Errata for class notes

The following were colleceted together in Fall 2024.

• p. 7 – Perhaps note that \(I_1\) and \(I_2\) are moments of inertia of the two links
• p. 24 – Vector space axioms should include distributive properties: \[a*(x+y)=a*x + a*y \quad \textrm{and} \quad (a+b)*x=a*x+b*x \]
• p. 31 – At the very bottom where the closed-form expression of \(\beta\) is shown, the top-right variable in the matrix, \(\alpha_{in}\), should be \(\alpha_{1n}\), and the bottom-left variable should be \(\alpha_{m1}\)
• p. 37 – Sixth line in the Section 2.9. “By orthogonality we then have” should be “By linearity we have” (No orthogonality is assumed/needed at that point)
• p. 38 – Seven lines before Section 2.10, the line contains “to to”
• p. 45 – Problem 2.11.20. the vector \(x\) should be from complex \(n\) dimensional space not \(\mathbb{R}^n\)
• p. 77 – In the seventh line of the proof. It is stated that the gradient of \(V\) is a row vector. Most other places in the book (and in the literature) gradients are column vectors. There are some transposes taken in some of the lines that follow which would not be there if the gradient were a column vector.
• p. 90 – In the equation in line 8 just after “Equivalently, we must have” the \(\lambda_i\) in the exponent should have a complex conjugate symbol over it (or a star).
• p. 171 – The variable \(y\) in (10.9) should be replaced by \(t\) and supescript \(^o\) should be on the \(x\) variables on the left hand . Moreover it would be better to add more to the notation to make clear that each side of the inequality is evaluated along a trajectory and the two derivatives are total derivatives. So it would be \[ 0 = \ell(x^o(t),u^o(t),t) + \frac{d}{dt} \left( V^o(x^o(t),t)\right) \leq \ell(x(t),u(t),t) + \frac{d}{dt} \left( V^o(x(t),t)\right)\]
• p. 171 – Four lines before Example 10.2.1, the apostrophe superscript on \(u\) in the integral should be removed.
• p. 173 – First line – should be “Since \(BR^{-1}B^T\) is semi positive definite …”
• p. 176 – About 5-7 lines from bottom: The sentence “Note that if the eigenvalues of \(\mathcal{H}\) are distinct then by Lemma 10.4 (i.e. if \(\lambda\) is an eigenvalue then so is \(-\lambda\)) none can lie on the \(j\omega\) axis” isn’t true. (It is stated six lines before that that if the model is sabilizable and detectable then there are no purely imaginary eigenvalues.)
• p. 178 – Six lines from end ARE is not defined until p. 181.
• p. 205 – Lines 6, 10, and 14: It should be \(p^T(t)\) defined to equal the \(\partial V^o / \partial x\). Later text switches correctly to gradients.
• p. 207 – Figure 11.1: It could be a problem with the PDF viewer but the gradient vectors don’t show up well in Figure 11.1.
• p. 212 – Something is wrong at the end of (11.14). A fix is to delete \(\cal H\) and vector at end with \(\dot{x}\) and \(\dot{\lambda}\). OR insert an equal sign before \(\cal H\) and remove dots on \(x\) and \(\lambda\) in the vector at the end.
• p. 218 – The equations in part (b): \(x_i(t_i)\) should be \(x_i(t_1).\) In the next line it should be for \(j\in I^c\) not for \(i\in I^c.\). In (11.15) the second argument of \((\partial m)/(\partial t)\) should be \(t_1\) (i.e. add subscript 1 to \(t\)). In last equation five lines from bottom of the page the superscript \(T\) should be just after \(p\): i.e. \(p^T(t)\) and not \(p(t)^T\).