State-Constrained Optimal Control of Semilinear Elliptic Equations with Nonlocal Radiation Interface Conditions

Abstract

We consider a control-and state-constrained optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions. These conditions occur during the modeling of diffuse-gray conductiveradiative heat transfer. The nonlocal radiation interface condition and the pointwise state-constraints represent the particular features of this problem. To deal with the state-constraints, continuity of the state is shown which allows to derive first-order necessary conditions. Afterwards, we establish second-order sufficient conditions that account for strongly active sets and ensure local optimality in an L 2 -neighborhood.

Regularization of state-constrained elliptic optimal control problems with nonlocal radiation interface conditions

“…As these additional constraints represent pointwise state constraints, the extension significantly increases the complexity of the problem. The first-and second-order analysis for the associated control problem is performed in [23]. Based on these results, we here focus on the numerical treatment of the problem.…”

“…We only refer to [21,22], where the temperature gradient inside the cavity is optimized by directly controlling the heat sources in the crucible. In [23], the corresponding model is extended by including pointwise inequality constraints on the temperature to ensure sublimation of the source powder and crystallization at the seed. As these additional constraints represent pointwise state constraints, the extension significantly increases the complexity of the problem.…”

“…As in [21][22][23], we optimize the gradient temperature in the gas phase g by controlling the heat source u in the solid phase s . The objective functional, considered here, reads as follows:…”

“…Finally, as mentioned above, the state-constraints in g are required to ensure sublimation of the polycrystalline powder and crystallization at the seed, respectively. The first-and second-order analysis for (P) has been carried out quite recently in [23]. In order to obtain the Karush-Kuhn-Tucker (KKT) type optimality conditions, the constraints, imposed on the state y in (1.1), have to be considered in the space of continuous functions, denoted by C( ).…”

Noncompactness of Integral Operators Modeling Diffuse-Gray Radiation in Polyhedral and Transient Settings

“…IEOT settings, developing the corresponding existence theory. More regular solutions of the radiosity equation are desirable in the context of optimal control [15,16]. Under suitable hypotheses, one can obtain solutions in Sobolev spaces and in spaces of continuous functions (see [9] and references therein).…”