Homework 4 - Due: 02/15

Problem 1

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

A¯(t)=P˙1(t)P(t)+P1(t)A(t)P(t)

For this problem

  1. Extend the coordinate change to the full system: x˙=A(t)x(t)+B(t)u(t)
  2. Show that if P(t) is a fundamental matrix for the system P˙(t)=A(t)P(t),P(t0)=CRn×n where C is invertible then x¯˙(t)=P1B(t)u(t)
  3. Using this result re-derive the variation of constants formula x(t)=ϕ(t,t0)x(t0)+t0tϕ(t,s)B(s)u(s)ds where ϕ(t,s)=P(t)P1(s).

Problem 2

Prove that the Euclidean norm |x|:=x,x=x12++xn2 satisfies the triangle inequality.

Hint: Use the Cauchy-Schwarz inequality |x,y|2x,xy,y.

Problem 3

Let A be a symmetric real-valued square matrix.

  1. Show that if λ+iμ is an eigenvalue of A and z=x+iy is a corresponding eigenvector, then μ=0 and x is an eigenvector (you can assume x0 for simplicity). In other words, eigenvalues of symmetric matrices are always real and eigenvectors can always be chosen to be real.

    Hint: Show that zTAz 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 R equipped with inner products ,X and ,Y, respectively. Further, let L:XY be a linear operator.

We define the adjoint of L to be a linear operator L:YX with the property that y,LxY=Ly,xXxX, yY Assume that the map LL:YY is invertible. Then the equation Lx=y0 has a solution x0=L(LL)1y0 for each y0Y.

Prove that if x1 is any other solution of Lx=y0, then x1,x1x0,x0.

Hint: Let y1:=(LL)1y0. Using the definition of adjoint, show that y1,Lx0=x0,x0 and also that x0,x1=y1,Lx0 Complete the proof by using the fact that x1x0,x1x00.

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 R2, the last is in R.

  1. x˙1=0 and x˙2=x2
  2. x˙1=x2 and x˙2=0
  3. x˙=0 if |x|>1 and 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:XX, a subspace YX is called A-invariant if AyY yY.

  • For a linear system x˙=Ax on X=Rn, this means that x0Y implies x(t)Y t (reason: x(t)=eAtx0=(I+A+A2/2+)x0).

  • If v is an eigenvector of A with a real eigenvalue, then span{v} is a 1-dimensional invariant subspace. For a k×k Jordan block, the eigenvector v1 and the generalized eigenvectors v2,,vk 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 Re(λ)<0 is called the stable invariant subspace; the corresponding object for Re(λ)0 is the unstable invariant subspace; together these two subspaces span Rn.

Consider the LTI system x˙=Ax where A=(100021012) Identify the stable and unstable invariant subspaces by giving a real basis for each of them.