Learning Dynamical Systems with Side Information

We present a mathematical and computational framework for the problem of learning a dynamical system from noisy observations of a few trajectories and subject to side information. Side information is any knowledge we might have about the dynamical system we would like to learn besides trajectory data. It is typically inferred from domain-specific knowledge or … Read more

Semidenite Approximations of Invariant Measures for Polynomial Systems

We consider the problem of approximating numerically the moments and the supports of measures which are invariant with respect to the dynamics of continuousand discrete-time polynomial systems, under semialgebraic set constraints. First, we address the problem of approximating the density and hence the support of an invariant measure which is absolutely continuous with respect to … Read more

Positive and Z-operators on closed convex cones

Let K be a closed convex cone with dual K-star in a finite-dimensional real Hilbert space V. A positive operator on K is a linear operator L on V such that L(K) is a subset of K. Positive operators generalize the nonnegative matrices and are essential to the Perron-Frobenius theory. We say that L is … Read more

Fast convergence of inertial dynamics and algorithms with asymptotic vanishing damping

In a Hilbert space setting $\mathcal H$, we study the fast convergence properties as $t \to + \infty$ of the trajectories of the second-order differential equation \begin{equation*} \ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \nabla \Phi (x(t)) = g(t), \end{equation*} where $\nabla\Phi$ is the gradient of a convex continuously differentiable function $\Phi: \mathcal H \to \mathbb R$, … Read more

A forward-backward dynamical approach to the minimization of the sum of a nonsmooth convex with a smooth nonconvex function

We address the minimization of the sum of a proper, convex and lower semicontinuous with a (possibly nonconvex) smooth function from the perspective of an implicit dynamical system of forward-backward type. The latter is formulated by means of the gradient of the smooth function and of the proximal point operator of the nonsmooth one. The … Read more

Convergence rates for forward-backward dynamical systems associated with strongly monotone inclusions

We investigate the convergence rates of the trajectories generated by implicit first and second order dynamical systems associated to the determination of the zeros of the sum of a maximally monotone operator and a monotone and Lipschitz continuous one in a real Hilbert space. We show that these trajectories strongly converge with exponential rate to … Read more

Second order forward-backward dynamical systems for monotone inclusion problems

We begin by considering second order dynamical systems of the from $\ddot x(t) + \Gamma (\dot x(t)) + \lambda(t)B(x(t))=0$, where $\Gamma: {\cal H}\rightarrow{\cal H}$ is an elliptic bounded self-adjoint linear operator defined on a real Hilbert space ${\cal H}$, $B: {\cal H}\rightarrow{\cal H}$ is a cocoercive operator and $\lambda:[0,+\infty)\rightarrow [0,+\infty)$ is a relaxation function depending … Read more

Formal property verification in a conformance testing framework

In model-based design of cyber-physical systems, such as switched mixed-signal circuits or software-controlled physical systems, it is common to develop a sequence of system models of different fidelity and complexity, each appropriate for a particular design or verification task. In such a sequence, one model is often derived from the other by a process of … Read more