New Trends In Differential Equations, Control Theory And Optimization - Proceedings Of The 8th Congress Of Romanian Mathematicians

Proceedings of the 8th Congress of Romanian Mathematicians

Viorel Barbu, Cătălin Lefter, Ioan I Vrabie

• Publisher: WSPC
• Year: 2016
• ISBN:
  9789813142879
• Topic: Mathématiques
• Subtopic: Équations différentielles

The volume contains a collection of original papers and surveys in various areas of Differential Equations, Control Theory and Optimization written by well-known specialists and is thus useful for PhD students and researchers in applied mathematics.

Contents:

• Dirichlet Problems with Mean Curvature Operator in Minkowski Space (Cristian Bereanu, Petru Jebelean and Călin Şerban)
• Free Boundary Fluid-Elasticity Interactions: Adjoint Sensitivity Analysis (Lorena Bociu and Kristina Martin)
• Non-Smooth Regularization of a Forward-Backward Parabolic Equation (Elena Bonetti, Pierluigi Colli and Giuseppe Tomassetti)
• Approaching Monotone Inclusion Problems via Second Order Dynamical Systems with Linear and Anisotropic Damping (Radu Ioan Boţ and Ernö Robert Csetnek)
• On the Solutions of a Quadratic Integral Inclusion (Aurelian Cernea)
• On the Bounded and Stabilizing Solution of a Generalized Riccati Differential Equation with Periodic Coefficients Arising in Connection with a Zero Sum Linear Quadratic Stochastic Differential Game (Vasile Dragan and Toader Morozan)
• A Maximum Principle for a Class of First Order Differential Operators (Maria Fărcăşeanu, Mihai Mihăilescu and Denisa Stancu-Dumitru)
• Differentiability and Integrability Properties for Solutions to Nonlocal Equations (Mikil Foss and Petronela Radu)
• Ferroelectric Thin Structures (Antonio Gaudiello and Kamel Hamdache)
• Sliding Modes for a Phase-Field System (Gianni Gilardi)
• Uniformly Hyperbolic Viable Sets in Affine IFS (Vasile Glavan and Valeriu Guţu)
• Some Support Considerations in the Asymptotic Optimality of Two-Scale Controlled PDMP (Dan Goreac and Oana Silvia Serea)
• Inverse Problems for Control Theory (Mohammed Al Horani and Angelo Favini)
• On the Ill-Posedness of Active Scalar Equations with Odd Singular Kernels (Igor Kukavica, Vlad Vicol and Fei Wang)
• Equilibrium in an Individual — Societal SIR Vaccination Model in Presence of Discounting and Finite Vaccination Capacity (Laetitia Laguzet, Gabriel Turinici and Ghozlane Yahiaoui)
• On Some Minimization Problems in R N (Mihai Mariş)
• Recent Results on Multiple Periodic Solutions of Forced Relativistic Pendulum-Type Continuous and Discrete Systems (Jean Mawhin)
• On the Anisotropic Caginalp Phase-Field System with Singular Nonlinear Terms (Alain Miranville)
• Space, Time, Similarity (Umberto Mosco)
• Singularly Perturbed Problems for Abstract Differential Equations of Second Order in Hilbert Spaces (Andrei Perjan and Galina Rusu)
• Global Controllability and Mixing for the Burgers Equation with Localised Finite-Dimensional External Force (Armen Shirikyan)
• Boundary Observation in Shape Optimization (Dan Tiba)
• Recent Progress on Steady Gravity Water Waves (Eugen Vărvărucă)

Readership: Researchers in partial differential equations, calculus of variations and optimal control, difference and functional equations.

Approaching monotone inclusion problems via second order dynamical systems with linear and anisotropic damping

Radu Ioan Boţ¸^∗ and Ernö Robert Csetnek^∗∗

University of Vienna
Faculty of Mathematics
Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
E-mail:^∗[email protected],
∗∗ [email protected]

We investigate second order dynamical systems of the from

λ(t)B(x(t)) = 0, where Γ :

→

is an elliptic bounded self-adjoint linear operator defined on a real Hilbert space

,B :

→

is a cocoercive operator andψλ : [0, +∞) → (0, +∞) is a relaxation function depending on time. We prove via Lyapunov analysis that the generated trajectories weakly converge to a zero of the operatorB. This opens the gate towards approaching through a second order dynamical system the problem of determining zeros of the sum of a maximally monotone operator and a cocoercive one, which captures as particular case the minimization of the sum of a nonsmooth convex function with a smooth convex one. Finally, whenB is the gradient of a smooth convex function, we prove a rate of

(1/t) for the convergence of the function values along the ergodic trajectory to its minimum value.

Keywords: Dynamical systems; Lyapunov analysis; Monotone inclusions; Convex optimization problems; Continuous forward-backward method.

1.Introduction and preliminaries

Let

a real Hilbert space endowed with inner product 〈·, ·〉 and corresponding norm

The problem of approaching the minimization of a potential function

supposed to be convex and differentiable, has been considered by several authors see^3,7,8,11). These investigations addressed either the convergence of the generated trajectories to a critical point of f or the convergence of the function along the trajectories to its global minimum value. We recall in this context the heavy ball with friction dynamical system

which is a nonlinear oscillator with constant damping parameter γ > 0. When

= ℝ², this system describes the motion of a heavy ball that keeps rolling over the graph ...