\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\let \LWRorighspace \hspace \)
\(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\)
\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\arabic }[1]{}\)
\(\newcommand {\number }[1]{}\)
\(\newcommand {\noalign }[1]{\text {#1}\notag \\}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\def \oe {\unicode {x0153}}\)
\(\def \OE {\unicode {x0152}}\)
\(\def \ae {\unicode {x00E6}}\)
\(\def \AE {\unicode {x00C6}}\)
\(\def \aa {\unicode {x00E5}}\)
\(\def \AA {\unicode {x00C5}}\)
\(\def \o {\unicode {x00F8}}\)
\(\def \O {\unicode {x00D8}}\)
\(\def \l {\unicode {x0142}}\)
\(\def \L {\unicode {x0141}}\)
\(\def \ss {\unicode {x00DF}}\)
\(\def \SS {\unicode {x1E9E}}\)
\(\def \dag {\unicode {x2020}}\)
\(\def \ddag {\unicode {x2021}}\)
\(\def \P {\unicode {x00B6}}\)
\(\def \copyright {\unicode {x00A9}}\)
\(\def \pounds {\unicode {x00A3}}\)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\let \Hat \hat \)
\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
\(\let \Breve \breve \)
\(\let \Bar \bar \)
\(\let \Vec \vec \)
\(\require {mathtools}\)
\(\newenvironment {crampedsubarray}[1]{}{}\)
\(\newcommand {\smashoperator }[2][]{#2\limits }\)
\(\newcommand {\SwapAboveDisplaySkip }{}\)
\(\newcommand {\LaTeXunderbrace }[1]{\underbrace {#1}}\)
\(\newcommand {\LaTeXoverbrace }[1]{\overbrace {#1}}\)
\(\newcommand {\LWRmultlined }[1][]{\begin {multline*}}\)
\(\newenvironment {multlined}[1][]{\LWRmultlined }{\end {multline*}}\)
\(\let \LWRorigshoveleft \shoveleft \)
\(\renewcommand {\shoveleft }[1][]{\LWRorigshoveleft }\)
\(\let \LWRorigshoveright \shoveright \)
\(\renewcommand {\shoveright }[1][]{\LWRorigshoveright }\)
\(\newcommand {\shortintertext }[1]{\text {#1}\notag \\}\)
\(\newcommand {\vcentcolon }{\mathrel {\unicode {x2236}}}\)
\(\newcommand {\mC }{\mathbb C}\)
\(\newcommand {\mR }{\mathbb R}\)
\(\newcommand {\mN }{\mathbb N}\)
\(\newcommand {\mZ }{\mathbb Z}\)
\(\newcommand {\mL }{\mathcal L}\)
\(\newcommand {\mF }{\mathcal F}\)
\(\newcommand {\ipd }[2]{\langle #1 , #2 \rangle }\)
\(\newcommand {\Ipd }[2]{\left \langle #1 , #2 \right \rangle }\)
\(\newcommand {\sbm }[1]{\left [\begin {smallmatrix}#1\end {smallmatrix}\right ]}\)
\(\newcommand {\bbm }[1]{\begin {bmatrix}#1\end {bmatrix}}\)
\(\newcommand {\re }{{\rm Re}}\)
\(\newcommand {\imag }{{\rm Im}}\)
\(\newcommand {\e }{{\rm e}}\)
\(\newcommand {\HS }{{\rm HS}}\)
\(\newcommand {\cl }{{\rm cl}}\)
\(\newcommand {\wt }{\widetilde {w}}\)
\(\newcommand {\zt }{\widetilde {z}}\)
\(\newcommand {\xu }{\underline {x}}\)
\(\newcommand {\uu }{\underline {u}}\)
\(\DeclareMathOperator {\vecc }{vec}\)
\(\DeclareMathOperator {\trace }{trace}\)
\(\)
Chapter D Problem Sheet 4 (Lectures 8–9)
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1. Consider the first order scalar differential equation \(\dot {x}(t)+6x(t)=2u(t)\). Find a control \(u\) such that for a given \(r\in \mR \) we have \(\lim _{t\to \infty }x(t)=r\) no matter what \(x(0)\) is.
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2. Consider the first order scalar differential equation \(\dot {x}(t)=x(t)+u(t)\). Find a control \(u\) such that for a given \(r\in \mR \) we have \(\lim _{t\to \infty }x(t)=r\) no matter what \(x(0)\) is.
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3. Consider the first order scalar differential equation \(\dot {x}(t)+x(t)=w(t)+u(t)\). Find a control \(u\) such that \(\lim _{t\to \infty }x(t)=r\) no matter what \(x(0)\) is when \(w(t)=r\), where \(r\in \mR \).
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4. Consider the second order scalar differential equation
\[ \ddot {q}+2\zeta \dot {q}+q=u, \]
where \(\zeta >0\). Find a control \(u\) such that for a given \(r\in \mR \) we have \(\lim _{t\to \infty }q(t)=r\) no matter what the initial conditions are.