On (Excessive) Transverse Coordinates for Orbital Stabilization of Periodic Motions

**dynamical**

**systems**. Expand abstract.

**dynamical**

**systems**. It is shown that the dynamics of any (minimal or excessive) set of transverse coordinates, which are defined in terms of a particular parameterization of the motion and a state-dependent projection operator recovering the parameterizing variable, admits a (transverse) linearization along the target motion, with explicit expressions stated. We then focus on a generic excessive set of orthogonal coordinates, and use these to illustrate a certain limitation of the corresponding excessive transverse linearization in regards to control design. To overcome this limitation, we introduce a new linear comparison

**system**of the linearized transverse dynamics, and state conditions for when the asymptotic stability of its origin corresponds to the asymptotic stability of the origin of linearized transverse dynamics. This consequently allows for the construction of feedback controllers utilizing this comparison

**system**which, when implemented on the

**dynamical**system, renders the desired motion asymptotically orbitally stable.

6/10 relevant

arXiv

Revisiting IRKA: Connections with pole placement and backward stability

**dynamical**

**systems**. Overall, \textsf{IRKA} has seen significant practical success in computing high fidelity (locally) optimal reduced models and has been successfully applied in a variety of large-scale settings. Moreover, \textsf{IRKA} has provided a foundation for recent extensions to the systematic model reduction of bilinear and nonlinear

**dynamical**

**systems**. Convergence of the basic \textsf{IRKA} iteration is generally observed to be rapid --- but not always; and despite the simplicity of the iteration, its convergence behavior is remarkably complex and not well understood aside from a few special cases. The overall effectiveness and computational robustness of the basic \textsf{IRKA} iteration is surprising since its algorithmic goals are very similar to a pole assignment problem, which can be notoriously ill-conditioned. We investigate this connection here and discuss a variety of nice properties of the \textsf{IRKA} iteration that are revealed when the iteration is framed with respect to a primitive basis. We find that the connection with pole assignment suggests refinements to the basic algorithm that can improve convergence behavior, leading also to new choices for termination criteria that assure backward stability.

5/10 relevant

arXiv

Inducing strong convergence of trajectories in **dynamical** **systems**
associated to monotone inclusions with composite structure

**dynamical**

**systems**are either of Krasnoselskii-Mann, of forward-backward type or of forward-backward-forward type, and with the help of injected regularization we demonstrate seminal results on the strong convergence of Hilbert space valued evolutions designed to solve monotone inclusion and equilibrium... Expand abstract.

**dynamical**

**systems**designed to approach the solution sets of inclusion problems involving the sum of two maximally monotone operators. Our aim is to design methods which guarantee strong convergence of trajectories towards the minimum norm solution of the underlying monotone inclusion problem. To that end, we investigate in detail the asymptotic behavior of

**dynamical**

**systems**perturbed by a Tikhonov regularization where either the maximally monotone operators themselves, or the vector field of the

**dynamical**

**system**is regularized. In both cases we prove strong convergence of the trajectories towards minimum norm solutions to an underlying monotone inclusion problem, and we illustrate numerically qualitative differences between these two complementary regularization strategies. The so-constructed

**dynamical**

**systems**are either of Krasnoselskii-Mann, of forward-backward type or of forward-backward-forward type, and with the help of injected regularization we demonstrate seminal results on the strong convergence of Hilbert space valued evolutions designed to solve monotone inclusion and equilibrium problems.

10/10 relevant

arXiv

Synthesis of Feedback Controller for Nonlinear Control **Systems** with
Optimal Region of Attraction

**dynamical**

**systems**. Expand abstract.

**systems**and control theory. By virtue here comes the connections to Lyapunov functions that are considered as the centerpiece of stability theory for a non linear

**dynamical**

**systems**. The agents may be imperfect because of several limitations in the sensors which ultimately restrict to fully observe the potential adversaries in the environment. Therefore while interacting with human life an autonomous robot should safely explore the outdoor environment by avoiding the dangerous states that may cause physical harm both the

**systems**and environment. In this paper we address this problem and propose a framework of learning policies that adapt to the shape of largest safe region in the state space. At the inception the model is trained to learn an accurate safety certificate for non-linear closed loop dynamics

**system**by constructing Lyapunov Neural Network. The current work is also an extension of the previous work of computing ROA under a fixed policy. Specifically we discuss how to design a state feedback controller by using a typical kind of performance objective function to be optimized and demonstrates our method on a simulated inverted pendulum which clearly shows that how this model can be used to resolve issues of trade-offs and extra design freedom.

