Coppel-Conti sets of linear unstable systems (Q1603156)

The authors consider linear systems \(\dot{x} = A(t) x\) with piecewise continuous coefficient matrix and Cauchy matrix \(X_A(t,\tau)\) and investigate properties of the sets \(M^p S\), \(p > 0\), consisting of all \(A\) such that \[ \int_t^\infty \|X_A(t,\tau)\|^p d\tau \leq C_p(A),\qquad t \geq 0, \] and \[ \|X_A(t,\tau)\|\leq C_\infty(A),\qquad 0 \leq t \leq \tau < \infty, \] with constants \(C_p(A)\) and \(C_\infty(A)\). The inclusions \(M^q S \subset M^p S\) for \(q > p > 0\) and the structural stability under uniformly small perturbations for \(p\geq 1\) are known. Here, the properties of \(M^p S\) treated as a function of the parameter \(p>0\) are studied, and the influence of linear perturbations of various smallness on systems of the class \(M^p S\) is analyzed.