Pde

% 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 $\|\phi_n\|=1$. ...