The transmission of FM signals is based on the modulation of frequency based on an input voltage. A simple circuit generating such modulated signals is the one satisfying the differential equation $$ \ddot{y}(t)+\left[\lambda^{2}+u(t)\right] y(t)=0, $$which corresponds to a harmonic oscillator with frequency controlled about the nominal value \(\lambda\). We model this as a time-invariant bilinear system $$ \begin{aligned} \dot{x} &=\left(\begin{array}{cc} 0 & 1 \\ -\lambda^{2} & 0 \end{array}\right) x+u\left(\begin{array}{cc} 0 & 0 \\ -1 & 0 \end{array}\right) x \\ y &=\left(\begin{array}{ll} 1 & 0 \end{array}\right) x \end{aligned} $$ with \(x=\mathbb{R}^{2}, u=\mathbb{R}\), and \(y=\mathbb{R}\). (The " \(B\) " matrix is identically zero.) When \(u \equiv 0\) one has a standard harmonic oscillator, and $$ x(t)=\left(\begin{array}{c} \sin \lambda t \\ \lambda \cos \lambda t \end{array}\right) $$ is a solution, with $$ x(0)=\left(\begin{array}{l} 0 \\ \lambda \end{array}\right) $$ Find the first three terms of the Volterra representation of \(y(t)\) for \((2.68)\), with this initial state.

Let \(m, n\) be nonnegative integers, $x:=\mathbb{K}^{n}, \mathcal{U}:=\mathbb{K}^{m} .$ Assume that $$ f(t, x, u)=A(t) x+B(t) u $$ where \(A(t)\) is an \(n \times n\) matrix and \(B(t)\) is an \(n \times m\) matrix, each of whose entries is a locally essentially bounded (measurable) function \(\mathbb{R} \rightarrow \mathbb{K}\). Equivalently, $$ f: \mathbb{R} \times \mathcal{X} \times \mathcal{U} \rightarrow \mathcal{X} $$ is linear in \((x, u)\) for each fixed \(t\), and is locally bounded in \(t\) for each fixed \((x, u)\). Conclude that \(f\) is a rhs.

If \(\Sigma\) is linear and \(\Lambda\) is an integral behavior, then $$ \Lambda_{\Sigma, 0}=\Lambda $$ if and only if $$ \widetilde{K}(t, \tau)=C(t) \Phi(t, \tau) B(\tau) $$ for almost all \((t, \tau)\). Thus, the integral behavior \(\Lambda\) is realizable by some linear system if and only if there exist time-varying matrices $$ (A(t), B(t), C(t)) $$ such that the above conditions hold.

Exercise 2.7.15 simplifying assumptions, and choosing appropriate units and time scales, we have an equation $$ \ddot{\theta}=\sin \theta-u \cos \theta $$

Assume that \(K\) in formula (2.27) is continuous. Pick any fixed $i \in\\{1, \ldots, m\\}\( and any \)T>0\(. For each \)\varepsilon>0\(, let \)\omega_{e}$ be the control defined on the interval \([0, T]\) by $$ \omega_{\varepsilon}(t):= \begin{cases}\frac{1}{\varepsilon} e_{i} & \text { if } 0 \leq t<\varepsilon \\ 0 & \text { otherwise }\end{cases} $$ where \(e_{i}=(0, \ldots, 0,1,0, \ldots, 0)^{\prime}\) is the \(i\) th canonical basis vector. Prove that $\lim _{\varepsilon \rightarrow 0} y_{\varepsilon}(T)=K_{i}(T)$,

Prove that, for any linear discrete-time system and any pair of integers \(\sigma<\tau, \phi(\tau, \sigma, \cdot, \cdot)\) is linear.