On the existence of stabilizable solutions for linear differential systems with periodic parameters
The author considers the initial value problem (IVP) NEWLINE\[NEWLINE\dot x(t)-P (t)x\bigl(h(t) \bigr)=f(t),\;t\in[0, \infty),\quad x(\xi)=\varphi(\xi), \text{ as }\xi<0, \quad x(0)=\alpha, \tag{1}NEWLINE\]NEWLINE for the system of differential equations with retarded argument. Since, generally speaking, the solution to (1) is not periodic the author is interested when a solution to (1) is stabilizable to a periodic function. The precise definition of stabilization he works with is: A solution \(x(t,\alpha)\) to (1) is said to be stabilizable to a \(T\)-periodic function \(y:[0,\infty) \to\mathbb{R}^n\), if NEWLINE\[NEWLINE\lim_{N\to \infty} \max_{t\in \bigl[NT, (N+1)T\bigr]} \bigl\|x(t,\alpha) -y(t)\bigr \|_{\mathbb{R}^n} =0.NEWLINE\]NEWLINE The main result in the paper is a theorem in which the author obtains conditions such that for each \(\alpha\) the solution \(x(t,\alpha)\) to IVP (1) is stabilizable to a \(T\)-periodic function \(y:[0,\infty) \to\mathbb{R}^n\). An illustrative example is given.NEWLINENEWLINEFor the entire collection see [Zbl 0940.00021].
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