Optimal control of a rate-independent evolution equation via viscous regularization

Stefanelli, Ulisse; Wachsmuth, Daniel; Wachsmuth, Gerd

doi:10.3934/dcdss.2017076

Cited by 12 publications

(31 citation statements)

References 42 publications

(41 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…According to Proposition 6 and the chain rule, Ψ is Fréchet differentiable. If we denote by (v, τ, η) the solution of (20) and the adjoint state by (w ϕ , ϕ, w T ), then a lengthy, but straightforward computation using the relations in (20) and (22)…”

Section: Adjoint Equationmentioning

confidence: 99%

“…An optimality condition for the original non-smooth optimal control problem (P) could be derived by passing to the limit λ, δ 0 in the regularized optimality system (22) and (26). This has been done for the case with hardening in [26] and for a scalar rate-independent system with uniformly convex energy in [20]. The optimality systems obtained in the limit are comparatively weak compared to what can be derived by regularization in the static case, see [12] for the latter.…”

Section: Adjoint Equationmentioning

confidence: 99%

See 1 more Smart Citation

Optimal control of perfect plasticity part I: Stress tracking

Meyer¹,

Walther²

2022

MCRF

View full text Add to dashboard Cite

The paper is concerned with an optimal control problem governed by the rate-independent system of quasi-static perfect elasto-plasticity. The objective is to optimize the stress field by controlling the displacement at prescribed parts of the boundary. The control thus enters the system in the Dirichlet boundary conditions. Therefore, the safe load condition is automatically fulfilled so that the system admits a solution, whose stress field is unique. This gives rise to a well defined control-to-state operator, which is continuous but not Gâteaux differentiable. The control-to-state map is therefore regularized, first by means of the Yosida regularization resp. viscous approximation and then by a second smoothing in order to obtain a smooth problem. The approximation of global minimizers of the original non-smooth optimal control problem is shown and optimality conditions for the regularized problem are established. A numerical example illustrates the feasibility of the smoothing approach.

show abstract

Section: Adjoint Equationmentioning

confidence: 99%

Section: Adjoint Equationmentioning

confidence: 99%

Optimal control of perfect plasticity part I: Stress tracking

Meyer¹,

Walther²

2022

MCRF

View full text Add to dashboard Cite

show abstract

“…Without claiming completeness, we refer the reader to [16] as well as to [7,17,37,38,40] for a collection of abstract results. Applications to plasticity [5,12,13], fracture [1,10,21,28,42], damage [11,22,29], adhesive contact [8,47], delamination [18,46,50], optimal control [58], and topology optimization [2,3] are also available. Recall that, although the artificial viscous term disappears as ε → 0, the choice of the specific form of viscosity actually influences the limit [26,55].…”

Section: Introductionmentioning

confidence: 99%

Rate-independent stochastic evolution equations: parametrized solutions

Scarpa¹,

Stefanelli²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

By extending to the stochastic setting the classical vanishing viscosity approach we prove the existence of suitably weak solutions of a class of nonlinear stochastic evolution equation of rate-independent type. Approximate solutions are obtained via viscous regularization. Upon properly rescaling time, these approximations converge to a parametrized martingale solution of the problem in rescaled time, where the rescaled noise is given by a general square-integrable cylindrical martingale with absolutely continuous quadratic variation. In absence of jumps, these are strong-in-time martingale solutions of the problem in the original, not rescaled time.

show abstract

“…e.g. [DDS11,BFM12,FrS13]), to fracture [KMZ08,LaT11,Neg14], damage and fatigue [KRZ13, CrL16, ACO19], and to optimal control [SWW17] to name a few. This paper revolves around a different, but still of vanishing-viscosity type, solution notion for system (1.2).…”

Section: Introductionmentioning

confidence: 99%

Balanced Viscosity (BV) solutions to infinite-dimensional rate-independent systems

Mielke¹,

Rossi²,

Savaré³

2016

J. Eur. Math. Soc.

111

View full text Add to dashboard Cite

Balanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-independent flows by adding a superlinear vanishing-viscosity dissipation.We address the main issue of proving the existence of such limits for infinite-dimensional systems and of characterizing them by a couple of variational properties that combine a local stability condition and a balanced energy-dissipation identity.A careful description of the jump behavior of the solutions, of their differentiability properties, and of their equivalent representation by time rescaling is also presented.Our techniques rely on a suitable chain-rule inequality for functions of bounded variation in Banach spaces, on refined lower semicontinuity-compactness arguments, and on new BVestimates that are of independent interest.whose support function is Ψ. Following [MRS12a], we say that a curve u ∈ BV([0, T ]; V ) is a BV solution to the RIS (V, E, Ψ, Φ), if it fulfills the following local stability conditionwhere J u is the jump set of u, and the Energy-Dissipation Balance Var f (u; [0, t]) + E t (u(t)) = E 0 (u(0)) + t 0

show abstract

Optimal control of a rate-independent evolution equation via viscous regularization

Cited by 12 publications

References 42 publications

Optimal control of perfect plasticity part I: Stress tracking

Optimal control of perfect plasticity part I: Stress tracking

Rate-independent stochastic evolution equations: parametrized solutions

Balanced Viscosity (BV) solutions to infinite-dimensional rate-independent systems

Contact Info

Product

Resources

About