Adjoint Energy Identity for the Kalman Covariance Riccati Equation

Adopt the setting and notation of the Kalman covariance Riccati existence theorem: real numbers $a<b$, a natural number $k\ge1$, assignments $A$, $C$ and $D$ of real matrices with $k$ rows and $k$ columns to each $t\in[a,b]$, all entries continuous in $t$, with every $C(t)$ and every $D(t)$ positive semidefinite, a positive semidefinite $P_{0}$, and the unique assignment $P$ of real $k\times k$ matrices with continuous entries such that

$$P(t)=P_{0}+\int_{a}^{t}\bigl(A(r)P(r)+P(r)A(r)^{\top}-P(r)D(r)P(r)+C(r)\bigr)\,dr\qquad(a\le t\le b),$$

every $P(t)$ being symmetric by that theorem. Integrals of matrix- or vector-valued functions are taken entrywise as Riemann integrals of continuous functions, which exist by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval; products are the matrix product and the matrix-vector product, $(\cdot)^{\top}$ is the transpose, and $x\cdot y$ is the dot product on the Euclidean space $\mathbb{R}^{k}$. Continuity of a real-valued function on an interval is understood as in that theorem. Let $s\in[a,b]$ and $x\in\mathbb{R}^{k}$.

Then the following hold.

1. (Adjoint path.) There is exactly one assignment $\lambda$ of a vector $\lambda(u)\in\mathbb{R}^{k}$ to each $u\in[a,b]$, with continuous components, such that $\lambda(s)=x$ and

$$\lambda(u)=\lambda(a)-\int_{a}^{u}\bigl(A(r)-P(r)D(r)\bigr)^{\top}\lambda(r)\,dr\qquad(a\le u\le b).$$

2. (Injection profile.) The assignment $\psi$ defined by $\psi(u)=P(u)\lambda(u)$ has continuous components and satisfies

$$\psi(u)=P_{0}\,\lambda(a)+\int_{a}^{u}\bigl(A(r)\psi(r)+C(r)\lambda(r)\bigr)\,dr\qquad(a\le u\le b).$$

3. (Energy identity.) $x\cdot\psi(s)=x\cdot\bigl(P(s)x\bigr)$; the function

$$r\ \longmapsto\ \lambda(r)\cdot\bigl(C(r)\lambda(r)\bigr)+\psi(r)\cdot\bigl(D(r)\psi(r)\bigr)$$

is continuous and nonnegative on $[a,b]$; the real number $\lambda(a)\cdot\bigl(P_{0}\lambda(a)\bigr)$ is nonnegative; and

$$\lambda(a)\cdot\bigl(P_{0}\lambda(a)\bigr)+\int_{a}^{s}\Bigl(\lambda(r)\cdot\bigl(C(r)\lambda(r)\bigr)+\psi(r)\cdot\bigl(D(r)\psi(r)\bigr)\Bigr)\,dr\ =\ x\cdot\bigl(P(s)x\bigr).$$