Analysis of optimal boundary control for radiative heat transfer modeled by the $SP_{n}$-system

Abstract

Abstract. We present an analytic study of an optimal boundary control problem for the diffusive SP 1 -system modeling radiative heat transfer. The cost functional is of tracking-type and the control problem is considered as a constrained optimization problem, where the constraint is given by the nonlinear parabolic/elliptic SP 1 -system. We prove the existence, uniqueness and regularity of bounded states, which allows for the introduction of the reduced cost functional. Further, we show the existence of an optimal control, derive the first-order optimality system and analyze the adjoint system, for which we prove existence, uniqueness and regularity of adjoint states.

Asymptotic and discrete concepts for optimal control in radiative transfer

“…This work extends recently published results [5,14] on such optimal control problems (1.1). Here, we essentially focus on the well-known SP N approximations for (1.3) (see e.g.…”

“…Similarly to [14], let us now consider the optimal control problem subject to the (forward) SP 1 approximation of the radiative transfer equation, i.e.,…”

“…Moreover, a similar discussion holds true for the SP 2 and SP 3 equations. For theoretical results on the optimality system we refer to [14].…”

“…In combination with the results of [5,14] we finally establish the following diagram ( Figure 1) for general tracking-type optimal control problems in radiative transfer.…”

“…• 'SP N approximations of RTE' → 'SP N of adjoints of the RTE' as been discussed in [14,15]. Here, the adjoint calculus was used for the derivation of first-and second-order optimality conditions for minimization problems constrained by the continuous SP N system coupled with a nonlinear heat equation.…”