5/10 relevant

arXiv

Non-intrusive model reduction of large-scale, nonlinear **dynamical**
**systems** using deep learning

**dynamical**

**systems**result from the semi-discretization of parametrized, nonlinear, hyperbolic partial differential equations---that show, in addition to non-intrusivity, the proposed approach provides more stable and accurate approximations to each

**dynamic**al... Expand abstract.

**dynamical**

**systems**so they can be used in many-query settings such as optimization and uncertainty quantification. For nonlinear systems, significant cost reduction is only possible with an additional layer of approximation to reduce the computational bottleneck of evaluating the projected nonlinear terms. Prevailing methods to approximate the nonlinear terms are code intrusive, potentially requiring years of development time to integrate into an existing codebase, and have been known to lack parametric robustness. This work develops a non-intrusive method to efficiently and accurately approximate the expensive nonlinear terms that arise in reduced nonlinear

**dynamical**

**system**using deep neural networks. The neural network is trained using only the simulation data used to construct the reduced basis and evaluations of the nonlinear terms at these snapshots. Once trained, the neural network-based reduced-order model only requires forward and backward propagation through the network to evaluate the nonlinear term and its derivative, which are used to integrate the reduced

**dynamical**

**system**at a new parameter configuration. We provide two numerical experiments---the

**dynamical**

**systems**result from the semi-discretization of parametrized, nonlinear, hyperbolic partial differential equations---that show, in addition to non-intrusivity, the proposed approach provides more stable and accurate approximations to each

**dynamical**

**system**across a large number of training and testing points than the popular empirical interpolation method.

10/10 relevant

arXiv

A Fej'{e}r theorem for boundary quotients arising from algebraic
**dynamical** **systems**

**dynamical**

**systems**. This makes it possible to strengthen a result on the structure of the relative commutant of a family of generating isometries in a boundary quotient.

10/10 relevant

arXiv

Surrogate Modeling of **Dynamic**s From Sparse Data Using Maximum Entropy
Basis Functions

**dynamical**

**systems**. Expand abstract.

**dynamical**

**systems**. A dynamics is approximated using basis functions, which are derived from maximization of the information-theoretic entropy, and can be generated directly from the data provided. This approach has advantages over other methods, where a dictionary of basis functions have to be provided by the user, which is non trivial in some applications. We compare the accuracy of the proposed data-driven modeling approach to existing methods in the literature, and demonstrate that for some applications the maximum entropy basis functions provide significantly more accurate models.

4/10 relevant

arXiv

Invariant states in inclined layer convection. Part 1. Temporal
transitions along **dynamical** connections between invariant states

**dynamical**

**systems**approach. We construct stable and unstable exact invariant states, including equilibria and periodic orbits of the fully nonlinear three-dimensional Oberbeck-Boussinesq equations. These invariant states underlie the observed convection patterns beyond their onset. We identify state-space trajectories that, starting from the unstable laminar flow, follow a sequence of

**dynamical**connections between unstable invariant states until the dynamics approaches a stable attractor. Together, the network of dynamically connected invariant states mediates temporal transitions between coexisting invariant states and thereby supports the observed complex time-dependent dynamics in inclined layer convection.

4/10 relevant

arXiv

**Dynamic** programming **systems** for modeling and control of the traffic in
transportation networks

**dynamic**programming

**system**s; 2) methods and approaches whose dynamic

**systems**are non-linear but are interpreted as stochastic dynamic programming systems. Expand abstract.

**Dynamic**programming

**systems**for modeling and control of the traffic in transportation networks. Two parts are distinguished in this dissertation: 1) methods and approaches based on min-plus or max-plus algebra, where the dynamics are deterministic

**dynamic**programming systems; 2) methods and approaches whose

**dynamic**

**systems**are non-linear but are interpreted as stochastic

**dynamic**programming

**systems**. Each of the two parts includes a chapter of necessary reviews, two main chapters and a chapter summarizing other works related to the concerned part. Part 1 includes a first chapter containing an introduction and some necessary reviews; two main chapters, one on the max-plus algebra model for the train dynamics on a metro line, the other one on the network calculus approach for modeling and calculating performance bounds on road networks; and a final chapter summarizing my other contributions on the topic of this part. Part 2 includes a first chapter containing an introduction and some necessary reviews; two main chapters, one on the microscopic modeling of traffic taking into account anticipation in driving, the other one on the modeling of the train dynamics on a metro line taking into account the passenger travel demand; and a final chapter summarizing my other contributions on the topic of this part.

4/10 relevant

arXiv

Non-coercive Lyapunov functions for input-to-state stability of
infinite-dimensional **systems**

**dynamical**

**systems**with inputs. Expand abstract.

**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.

6/10 relevant

arXiv