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The existence theory is developed for solutions of the inhomogeneous linearized field equations for causal variational principles. These equations are formulated weakly with an integral operator which is shown to be bounded and symmetric on…
This paper develops methods for proving Lyapunov stability of dynamical systems subject to disturbances with an unknown distribution. We assume only a finite set of disturbance samples is available and that the true online disturbance…
This paper introduces a second-order differential inclusion for unconstrained convex optimization. In continuous level, solution existence in proper sense is obtained and exponential decay of a novel Lyapunov function along with the…
Polyhedral Lyapunov functions can approximate any norm arbitrarily well. Because of this, they are used to study the stability of linear time varying and linear parameter varying systems without being conservative. However, the…
The Numerov method for linear second-order differential equations is generalized to include equations containing a first derivative term. The method presented has the same degree of accuracy as the ordinary Numerov sixth-order method. A…
Recently, many machine learning optimizers have been analysed considering them as the asymptotic limit of some differential equations when the step size goes to zero. In other words, the optimizers can be seen as a finite difference scheme…
We analyse the Krylov solvability of inverse linear problems on Hilbert space $\mathcal{H}$ where the underlying operator is compact and normal. Krylov solvability is an important feature of inverse linear problems that has profound…
Linear matrix equations, such as the Sylvester and Lyapunov equations, play an important role in various applications, including the stability analysis and dimensionality reduction of linear dynamical control systems and the solution of…
Let $L$ be the $n$-th order linear differential operator $Ly = \phi_0y^{(n)} + \phi_1y^{(n-1)} + \cdots + \phi_ny$ with variable coefficients. A representation is given for $n$ linearly independent solutions of $Ly=\lambda r y$ as power…
This paper introduces new solvers for the computation of low-rank approximate solutions to large-scale linear problems, with a particular focus on the regularization of linear inverse problems. Although Krylov methods incorporating explicit…
This is the third part of four series papers, aiming at the delay compensation for the abstract linear system (A,B,C). Both the input delay and output delay are investigated. We first propose a full state feedback control to stabilize the…
Non-self-adjoint second-order ordinary differential operators on a finite interval with complex weights are studied. Properties of spectral characteristics are established and the inverse problem of recovering operators from their spectral…
A symmetric characteristic singular integral equation with two fixed singularities at the endpoints in the class of functions bounded at the ends is analyzed. It reduces to a vector Hilbert problem for a half-disc and then to a vector…
This paper deals with a special type of Lyapunov functions, namely the solution of Zubov's equation. Such a function can be used to characterize the domain of attraction for systems of ordinary differential equations. We derive and prove an…
We consider small nonlinear perturbations of linear systems on a time scale with the phase space being finite or infinite-dimensional. For $\Delta$-differential operators, corresponding to linear dynamic systems we consider their…
We suggest and compare different methods for the numerical solution of Lyapunov like equations with application to control of Markovian jump linear systems. First, we consider fixed point iterations and associated Krylov subspace…
The aim of the paper is to find representation for solutions of $2\times 2$ system of ordinary differential equations $$ \mathbf{y^\prime} - B(x)\mathbf{y} = \lambda A(x)\mathbf{y}, \quad \ x \in [0, 1], $$ where $A(x) = diag\{a_1(x),…
This paper considers the stability problem of a linear time invariant system in feedback with a string equation. A new Lyapunov functional candidate is proposed based on the use of augmented states which enriches and encompasses the…
Particle-based simulations of the Vlasov equation typically require a large number of particles, which leads to a high-dimensional system of ordinary differential equations. Solving such systems is computationally very expensive, especially…
The numerical solution of linear discrete ill-posed problems typically requires regularization, i.e., replacement of the available ill-conditioned problem by a nearby better conditioned one. The most popular regularization methods for…