Homework 4 - Due: 02/15

Problem 1

In class last week, we discussed changes to the LTV system \(\dot x = A(t) x(t)\) under a time varying coordinate change \(x(t) = P(t) \bar{x}(t)\) and derived a form for the \(\bar{A}(t)\) matrix in the equivalent representation: \(\dot {\bar{x}} = \bar{A}(t) \bar{x}(t)\) under the assumption that \(P(t)\) is invertible for all \(t\).

\[ \bar{A}(t) = \dot{P}^{-1}(t) P(t) + P^{-1}(t)A(t)P(t) \]

For this problem

1. Extend the coordinate change to the full system: \[ \dot x = A(t) x(t) + B(t) u (t) \]
2. Show that if \(P(t)\) is a fundamental matrix for the system \[ \dot{P}(t) = A(t) P(t), \qquad P(t_0) = C \in \mathbb{R}^{n\times n} \] where \(C\) is invertible then \[ \dot{\bar{x}} (t) = P^{-1}B(t) u(t) \]
3. Using this result re-derive the variation of constants formula \[ x(t) = \phi(t, t_0)x(t_0) + \int \limits _{t_0} ^{t} \phi (t, s) B(s) u(s) ds \] where \(\phi (t, s) = P(t)P^{-1}(s)\).

Problem 2

Prove that the Euclidean norm \[|x|:=\sqrt{\langle x,x\rangle} =\sqrt{x_1^2+\dots+x_n^2}\] satisfies the triangle inequality.

Hint: Use the Cauchy-Schwarz inequality \(|\langle x,y\rangle|^2\le \langle x,x\rangle\cdot \langle y,y\rangle\).

Problem 3

Let \(A\) be a symmetric real-valued square matrix.

1. Show that if \(\lambda+i\mu\) is an eigenvalue of \(A\) and \(z=x+iy\) is a corresponding eigenvector, then \(\mu=0\) and \(x\) is an eigenvector (you can assume \(x\ne 0\) for simplicity). In other words, eigenvalues of symmetric matrices are always real and eigenvectors can always be chosen to be real.

   Hint: Show that \(\overline z^T\!Az\) is real.

2. Show that eigenvectors of \(A\) corresponding to distinct eigenvalues are orthogonal.

Problem 4

Let \(X\) and \(Y\) be linear vector spaces over \(\mathbb R\) equipped with inner products \(\langle\cdot,\cdot\rangle_X\) and \(\langle\cdot,\cdot\rangle_Y\), respectively. Further, let \(L:X\to Y\) be a linear operator.

We define the adjoint of \(L\) to be a linear operator \(L^*:Y\to X\) with the property that \[ \langle y,Lx\rangle_Y=\langle L^*y,x\rangle_X\qquad \forall\, x\in X,\ y\in Y \] Assume that the map \(LL^*:Y\to Y\) is invertible. Then the equation \(Lx=y_0\) has a solution \[ x_0=L^*(LL^*)^{-1}y_0 \] for each \(y_0\in Y\).

Prove that if \(x_1\) is any other solution of \(Lx=y_0\), then \(\langle x_1,x_1\rangle\ge \langle x_0,x_0\rangle\).

Hint: Let \(y_1:=(LL^*)^{-1}y_0\). Using the definition of adjoint, show that \(\langle y_1,Lx_0\rangle=\langle x_0,x_0\rangle\) and also that \(\langle x_0,x_1\rangle=\langle y_1,Lx_0\rangle\) Complete the proof by using the fact that \(\langle x_1-x_0,x_1-x_0\rangle\ge 0\).

Problem 5

Using the stability definitions given in class, determine if the systems below are stable, asymptotically stable, globally asymptotically stable, or neither. The first two systems are in \(\mathbb{R}^2\), the last is in \(\mathbb{R}\).

1. \(\dot x_1 = 0\) and \(\dot x_2 = -x_2\)
2. \(\dot x_1 = -x_2\) and \(\dot x_2 = 0\)
3. \(\dot x = 0\) if \(|x|>1\) and \(\dot x = -x\) otherwise

Justify your answers using only the definitions of stability (not eigenvalues or Lyapunov’s method).

Problem 6

First, some definitions.

• Given a linear operator \(A:X\to X\), a subspace \(Y\subset X\) is called \(A\)-invariant if \(Ay\in Y\) \(\forall y\in Y\).

• For a linear system \(\dot x=Ax\) on \(X=\mathbb{R}^n\), this means that \(x_0\in Y\) implies \(x(t)\in Y\) \(\forall t\) (reason: \(x(t)=e^{At}x_0=(I+A+A^2/2+\dots)x_0\)).

• If \(v\) is an eigenvector of \(A\) with a real eigenvalue, then \(\operatorname{span}\{v\}\) is a 1-dimensional invariant subspace. For a \(k\times k\) Jordan block, the eigenvector \(v_1\) and the generalized eigenvectors \(v_2,\dots, v_k\) together span an invariant subspace.

• The case of a pair of complex eigenvalues was discussed in Problem Set 2, Problem 3

• The sum of all invariant subspaces corresponding to eigenvalues with \(\text{Re}(\lambda)<0\) is called the stable invariant subspace; the corresponding object for \(\text{Re}(\lambda)\ge 0\) is the unstable invariant subspace; together these two subspaces span \(\mathbb{R}^n\).

Consider the LTI system \(\dot x =Ax\) where \[ A= \begin{pmatrix} -1 & 0 & 0 \\ 0 & 2 & 1\\ 0 & -1 & 2 \end{pmatrix} \] Identify the stable and unstable invariant subspaces by giving a real basis for each of them.