Special Session 30: Evolution Equations, their Control and Applications

Mathematical analysis of a nonlinear contact problem for a viscoelastic rod

Krzysztof Bartosz Jagiellonian University Poland Co-Author(s):

Abstract: We consider a nonlinear viscoelastic rod that is in contact with a foundation along its length and with an obstacle at its end. The rod is subjected to body forces and, as a result, its mechanical state evolves. We assume that the internal stress at the end of the rod depends on both its displacement and velocity, and that this relationship takes the form of a Clarke subdifferential inclusion. In a specific case, it coincides with the so-called damped normal compliance condition introduced recently in contact mechanics. This generalizes our previous model, in which the stress depends only on the velocity. Our aim is twofold. The first is to construct an appropriate mathematical model that describes the evolution of the rod. The second is to prove the existence of a weak solution to the problem. To this end, we use a time discretization method for second-order inclusion with multivalued pseudomonotone operators, which constitutes the weak formulation of the problem.

Nonlinear impulsive evolution inclusions with history-dependent operators

Anna Ochal Jagiellonian University in Krakow Poland Co-Author(s):

Abstract: In this talk, we present recent advancements in the mathematical analysis and control of nonlinear impulsive evolution inclusions formulated within the framework of an evolution triple of spaces. Our research focuses on a class of variational-hemivariational inequalities involving both convex and nonconvex potentials, as well as history-dependent operators. These systems represent a novel integration of impulsive effects - which introduce discontinuities and state jumps - with nonsmooth potentials described by the Clarke generalized gradient. Using tools from nonsmooth and nonconvex analysis, specifically the Clarke generalized gradient, we establish the existence of solutions to these impulsive systems. A significant part of the presentation is devoted to an optimal control problem, where we prove the existence of optimal control-state triples under general hypotheses on the cost functional and control sets. Furthermore, we illustrate the theoretical results with applications to physical models, such as semipermeability problems and frictional contact, where surface traction is governed by impulsive differential equations. This work highlights the synergy between nonsmooth analysis, impulsive dynamics, and memory effects in solving complex problems in mechanics and physical sciences.

Hemivariational inequality with state-dependent delay

Shashank Pandey IIT Roorkee India Co-Author(s):    Dwijendra N. Pandey

Abstract: We develop a new class of hemivariational inequality that explicitly accounts for state-dependent memory. Our analysis establishes the existence and approximate controllability of mild solutions by recasting the problem as a differential inclusion. To demonstrate scope, we conclude with a heat equation featuring distributed memory that meets the hypotheses of our main results.