On the set \(L^ pS\) (Q1915897)

Denote by \(L^p S\) the set of all matrices \(A(t) \in C^0 (\mathbb{R}_+, \mathbb{R}^n)\) for which the fundamental matrix of solutions \(X(t, \tau) = X(t) X^{-1} (\tau)\) of the linear differential system \(x' = A(t)x\) satisfies the inequality \(\sup_{t \geq 0} (\int^t_0 |X(t, \tau) |^p d \tau)^{{1 \over p}} = C_p < \infty\) [see \textit{R. Conti}, Linear Differential Equations and Control, New York (1976; Zbl 0356.34007)]. \textit{N. A. Izobov} and \textit{R. A. Prokhorova} [Differ. Uravn. 23, 775-791 (1987; Zbl 0657.34014)] proved that the set \(L^pS\) is open if an only if \(p \geq 1\). They indicated conditions on \(Q(t)\) under which \(A(t) + Q(t) \in L^pS\) if \(A(t) \in L^p S\). The author proves: If \(A(t) \in L^pS\) and the continuous matrix \(Q(t)\) satisfies the inequality \[ \sup_{t \geq 0} {1 \over T} \int^{t + T}_t \bigl |Q(s) \bigr |ds < {C_p^{1/p} \over pC^2_p} \] for some \(T > 0\), then \(A(t) + Q(t) \in L^pS\) (Theorem 1). This result improves the one of N. A. Izobov and R. A. Prokhorova. Moreover, we say that the system (1) \(x' = A(t)x + f(t,x)\) belongs to class \(L^pS\) if for any \(x_0 \in \mathbb{R}^n\) its solution \(x(t, \tau, x_0)\) with the condition \(x(\tau, \tau, x_0) = x_0\) satisfies \(\sup_{t \geq 0} (\int^t_0 |x(t, \tau, x_0) |^p d \tau)^{{1 \over p}} = C_p < \infty\). By Theorem 2, system (1) belongs to \(L^pS\) provided \(A(t) \in L^pS\) and the continuous vector function \(f\) satisfies the condition \(|f(t,x) |\leq g(t) |x |\), where \(\sup_{t \geq 0} {1 \over T} \int^{t + T}_t g(\tau)d \tau < {C_p^{1/p} \over pC^2_p}\) with a \(T > 0\).