Abstract
A class of optimal control problems for quasilinear elliptic equations is considered, where the coefficients of the elliptic differential operator depend on the state function. First- and second-order optimality conditions are discussed for an associated control-constrained optimal control problem. Main emphasis is laid on second-order sufficient optimality conditions. To this aim, the regularity of the solutions to the state equation and its linearization is studied in detail and the Pontryagin maximum principle is derived. One of the main difficulties is the nonmonotone character of the state equation.
Highlights
In this paper, we consider optimal control problems for a quasilinear elliptic equation of the type (1.1)−div [a(x, y(x)) ∇y(x)] + f (x, y(x)) = u(x) in Ω, y(x) = 0 on Γ.Equations of this type occur, for instance, in models of heat conduction, where the heat conductivity a depends on the spatial coordinate x and on the temperature y.For instance, the heat conductivity of carbon steel depends on the temperature and on the alloying additions contained; cf. Bejan [2]
The heat conductivity of carbon steel depends on the temperature and on the alloying additions contained; cf
Associated error estimates for local solutions of the FEM-approximated optimal control problem are based on second-order sufficiency
Summary
We consider optimal control problems for a quasilinear elliptic equation of the type (1.1). −div [a(x, y(x)) ∇y(x)] + f (x, y(x)) = u(x) in Ω, y(x) = 0 on Γ Equations of this type occur, for instance, in models of heat conduction, where the heat conductivity a depends on the spatial coordinate x and on the temperature y. The heat conductivity of carbon steel depends on the temperature and on the alloying additions contained; cf Bejan [2]. In the more general case a = a(x, y), a Kirchhoff-type transformation can be applied. Y 0 a(x, z)dz and set θ(x) := b(x, y(x)) Under this transformation, we obtain a semilinear equation of the type −Δ θ + div [(∇xb)(x, b−1(x, θ))] + f (x, b−1(x, θ)) = u. The form (1.1) seems to be more directly accessible to a numerical solution
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