the case where the feedthrough term fails to have an invertible symmetric part. The resulting construction is able to identify infeasibility when the system fails to be port-Hamiltonian, and allows for the incorporation of perturbations in order to arrive at a nearby port-Hamiltonian system. Results are illustrated via numerical examples.
In this paper, we study a specific class of integro-differential systems within Hilbert spaces that aligns with the Coleman-Gurtin model of heat conduction incorporating memory effects. Motivated by the recent results in [1], we relax the regularity assumptions on the kernels compared to previous studies. The well-posedness of the system’s state equation is demonstrated through the application of semigroups and resolvents operator theories. By relying on Laplace transform methods, we derive sufficient conditions under which the finite-time (and infinite-time) admissibility of the system’s observation operator can be inferred from the corresponding finite-time admissibility of the same operator for the related first-order Cauchy system, which lacks convolution terms. The finite-time admissibility is established by integrating a perturbation semigroup approach with admissible observation operators, while the infinite-time admissibility is achieved using the semigroup method combining with the Hardy space technique. Finally, illustrative examples are presented.
