Optimal Control of Nonsmooth Distributed Parameter Systems

Abstract. In this paper we propose a new algorithm for the well known elliptic obstacle problem and for parabolic variational inequalities like one and two phase Stefan problem and of obstacle type. Our approach enters the category of fixed domain methods and solves just linear elliptic or parabolic equations and their discretization at each iteration. We prove stability and convergence properties. The approximating coincidence set is explicitly computed and it converges in the Hausdorff-Pompeiu sense to the searched geometry. In the numerical examples, the algorithm has a very fast convergence and the obtained solutions (including the free boundaries) are accurate.

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“…By passing to the limit, it would satisfy the equation (43). The analysis of (43) for k → ∞ and ϵ → 0 is known [10], [33].…”

Section: Parabolic Casementioning

confidence: 99%

“…Applications, including optimal control problems are investigated in the books [4,9,10,33]. From the point of view of the numerical approximation, we quote just the monographs [10,12,27].…”

Section: Introductionmentioning

confidence: 99%

A direct algorithm in some free boundary problems

Murea

Tiba

2016

Journal of Numerical Mathematics

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“…This is quite a classical philosophy in connection to Lagrange multipliers techniques (see for instance the monograph of Tikhomirov [15]). In the setting of partial differential equations, it has been extensively exploited in the books of Lions [12] or Tiba [14], Ch.2, in connection to nonlinear singular control problems. Recently Bergounioux [5,6] has applied this method to control problems with state constraints governed by elliptic systems and Bergounioux, Männikkö and Tiba [7] have studied some examples of parabolic control problems.…”

Section: Condition (H) or Its Weaker Variant (H ′mentioning

confidence: 99%

General Optimality Conditions for Constrained Convex Control Problems

Bergounioux¹,

Tiba²

1996

SIAM J. Control Optim.

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“…Before studying the necessary condition of (ocp), we need the definition of the Generalized Gradient and the Yosida regularization of the Lipschitz function (Tiba D. [ 10] From Lemma 2 we get y, y, in L2(f/). By Egorov Theorem we know Va > 0,3f2o C f, such that rn(f f2o) < a, and y, yx, uniformly in L2(f2).…”

Section: Necessary Condition Of the Solution Of (Ocp)mentioning

confidence: 99%