Pontryagin's maximum principle of optimal control governed by some non-well-posed semilinear parabolic differential equations
The systems are of the form \[ {\partial y(t, x) \over \partial t} = Ay(t, x) + f(t, x, y(t, x), u(t, x)) \quad (0 < t \leq T, x \in \Omega) \tag{1} \] with \(A\) a uniformly elliptic operator in an \(n\)-dimensional domain \(\Omega\) and the Dirichlet boundary condition on \(\Gamma =\) boundary of \(\Omega.\) It is not assumed that (1) has the global existence property, and the optimal problems involve minimization of an integral functional under control constraints and a two point state constraint. The results in this paper are versions of Pontryagin's maximum principle. For systems of similar type where the control appears linearly, see [\textit{G. Wang} and \textit{L. Wang}, SIAM J. Control Optimization 40, 1517-1539 (2002; Zbl 1013.49016)].
non well-posed semilinear parabolic equations
scientific article; zbMATH DE number 1903493