Real stability radius of bidimensional systems for linear time-dependent perturbations (Q2721519)

Let \(A\) be a real \(k \times k\)-matrix with the eigenvalues \(\sigma_i (A)\), and assume \(Re [\sigma_i (A)] < 0\) for all \(i\), so that the origin is an asymptotically stable equilibrium for the system \({\dot x} = A x\). One wants to consider a perturbed system, still in linear form, written as \( {\dot x} = \widetilde{A} x\) with \(\widetilde{A} = A + B \Delta C\); here, \(B,C\) are real constant matrices, respectively \(k \times p\) and \(q \times k\)-dimensional; \(\Delta = \Delta (t)\) is a real \(p\times q\)-matrix, smoothly dependent on \(t \in \mathbb{R}_+\). We define as norm of the perturbation \(\Delta (t)\), \(\|\Delta\|_\infty\), the sup of \(\sqrt{\text{Tr} \Delta^T \Delta}\) over \(\mathbb{R}_+\). The inf of the values of \(\|\Delta\|_\infty\) for which the origin is not asymptotically stable for \(\widetilde{A} = A + B \Delta C\) is called the real stability radius of \(A\) for time-dependent linear perturbations of structure \((B,C)\). This is denoted as \(r^-_t (A,B,C)\). In the case where \(\Delta\) does not actually depend on time, the stability radius \(r^-_0\) is the \textit{inf} of the value of \(\sqrt{\text{Tr} \Delta^T \Delta}\) for which the spectrum of \(\widetilde{A}\) does not satisfy \(Re [\sigma_i (\widetilde{A})] < 0\). This case is obviously easier to analyze than the time-dependent one, and general methods for the computation of \(r^-_0\) do indeed exist. NEWLINENEWLINENEWLINEThe authors consider the time-dependent problem and focus on the case of \(k=p=q=2\). After giving a formula for \(r^-_0 (A,B,C)\) in this case, they relate the problem of determining \(r^-_t\) to a pair of one-parameter families of homogeneous ODEs, called Barabanov extremal systems . One can determine the values of \(r \in [0,r^-_0 (A,B,C)]\) for which both systems are asymptotically stable, and this leads finally to \(r^-_t (A,B,C) \) in terms of these. A simple example is considered. The result is not in closed form, but it does provide an algorithmic method for the computation of \(r^-_t (A,B,C)\).