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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, CÄtÄlin Lefter, Ioan I Vrabie
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eBook - ePub
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, CÄtÄlin Lefter, Ioan I Vrabie
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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.
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Sujet
MathématiquesSous-sujet
Ăquations diffĂ©rentiellesApproaching monotone inclusion problems via second order dynamical systems with linear and anisotropic damping
Radu Ioan Boƣžâ and Ernö Robert Csetnekââ
University of Vienna
Faculty of Mathematics
Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
E-mail:â[email protected],
ââ [email protected]
Faculty of Mathematics
Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
E-mail:â[email protected],
ââ [email protected]
We investigate second order dynamical systems of the from λ(t)B(x(t)) = 0, where Î : â is an elliptic bounded self-adjoint linear operator defined on a real Hilbert space ,B : â is a cocoercive operator andÏλ : [0, +â) â (0, +â) 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 â©Â·, ·âȘ 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 see3,7,8,11). 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 Îł > 0. When = â2, this system describes the motion of a heavy ball that keeps rolling over the graph ...