On the optimal control for a semilinear equation with cost
depending on the free boundary

Díaz, Jesús Ildefonso Díaz; Mingazzini, Tommaso; Ramos, Ángel

doi:10.3934/nhm.2012.7.605

Cited by 3 publications

(3 citation statements)

References 25 publications

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“…Thus, δ satisfies (9) as claimed. By revisiting (11), by noting that the first term on the right-hand side of ( 4) is nonnegative (again due to the properties of difference quotients of convex functions), and by invoking Fatou's lemma [5, Corollary 2.8.4] (keeping in mind that all of the involved integrands are nonnegative and that δ k → δ holds in L 2 (Ω)), we further obtain that…”

Section: Lemma 41 (Taylor-like Expansion For Powers Of Absolute Value...mentioning

confidence: 99%

Multiobjective optimal control of a non-smooth semilinear elliptic partial differential equation

Christof

Müller

2021

ESAIM: COCV

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This paper is concerned with the derivation and analysis of first-order necessary optimality conditions for a class of multiobjective optimal control problems governed by an elliptic non-smooth semilinear partial differential equation. Using an adjoint calculus for the inverse of the non-linear and non-differentiable directional derivative of the solution map of the considered PDE, we extend the concept of strong stationarity to the multiobjective setting and demonstrate that the properties of weak and proper Pareto stationarity can also be characterized by suitable multiplier systems that involve both primal and dual quantities. The established optimality conditions imply in particular that Pareto stationary points possess additional regularity properties and that mollification approaches are - in a certain sense - exact for the studied problem class. We further show that the obtained results are closely related to rather peculiar hidden regularization effects that only reveal themselves when the control is eliminated and the problem is reduced to the state. This observation is also new for the case of a single objective function. The paper concludes with numerical experiments that illustrate that the derived optimality systems are amenable to numerical solution procedures.

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Section: Lemma 41 (Taylor-like Expansion For Powers Of Absolute Value...mentioning

confidence: 99%

Multiobjective optimal control of a non-smooth semilinear elliptic partial differential equation

Christof

Müller

2021

ESAIM: COCV

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“…By iii) of the above theorem it is enough to construct a (local) subsolution satisfying the required boundary behavior. We shall carry out such construction by adapting the techniques presented in [24] (see also some related local subsolutions in [1], [30] and [23]). From the assumption (23) for any…”

Section: Proposition 1 For Anymentioning

confidence: 99%

Flat solutions of some non-Lipschitz autonomous semilinear equations may be stable for N ≥ 3

Díaz

Hernández

Il’yasov

2017

Chin. Ann. Math. Ser. B

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To a master, Haïm Brezis, with admiration. AbstractWe prove that flat ground state solutions (i.e. minimizing the energy and with gradient vanishing on the boundary of the domain) of the Dirichlet problem associated to some semilinear autonomous elliptic equations with a strong absorption term given by a non-Lipschitz function are unstable for dimensions N = 1, 2 and they can be stable for N ≥ 3 for suitable values of the involved exponents. * J.I. Díaz and J.

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“…), economic, and biological, [12]. In general, there are many optimal control problems are governed either by ODEs as Orpel in 2009 [11] or by different types of PDEs and are subject to control and state constraints, as El-Borari and et al in 2013 [9], and Wang, Y. and et al in 2015 [15], which are studied an optimal control of parabolic partial differential equations, Farag, M. H. in 2014 [10] studied classical optimal control of hyperbolic partial differential equations, Diaz and et al in 2012 [7] studied a optimal control of elliptic partial differential equations, Al-Rawdanee, E. in 2014 [3] studied an a classical optimal control of a coupled of nonlinear elliptic partial differential equations and M. K. Ghufran in 2016 [4] studied a classical optimal control of a coupled of nonlinear parabolic partial differential equations while, Al-Hawasy, J. in 2016 [2] studied a classical optimal control of a coupled of nonlinear hyperbolic partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

The Continuous Classical Boundary Optimal Control of a Couple Nonlinear Parabolic Partial Differential Equations

Al-Hawasy¹,

Naeif²

2018

ANJS

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In this paper the continuous classical boundary optimal control problem of a couple nonlinear partial differential equations of parabolic type is studied. The Galerkin method is used to prove the existence and uniqueness theorem of the state vector solution of a couple nonlinear parabolic partial differential equations for given (fixed) continuous classical boundary control vector. The theorem of the existence of a continuous classical optimal boundary control vector associated with the couple of nonlinear parabolic partial differential equations is proved. The existence of a unique vector solution of the adjoint equations is studied. The Fréchet derivative is derived; Finally The Kuhn-Tucker-Lagrange multipliers theorems is developed and then is used to prove the necessary conditions theorem and the sufficient conditions theorem of optimality of a couple of nonlinear parabolic equations with equality and inequality constraints.

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On the optimal control for a semilinear equation with cost depending on the free boundary

Cited by 3 publications

References 25 publications

Multiobjective optimal control of a non-smooth semilinear elliptic partial differential equation

Multiobjective optimal control of a non-smooth semilinear elliptic partial differential equation

Flat solutions of some non-Lipschitz autonomous semilinear equations may be stable for N ≥ 3

The Continuous Classical Boundary Optimal Control of a Couple Nonlinear Parabolic Partial Differential Equations

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