First- and Second-Order Conditions at a Local Extremum of a $C^2$ Function Penalised by $h$ on the Small Space

In the setting of Hilbert Triples: Standing Notation and Background, let $U\subseteq H$ be open in $H$, with $W=D(A)\cap U$, the class $C^{2}(U)$ and local extrema relative to $V\cap U$ as fixed there, let $h$ be the penalty function, let $Y\mapsto Y|_{V}$, $I_{V}$ and $\preceq$ be as in Hilbert Triples: Standing Notation and Background (vi), let $\psi\in C^{2}(U)$, let $\lambda\in\mathbb{R}$ satisfy $0<\lambda$, and let $\hat{x}\in V\cap U$. Then the following hold.

1. (Local maximum of $\psi-\lambda h$)(i) If the function $V\cap U\to\mathbb{R}$ with value $\psi(x)-\lambda h(x)$ at $x$ has a local maximum at $\hat{x}$ relative to $V\cap U$, then $\hat{x}\in D(A)$, so that $\hat{x}\in W$, and

$$\lambda A\hat{x}=D\psi(\hat{x})\qquad\text{and}\qquad D^{2}\psi(\hat{x})|_{V}\preceq\lambda I_{V}.$$

2. (Local minimum of $\psi+\lambda h$)(ii) If the function $V\cap U\to\mathbb{R}$ with value $\psi(x)+\lambda h(x)$ at $x$ has a local minimum at $\hat{x}$ relative to $V\cap U$, then $\hat{x}\in D(A)$, so that $\hat{x}\in W$, and

$$\lambda A\hat{x}=-D\psi(\hat{x})\qquad\text{and}\qquad -\lambda I_{V}\preceq D^{2}\psi(\hat{x})|_{V}.$$