Connections between exponential stability and boundedness of solutions of a couple of differential time depending and periodic systems

Abstract

Among others, we prove that the vectorial time dependent $q$-periodic differential system $$\dot x(t)=A(t)x(t),\quad t\in\mathbb{R}, \quad x(t)\in\mathbb{C}^n\tag{A(t)}$$ is uniformly exponentially stable (i.e. all its solutions decay exponentially at infinity) if and only if for each vector $b\in \mathbb{C}^n$, the solution of the Cauchy Problem $$\dot{y}(t)=A(t)y(t)+e^{i\mu t}b,\quad t\ge 0,\quad b\in\mathbb{C}^n,\quad y(0)=0$$ is bounded on $\mathbb{R}_+,$ uniformly in respect with the parameter $\mu$ on the entire real axis. As a consequence, we get that the system $(A(t))$ is uniformly exponentially stable if and only if for each vector $x\in \mathbb{C}^n,$ the map $$t\mapsto\int\limits_0^t||ds$$ is bounded on $\mathbb{R}_+.$ This latter result is a weak version of the Barbashin theorem which seems to be new. Here $\Phi(t)$ is the fundamental matrix associated to the system $(A(t)).$

Keywords

• uniform exponential stability
• boundedness for nonautonomous systems
• spectral radius of matrices and bounded linear operators
• barbashin’s type theorems