Sharp Tunneling Estimates for a Double-Well Model in **Infinite** **Dimension**

**infinite**-

**dimensional**version of some sharp tunneling estimates known in finite dimensions for semiclassical Witten Laplacians in degree zero. From a stochastic point of view it proves that the $L^2$ spectral gap of the stochastic one-

**dimensional**Allen-Cahn equation in finite volume satisifies a Kramers-type formula in the limit of vanishing noise. We work with finite-

**dimensional**lattice approximations and establish semiclassical estimates which are uniform in the

**dimension**. Our key estimate shows that the constant separating the two exponentially small eigenvalues from the rest of the spectrum can be taken independently of the

**dimension**.

5/10 relevant

arXiv

A Survey on Invariant Cones in **Infinite** **Dimensional** Lie Algebras

**infinite**-

**dimensional**Lie group~$G$, there is a natural duality between so-called semi-equicontinuous weak-*-closed convex Ad^*(G)-invariant subsets of the dual space $\g'$ and Ad(G)-invariant lower semicontinuous positively homogeneous convex functions on open convex cones in $\g$. In this survey, we discuss various aspects of this duality and some of its applications to a more systematic understanding of open invariant cones and convexity properties of coadjoint orbits. In particular, we show that root decompositions with respect to elliptic Cartan subalgebras provide powerful tools for important classes of

**infinite**Lie algebras, such as completions of locally finite Lie algebras, Kac--Moody algebras and twisted loop algebras with

**infinite**-

**dimensional**range spaces. We also formulate various open problems.

10/10 relevant

arXiv

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

**infinite**-

**dimensional**dynamical systems with inputs. Expand abstract.

**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.

10/10 relevant

arXiv

On **infinite**-**dimensional** graded modular Lie algebras of maximal class
generated by two elements of different weights

**infinite**-

**dimensional**graded modular Lie algebras of maximal class generated by two elements of weights $1$ and $n$. Such Lie algebras, which we call algebras of type $n$, can be viewed as a generalization of certain graded Lie algebras of maximal class generated by two elements of weights $1$ and $2$ investigated in 2000 by Caranti and Vaughan-Lee.

10/10 relevant

arXiv

Entanglement Wedge Reconstruction of **Infinite**-**dimensional** von Neumann
Algebras using Tensor Networks

**dimensional**Hilbert spaces have yielded new insights on bulk reconstruction in AdS/CFT. In this paper, we give an explicit construction of a quantum error correcting code where the code and physical Hilbert spaces are

**infinite**-dimensional. We define a von Neumann algebra of type II$_1$ acting on the code Hilbert space and show how it is mapped to a von Neumann algebra of type II$_1$ acting on the physical Hilbert space. This toy model demonstrates the equivalence of entanglement wedge reconstruction and the exact equality of bulk and boundary relative entropies in

**infinite**-

**dimensional**Hilbert spaces.

10/10 relevant

arXiv

Observer-based boundary control of distributed port-Hamiltonian systems

**infinite**-

**dimensional**system is asymptotically stable. Expand abstract.

**infinite**-

**dimensional**port-Hamiltonian system defined on 1D spatial domains is proposed. The design is based on an early-lumping approach in which a finite-

**dimensional**approximation of the

**infinite**-

**dimensional**system is used to design the observer and the controller. The main contribution is a constructive method which guarantees that the interconnection between the controller and the

**infinite**-

**dimensional**system is asymptotically stable. A Timoshenko beam model has been used to illustrate the approach.

6/10 relevant

arXiv

Computing Casimir invariants from Pfaffian systems

**infinite**-

**dimensional**examples, including a Poisson bracket embodying both finite and infinite-dimensional structure. Expand abstract.

**infinite**-

**dimensional**Poisson brackets. We apply the method to various finite and

**infinite**-

**dimensional**examples, including a Poisson bracket embodying both finite and

**infinite**-

**dimensional**structure.

6/10 relevant

arXiv

Stability Analysis of Perturbed **Infinite**-**dimensional** Sampled-data
Systems

**infinite**-

**dimensional**sampled-data systems under unbounded perturbations. Expand abstract.

**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.

10/10 relevant

arXiv

PIETOOLS: A Matlab Toolbox for Manipulation and Optimization of Partial Integral Operators

**infinite**-

**dimensional**systems such as Partial Differential equations (PDE) and Time-delay systems (TDS). Expand abstract.

**infinite**-

**dimensional**spaces that form a *-subalgebra with two binary operations (addition and composition) on the space RxL2. These operators frequently appear in analysis and control of

**infinite**-

**dimensional**systems such as Partial Differential equations (PDE) and Time-delay systems (TDS). Furthermore, PIETOOLS can: declare opvar decision variables, add operator positivity constraints, declare an objective function, and solve the resulting optimization problem using a syntax similar to the sdpvar class in YALMIP. Use of the resulting Linear Operator Inequalities (LOIs) are demonstrated on several examples, including stability analysis of a PDE, bounding operator norms, and verifying integral inequalities. The result is that PIETOOLS, packaged with SOSTOOLS and MULTIPOLY, offers a scalable, user-friendly and computationally efficient toolbox for parsing, performing algebraic operations, setting up and solving convex optimization problems on PI operators.

4/10 relevant

arXiv

Input-to-state stability of **infinite**-**dimensional** systems: recent results
and open questions

**infinite**-

**dimensional**systems. Expand abstract.

**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.

10/10 relevant

arXiv