Feedback Nash Equilibria for Linear Quadratic Descriptor Differential Games

In this note we consider the non-cooperative linear feedback Nash quadratic differential game with an infinite planning horizon for descriptor systems of index one. The performance function is assumed to be indefinite. We derive both necessary and sufficient conditions under which this game has a Nash equilibrium.linear-quadratic games;linear feedback Nash equilibrium;affine systems;solvability conditions;Riccati equations

A Bound for the Eigenvalue Counting Function for Higher-Order Krein Laplacians on Open Sets

For an arbitrary nonempty, open set $\Omega \subset \mathbb{R}^n$, $n \in \mathbb{N}$, of finite (Euclidean) volume, we consider the minimally defined higher-order Laplacian $(- \Delta)^m\big|_{C_0^{\infty}(\Omega)}$, $m \in \mathbb{N}$, and its Krein--von Neumann extension $A_{K,\Omega,m}$ in $L^2(\Omega)$. With $N(\lambda,A_{K,\Omega,m})$, $\lambda > 0$, denoting the eigenvalue counting function corresponding to the strictly positive eigenvalues of $A_{K,\Omega,m}$, we derive the bound $$N(\lambda,A_{K,\Omega,m}) \leq (2 \pi)^{-n} v_n |\Omega| \{1 + [2m/(2m+n)]\}^{n/(2m)} \lambda^{n/(2m)}, \quad \lambda > 0,$$ where $v_n := \pi^{n/2}/\Gamma((n+2)/2)$ denotes the (Euclidean) volume of the unit ball in $\mathbb{R}^n$. The proof relies on variational considerations and exploits the fundamental link between the Krein--von Neumann extension and an underlying (abstract) buckling problem.Comment: 22 pages. Considerable improvements mad

Asymptotic Expansions for Stationary Distributions of Perturbed Semi-Markov Processes

New algorithms for computing of asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes are presented. The algorithms are based on special techniques of sequential phase space reduction, which can be applied to processes with asymptotically coupled and uncoupled finite phase spaces.Comment: 83 page

Robust stability of differential-algebraic equations

This paper presents a survey of recent results on the robust stability analysis and the distance to instability for linear time-invariant and time-varying differential-algebraic equations (DAEs). Different stability concepts such as exponential and asymptotic stability are studied and their robustness is analyzed under general as well as restricted sets of real or complex perturbations. Formulas for the distances are presented whenever these are available and the continuity of the distances in terms of the data is discussed. Some open problems and challenges are indicated

Feedback Nash Equilibria for Descriptor Differential Games Using Matrix Projectors

In this article we address the problem of finding feedback Nash equilibria for linear quadratic differential games defined on descriptor systems. First, we decouple the dynamic and algebraic parts of a descriptor system using canonical projectors. We discuss the effects of feedback on the behavior of the descriptor system. We derive necessary and sufficient conditions for the existence of the feedback Nash equilibria for index 1 descriptor systems and show that there exist many informationally non-unique equilibria corresponding to a single solution of the game. Further, for descriptor systems with index greater than 1, we give a regularization based approach and discuss the associated drawbacks.

Homogenization of Steklov spectral problems with indefinite density function in perforated domains

The asymptotic behavior of second order self-adjoint elliptic Steklov eigenvalue problems with periodic rapidly oscillating coefficients and with indefinite (sign-changing) density function is investigated in periodically perforated domains. We prove that the spectrum of this problem is discrete and consists of two sequences, one tending to -{\infty} and another to +{\infty}. The limiting behavior of positive and negative eigencouples depends crucially on whether the average of the weight over the surface of the reference hole is positive, negative or equal to zero. By means of the two-scale convergence method, we investigate all three cases.Comment: 24 pages. arXiv admin note: substantial text overlap with arXiv:1106.390

A bound for the eigenvalue counting function for Krein--von Neumann and Friedrichs extensions

For an arbitrary open, nonempty, bounded set $\Omega \subset \mathbb{R}^n$, $n \in \mathbb{N}$, and sufficiently smooth coefficients $a,b,q$, we consider the closed, strictly positive, higher-order differential operator $A_{\Omega, 2m} (a,b,q)$ in $L^2(\Omega)$ defined on $W_0^{2m,2}(\Omega)$, associated with the higher-order differential expression $$\tau_{2m} (a,b,q) := \bigg(\sum_{j,k=1}^{n} (-i \partial_j - b_j) a_{j,k} (-i \partial_k - b_k)+q\bigg)^m, \quad m \in \mathbb{N},$$ and its Krein--von Neumann extension $A_{K, \Omega, 2m} (a,b,q)$ in $L^2(\Omega)$. Denoting by $N(\lambda; A_{K, \Omega, 2m} (a,b,q))$, $\lambda > 0$, the eigenvalue counting function corresponding to the strictly positive eigenvalues of $A_{K, \Omega, 2m} (a,b,q)$, we derive the bound $$N(\lambda; A_{K, \Omega, 2m} (a,b,q)) \leq C v_n (2\pi)^{-n} \bigg(1+\frac{2m}{2m+n}\bigg)^{n/(2m)} \lambda^{n/(2m)} , \quad \lambda > 0,$$ where $C = C(a,b,q,\Omega)>0$ (with $C(I_n,0,0,\Omega) = |\Omega|$) is connected to the eigenfunction expansion of the self-adjoint operator $\widetilde A_{2m} (a,b,q)$ in $L^2(\mathbb{R}^n)$ defined on $W^{2m,2}(\mathbb{R}^n)$, corresponding to $\tau_{2m} (a,b,q)$. Here $v_n := \pi^{n/2}/\Gamma((n+2)/2)$ denotes the (Euclidean) volume of the unit ball in $\mathbb{R}^n$. Our method of proof relies on variational considerations exploiting the fundamental link between the Krein--von Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of $\widetilde A_{2} (a,b,q)$ in $L^2(\mathbb{R}^n)$. We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension $A_{F,\Omega, 2m} (a,b,q)$ in $L^2(\Omega)$ of $A_{\Omega, 2m} (a,b,q)$. No assumptions on the boundary $\partial \Omega$ of $\Omega$ are made.Comment: 39 pages. arXiv admin note: substantial text overlap with arXiv:1403.373

A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq \varepsilon I_{\mathcal{H}}$ for some $\varepsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for $H_{K,\Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-\Delta+V$ (in short, the perturbed Krein Laplacian) defined on $C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$.Comment: 68 pages. arXiv admin note: extreme text overlap with arXiv:0907.144