% History: Linear Quadratic Regulator problem 1960s
Consider the following dynamical system
$$ \begin{align*} \dot x &= Ax + Bu, \qquad x(0)=x_0\\ y &= Cx, \qquad 0\leq t\leq T \end{align*} $$where $u\in L^2(0,T; U)$ and $y\in L^2(0,T; Y)$ and $U$ and $Y$ are Hilbert spaces.
% Hypothesis, well-posedness
% Differential Riccati equation
Examples
Parabolic
Consider the equation
$$ \dot x = Ax + Bu, \qquad y = Cx $$where $A$ is self-adjoint on a real Hilbert space. We assume that $A$ has a compact resolvent operator and that the spectrum of $A$ consists of a strictly decreasing sequence $\lambda_n, n\in \mathbb{N}$ of real eigenvalues with associated eigenvector $\phi_n\in H$ with $\|\phi_n\|=1$.
$$ \begin{align*} z_t &= z_{\xi\xi}, \qquad 0<\xi<1\\ z_\xi(t,0) &= u(t), \quad z_\xi(t,1)=0,\ t>0\\ y(t) &= \int_0^1 c(\xi) z(t,\xi) d\xi \end{align*} $$Hyperbolic
Consider the dynamical system
$$ \ddot z = Az + Bu, \qquad y = Cz $$where $A$ is self-adjoint on a real Hilbert space. We assume that $A$ has a compact resolvent operator and that the spectrum of $A$ consists of a strictly decreasing sequence $\lambda_n=-\omega_n^1, n\in \mathbb{N}$ satisfy $\omega_1\geq \delta$ and $\omega_{n+1} - \omega_{n}\geq\delta$ for all $n\in \mathbb{N}$.
$$ \begin{align*} z_{tt} &= z_{\xi\xi}, \qquad 0<\xi<1\\ z_\xi(t,0) &= u(t), \quad z_\xi(t,1)=0,\ t>0\\ y(t) &= \int_0^1 c(\xi) z(t,\xi) d\xi \end{align*} $$Delayed Differential Equations
Consider the following Retarded Differential Equation (RDF)
$$ \frac{d}{dt}(x(t) - Mx(t-\tau)) = Lx(t-\tau) + Bu(t), \qquad y = Cx $$where $x(t)\in \mathbb{R}^n$, $u\in \mathbb{R}^m$, $M, L\in \mathbb{R}^{n\times n}$ and $B\in \mathbb{R}^{n\times m}$.