Dual companions and improved regularity of Lagrange multipliers in optimal control

Abstract

We investigate optimal control problems whose state equation takes the form $\mathcal{A}x-\mathcal{B}u=b$, where $\mathcal{A}:\mathcal{X}\to\mathcal{P}'$ is an isomorphism from a Banach state space $\mathcal{X}$ onto the dual of a reflexive Banach space $\mathcal{P}$. The adjoint equation involves the Banach space adjoint $\mathcal{A}':\mathcal{P}\to\mathcal{X}'$, so that, in general, state and multiplier are of different regularity. For particular classes of partial differential equations, the typically weaker regularity of the multiplier is improved a posteriori by bootstrapping. In this work, we provide an abstract foundation for this methodology on the operator level. More precisely, we show that whenever the state cost extends to a differentiable function on a Hilbert space $\mathcal{H}$ in which $\mathcal{X}$ is densely embedded, as for $L^2$-tracking costs on Sobolev-type state spaces, the multiplier automatically lies in a subspace of $\mathcal{P}$, the dual companion, on which $\mathcal{A}'$ restricts to an isomorphism onto $\mathcal{H}'$. This yields an optimality system with balanced regularity, where state and multiplier solve equations of the same type. In applications, the dual companion can be identified by integration by parts in time or space. We illustrate the framework by linear elliptic and parabolic equations, Cauchy problems governed by general $C_0$-semigroups, and a wave equation with accumulated-state observation.