Regularity of Optimal Boundary Controls for Parabolic Equations, I. Analyticity
Thomas I. Seidman · SIAM Journal on Control and Optimization · 1982
Let $({\bf A},{\boldsymbol \beta} )$ be a second order elliptic operator on $\Omega \subset \mathbb{R}^d $ with a compatible boundary operator and let $\phi _ * $ be the optimal boundary control for $\dot u = Au + f$, ${\boldsymbol \beta} u = \phi $ on $(0,T)$, $u(0) = \omega _0 $ minimizing J of the form \[J(\phi ) = \int_0^T {\left| \phi \right|_{\boldsymbol \kappa }^2 } + \int_0^T {\left| {u - u_T } \right|_{\boldsymbol \lambda }^2 + \left| {u(T) - \omega _T } \right|_{\boldsymbol \mu }^2 .}\] Then, if f, $u_T $, etc., are analytic in t on $(0,T)$ so is $\phi _ * $. A similar result is obtained for the infinite horizon problem.