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Second order optimality conditions for optimal control of quasilinear parabolic equations

  • * Corresponding author: Ira Neitzel

    * Corresponding author: Ira Neitzel 
The first author is supported by the International Research Training Group IGDK, funded by the German Science Foundation (DFG) and the Austrian Science Fund (FWF).
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  • We discuss an optimal control problem governed by a quasilinear parabolic PDE including mixed boundary conditions and Neumann boundary control, as well as distributed control. Second order necessary and sufficient optimality conditions are derived. The latter leads to a quadratic growth condition without two-norm discrepancy. Furthermore, maximal parabolic regularity of the state equation in Bessel-potential spaces $H_D^{-\zeta,p}$ with uniform bound on the norm of the solution operator is proved and used to derive stability results with respect to perturbations of the nonlinear differential operator.

    Mathematics Subject Classification: 35K59, 49K20, 90C48.


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  • Table 1.  Summary of differentiability and integrability exponents

    Variable Description
    $p$ Given by isomorphism $-\nabla\cdot\mu\nabla + 1 \colon W_D^{1,p}\rightarrow W_D^{-1,p}$ and close to the spatial dimension $d$ ; see Assumption 3.
    $\zeta$ Differentiability exponent close to one defining $H_D^{-\zeta,p}$ .
    $s$ Integrability exponent for the controls determined by $p$ , $\zeta$ , and $d$ , possibly large; see Assumption 4.
    $r$ , $r'$ Integrability exponents for linearized and adjoint state equation introduced in Section 4.
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