Optimal control of quasilinear parabolic PDEs with gradient terms and pointwise constraints on the gradient of the state

Kurzfassung

We derived existence results and first order necessary optimality conditions for optimal control problems governed by quasi-linear parabolic partial differential equations (PDEs) with a class of first order nonlinearities that included, for instance, quadratic gradient terms. Pointwise in space and time or averaged in space and pointwise in time constraints on the gradient of the state controlled the growth of the nonlinear terms. We relied on and extended the improved regularity analysis for quasilinear parabolic PDEs on a whole scale of function spaces from [29]. In case of integral in space gradient-constraints, we derived first-order optimality conditions under rather general regularity assumptions for domain, coefficients, and boundary conditions, similar to e.g. [8]. In the case of pointwise in time and space gradient-constraints, we used slightly stronger regularity assumptions leading to a classical smoother W2,p -setting similar to [11].