arXiv subjects

Merlin Andreia

Publications and source records attributed to Merlin Andreia.

3 recordsLinked to original sources

arXiv2026

Viscous Approximation of Optimal Control Problems Governed by Rate-Independent Systems with Non-Convex Energies

We consider an optimal control problem governed by a rate-inde\-pendent system with non-convex energy. The state equation is approximated by means of viscous regularization w.r.t.\ to hierarchy of two different Hilbert spaces. The regularized problem corresponds to an optimal control problem subject to a non-smooth ODE in Hilbert space, which is substantially easier to solve than the original optimal control problem. The convergence properties of the viscous regularization are investigated. It is shown that every sequence of globally optimal solutions of the viscous problems admits a (weakly) converging subsequence whose limit is a globally optimal solution of the original problem, provided that the latter admits at least one optimal solution with an optimal state that is continuous in time.

math.OC↗

arXiv2023

On a Lack of Stability of Parametrized BV Solutions to Rate-Independent Systems with Non-Convex Energies and Discontinuous Loads

We consider a rate-independent system with nonconvex energy under discontinuous external loading. The underlying space is finite dimensional and the loads are functions in $BV([0,T];\mathbb{R}^d)$. We investigate the stability of various solution concepts w.r.t. a sequence of loads converging weakly$*$ in $BV([0,T];\mathbb{R}^d)$ with a particular emphasis on the so-called normalized, $\mathfrak{p}$-parametrized balanced viscosity solutions. By means of two counterexamples, it is shown that common solution concepts are not stable w.r.t. weak$*$ convergence of loads in the sense that a limit of a sequence of solutions associated with these loads need not be a solution corresponding to the load in the limit. We moreover introduce a new solution concept, which is stable in this sense, but our examples show that this concept necessarily allows "solutions" that are physically meaningless.

math.AP↗

arXiv2022

An Adaptive Time Stepping Scheme for Rate-Independent Systems with Non-Convex Energy

We investigate a local incremental stationary scheme for the numerical solution of rate-independent systems. Such systems are characterized by a (possibly) non-convex energy and a dissipation potential, which is positively homogeneous of degree one. Due to the non-convexity of the energy, the system does in general not admit a time-continuous solution. In order to resolve these potential discontinuities, the algorithm produces a sequence of state variables and physical time points as functions of a curve parameter. The main novelty of our approach in comparison to existing methods is an adaptive choice of the step size for the update of the curve parameter depending on a prescribed tolerance for the residua in the energy-dissipation balance and in a complementarity relation concerning the so-called local stability condition. It is proven that, for tolerance tending to zero, the piecewise affine approximations generated by the algorithm converge (weakly) to a so-called $\mathbb{V}$-parametrized balanced viscosity solution. Numerical experiments illustrate the theoretical findings and show that an adaptive choice of the step size indeed pays off as they lead to a significant increase of the step size during sticking and in viscous jumps.

math.NA↗