There are multiple similar questions, but I can't find any good news on this special case. I'm afraid the answer may just turn out to be, "Sorry, you're just out of luck", but I've got to ask.
I've got a multi-body spring-mass system with $n$ masses, $n$ springs, and no dampers. This is a time-invariant linear dynamical system: \begin{equation} \dot{x} = Ax \end{equation}
The matrix $A$ is tridiagonal, and all its off-diagonal elements are positive. $A$ is itself parametrized as an affine combination of sparse tridiagonal matrices. Denote the coefficients of the affine combination by the vector $z$.
\begin{equation} A = \sum_i A_i z_i \end{equation}
The eigenvalues of $A$ are all real. They're all negative. They will always be so, because each eigenvalue is the negative of the square of one of the fundamental frequencies of vibration. There is no damping. We want to modify $z$ to find an $A$ with desirable spectral characteristics. In particular, we are interested in the following optimization problem.
\begin{equation} \begin{array}{lrcl} \textrm{minimize} & c^T z \\ \textrm{w.r.t.} & z,\\ \textrm{s.t.} & \lambda_{\max}(A) &\leq& r,\\ & z_{\min} - z &\leq& 0. \end{array} \end{equation}
Unfortunately, $A$ is not symmetric. If we restrict ourselves to the case where all the masses and springs are the same, then everything is good, $A$ is symmetric, $\lambda_\max$ is convex in $z$, and we're done. In the general case, $A$ is not symmetric, but I'm still hopeful. There's so much structure here, something should work out. It's like a question from one of Steve Boyd's take-home finals.
Surely there must be some little concession, some cute trick, something I'm missing. Is this spectral constraint at least quasi-convex? If we're guaranteed that the eigenvalues remain real, then what's so special about symmetry?
Hoping someone has an a-ha moment.
p.s.: unfortunately Michael Grant does not appear to be a tag.
This is indeed a convex problem. The tridiagonal matrix can be made symmetric via a special similarity transformation. The equation
$A = \sum_i A_i z_i$
is replaced with another equation in the same variables with symmetric matrices:
$S = \sum_i S_i z_i.$
With that, this is a straightforward problem with a single linear matrix inequality, or, if your optimizer supports it, a max. eigenvalue constraint.