# final

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$
Due by noon 12.18.

Numbers in brackets indicate the points for each (sub)problem.

Questions? Misprints suspected? Post as a comment here!

1. Consider the vector field $$\bv$$ given by
$\dot{x}=-xy^2-x^5;\\ \dot{y}=-x^2y-y^5.$

1. [3] Find a Lyapunov function and prove thus that the origin is Lyapunov stable. Is it asymptotically stable? Is it exponentially stable?
2. [1] Show that one can find an arbitrarily small linear perturbation of the vector field $$\bv$$ (that is add arbitrarily small linear terms to $$\bv$$ so that the origin becomes an unstable equilibrium.
3. [3] What is the maximal degree of perturbation of the vector field that can make the origin unstable?
2. Consider the system given by
$\begin{array}{ccl} \dot{x}_1&=&a_1(t)x_2,\\ \dot{x}_2&=&u,\\ \dot{x}_3&=&a_3(t)x_2, \end{array}$
where
$a_1=\left\{\begin{array}{ccc} 0 &\mathrm{for}& 0\leq t<1;\\ 1 &\mathrm{for}& 1\leq t\leq 2; \end{array}\right. \mathrm{ and }\quad a_3=\left\{\begin{array}{ccc} 1 &\mathrm{for}& 0\leq t<1;\\ 0 &\mathrm{for}& 1\leq t\leq 2; \end{array}\right.$

1. [3] Find the controllability Grammian $$W(0,t), 0\leq t\leq 2$$.
2. [1] Is this system controllable on $$[0,1]$$? on $$[1,2]$$?
3. [2] Is this system controllable on $$[0,2]$$?
3. [5] Find the rank of the minimal realization of the following transfer function (in this $$101\times 101$$-matrix, the entries on the $k$-th anti-diagonal are $$1/(s+k-1)$$):
$P(s)=\left( \begin{array}{ccccc} \frac{1}{s}&\frac{1}{s+1}&\frac{1}{s+2}&\cdots&\frac{1}{s+100}\\ \frac{1}{s+1}&\frac{1}{s+2}&\frac{1}{s+3}&\cdots&\frac{1}{s+101}\\ \frac{1}{s+2}&\frac{1}{s+3}&\frac{1}{s+4}&\cdots&\frac{1}{s+102}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \frac{1}{s+100}&\frac{1}{s+101}&\frac{1}{s+102}&\cdots&\frac{1}{s+200} \end{array}\right)$
4. Consider the optimal control problem
$\dot{x}_1=x_2;\\ \dot{x}_2=x_1+u;\\ |u|\leq 1; L\equiv 1; (x_1(0),x_2(0))=(x_o,v_o).$

1. [4] For $$(x_o,v_o)=(1,-1)$$, find the trajectory stopping at $$(0,0)$$ at the shortest time. (This is the shortest time problem, so the terminal time is not fixed, hence the Hamiltonian is $$\equiv 0$$.)
2. [3]Find for which $$(x_o,v_o)$$ it is possible at all to stop at $$(0,0)$$.
5. [5] Consider the infinite horizon LQR
$\begin{array}{ccl} \dot{x}_1&=&x_2+b_1u;\\ \dot{x}_2&=&-x_1+b_2u; \end{array}\\ \int_0^\infty u^2+x_1^2 dt\to\min.$
Assume that the control vector $$b$$ has norm one, i.e.
$b_1^2+b_2^2=1.$
Among such vectors, find one providing the speediest convergence of the close-loop feedback LQR regulator, that is minimizing the (negative) maximal real part of $$\dot{x}=(A-bk)x$$.