MATH319 Exercises 5 Assessed Exercise 2 7 Assessed Exercise 3

6 Workshop Exercises 3

W3.1 Calculate the Laplace transform $\hat{Y}(s)$ of the matrix function

Y(t)=\left[\begin{pmatrix}3t^{2}e^{-t}&te^{2t}\cr 1&e^{t}\cr\end{pmatrix}% \right],

and locate any poles of $\hat{Y}(s)$ .

W3.2 Let $\omega,\nu>0$ be constants.

(i) Verify that the initial value problem

{{d^{2}y}\over{dt^{2}}}+\nu^{2}y=u(t)

y(0)=0

{{dy}\over{dt}}(0)=0

has solution

y(t)={{1}\over{\nu}}\int_{0}^{t}u(\tau)\sin\nu(t-\tau)\,d\tau.

(The addition formula for $\sin$ may be helpful.)

(ii) For $u(t)=\cos\omega t$ , evaluate the solution $y$ explicitly, treating the cases $\nu\neq\omega$ and $\nu=\omega$ separately.

W3.3 (i) Let $D$ be a $(n\times n)$ diagonal matrix with positive diagonal entries $\kappa_{1}\geq\kappa_{2}\geq\dots\geq\kappa_{n}$ . Show that

\kappa_{n}\|X\|^{2}\leq\langle DX,X\rangle\leq\kappa_{1}\|X\|^{2}\qquad(X\in{% \bf R}^{n\times 1}).

(ii) Let $K$ be a $(n\times n)$ real symmetric matrix with positive eigenvalues $\kappa_{1}\geq\kappa_{2}\geq\dots\geq\kappa_{n}$ . Show that

\kappa_{n}\|X\|^{2}\leq\langle KX,X\rangle\leq\kappa_{1}\|X\|^{2}\qquad(X\in{% \bf R}^{n\times 1}).

W3.4 Solve the integral equation

y(t)=e^{-2t}+\int_{0}^{t}e^{u-t}y(u)\,du,

where $y$ has property $(E)$ , by using Laplace transforms.

W3.5 A model for an electrical circuit is given by a $S I S O$ system $(A,B,C,D)$ where $L, c, R$ are positive constants and

A=\left[\begin{pmatrix}0&-1/c\cr 1/L&-R/L\cr\end{pmatrix}\right],\quad B=\left% [\begin{pmatrix}1/c\cr 0\cr\end{pmatrix}\right],\quad C=\left[\begin{pmatrix}0% &R\cr\end{pmatrix}\right],\quad D=0.

(i) Find the transfer function $T(s)$ .

(ii) Show that the eigenvalues $\lambda$ of $A$ have negative real parts.

W3.6 Find a SISO system $(A,B,C,D)$ that has transfer function

T(s)={{5s^{4}+7s^{3}-6s^{2}+s+2}\over{s^{4}-3s^{3}+4s^{2}+7s+6}},

and find numerical values for the eigenvalues of $A$ . Start by dividing numerator by denominator.

W3.7 Say that $f:(0,\infty)\rightarrow{\bf C}$ belongs to $L^{1}(0,\infty)$ if $f$ is integrable and $\int_{0}^{\infty}|f(x)|dx$ is finite. Say that $h:(0,\infty)\rightarrow{\bf C}$ is bounded if there exists $M$ such that $|h(t)|\leq M$ for all $t>0$ . Show that if $f\in L^{1}(0,\infty)$ and $h$ is bounded and continuous, then $f\ast h$ is bounded.

W3.8 Let $A$ be a real $(3\times 3)$ matrix.

(i) Show that $\det(sI-A)$ has either (a) three real zeros, or (b) one real root and a pair of complex conjugate zeros.

(ii) Show that, in both cases (a) and (b), $A$ has a real eigenvector.

W3.9 Let $A$ and $B$ be $(n\times n)$ complex matrices. By considering the $(2n\times 2n)$ block matrices

X=\left[\begin{pmatrix}I&A\cr 0&I\cr\end{pmatrix}\right],\quad Y=\left[\begin{% pmatrix}sI&-A\cr-B&I\cr\end{pmatrix}\right],\quad Z=\left[\begin{pmatrix}I&A/s% \cr 0&I\cr\end{pmatrix}\right],

and the determinants of $X Y$ and $Y Z$ , show that $A B$ and $B A$ have equal characteristic polynomials.

W3.10 (i) Describe or sketch the graph of $h(t)=H(t-a)-H(t-b)$ for $0<a<b$ , and compute the Laplace transform $\hat{h}(s)$ .

(ii) For $\varepsilon>0$ , let $f_{\varepsilon}(t)=\varepsilon^{-1}(H(t-\varepsilon)-H(t)).$ Calculate the Laplace transform $\hat{f}_{\varepsilon}(s)$ , and find the limit as $\varepsilon\rightarrow 0+$ .

W3.11 Let ${\cal R}$ be the set of functions of the form

f(s)=\sum_{j=1}^{n}{{a_{j}}\over{(1+s)^{j}}}

where $n\geq 0$ and $a_{j}\in{\bf C}$ .

(i) Show that $f(s)$ is differentiable, and $f^{\prime}(s)\in{\cal R}$ .

(ii) Show that, for all $f(s),g(s)\in{\cal R}$ , the sum $f(s)+g(s)$ and the product $f(s)g(s)$ also belong to ${\cal R}$ .

(iii) Show that $f(s)$ is the Laplace transform of

y(t)=\sum_{j=1}^{n}{{a_{j}t^{j-1}e^{-t}}\over{(j-1)!}}\qquad(t>0).