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Existence and uniqueness of solutions to rate independent systems with history variable

Livia Betz

TL;DR

The paper analyzes history-dependent, rate-independent evolutions described by $- abla_q\mathcal{E}(t,q(t)) \in \partial_2\mathcal{R}(\mathcal{H}(q)(t),\dot q(t))$, where the history enters via the integral operator $\mathcal{H}$ and the energy $\mathcal{E}$ is $\alpha$-uniformly convex. It employs a vanishing-viscosity regularization $\mathcal{R}_{\varepsilon}$ to establish existence of strong solutions and, under suitable convergence assumptions, existence of solutions for the rate-independent problem and optimal controls. A key, novel estimate exploits the history-operator’s structure to obtain uniform bounds and compactness, enabling the passage to the limit and the derivation of an energy balance. The work also proves a unique solvability result for a class with linear dependence on the velocity and unbounded subdifferentials, using a crucial bound and Grönwall-type arguments, together with a Lipschitz continuity of the solution map with respect to the data. These results lay the groundwork for future optimality conditions and control theory for history-dependent RIS, including explicit Lipschitz stability in the viscous regime and robust control-to-state mappings.

Abstract

This paper investigates rate-independent systems (RIS), where the dissipation functional depends not only on the rate but also on the history of the state. The latter is expressed in terms of an integral operator. We establish an existence result for the original problem and for the control thereof, without resorting to smallness assumptions. Under a smoothness condition, we prove the uniqueness of solutions to a certain class of history-dependent RIS where the subdifferential of the dissipation potential is an unbounded operator. In this context, we derive an essential estimate that opens the door to future research on the topic of optimization.

Existence and uniqueness of solutions to rate independent systems with history variable

TL;DR

The paper analyzes history-dependent, rate-independent evolutions described by , where the history enters via the integral operator and the energy is -uniformly convex. It employs a vanishing-viscosity regularization to establish existence of strong solutions and, under suitable convergence assumptions, existence of solutions for the rate-independent problem and optimal controls. A key, novel estimate exploits the history-operator’s structure to obtain uniform bounds and compactness, enabling the passage to the limit and the derivation of an energy balance. The work also proves a unique solvability result for a class with linear dependence on the velocity and unbounded subdifferentials, using a crucial bound and Grönwall-type arguments, together with a Lipschitz continuity of the solution map with respect to the data. These results lay the groundwork for future optimality conditions and control theory for history-dependent RIS, including explicit Lipschitz stability in the viscous regime and robust control-to-state mappings.

Abstract

This paper investigates rate-independent systems (RIS), where the dissipation functional depends not only on the rate but also on the history of the state. The latter is expressed in terms of an integral operator. We establish an existence result for the original problem and for the control thereof, without resorting to smallness assumptions. Under a smoothness condition, we prove the uniqueness of solutions to a certain class of history-dependent RIS where the subdifferential of the dissipation potential is an unbounded operator. In this context, we derive an essential estimate that opens the door to future research on the topic of optimization.
Paper Structure (9 sections, 21 theorems, 204 equations)

This paper contains 9 sections, 21 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. Standing assumptions and the viscous problem
  3. Uniform bounds
  4. Existence of solutions for the rate-independent system
  5. Existence of optimal solutions for the control problem
  6. Unique solvability for a class of rate-independent systems
  7. Lipschitz continuity of the solution map to \ref{['eq:n2']}
  8. Consequences of the uniqueness result and future work
  9. Fenchel conjugate functionals

Key Result

Lemma 2.10

\newlabellem:proj Let $\ell \in L^2(0,T;Y^\star)$. Then, $q\in H^1_0(0,T;Y)$ satisfies the evolution in eq:q1 if and only if where $\mathcal{V}_Y:Y \to Y^\ast$ is the Riesz isomorphism associated with the Hilbert space $Y$. For each $\zeta \in X$, the operator $P_{\partial_2 \mathcal{R}(\zeta, 0)}:Y^\ast \to Y^\ast$ is the (metric) projection onto the set with respect to the inner product $\lang

Theorems & Definitions (59)

  • Remark 2.2
  • Remark 2.5
  • Remark 2.6: Assumption \ref{['it:st1']}.\ref{['it:st12']}
  • Remark 2.7
  • Remark 2.9: Existence of viscous solutions
  • Lemma 2.10: The viscous problem as a non-smooth ODE, aos
  • Lemma 2.11: Lipschitz continuity of the projection, aos
  • Lemma 2.12: Another equivalent formulation
  • proof
  • Lemma 2.13
  • ...and 49 more