Narrow Results

The Cesàro theorem is extended to the cases: (1) higher order Cesàro mean for sequence (discrete case); and (2) higher order, multi-dimensional and continuous Cesàro mean for functions. Also, we study the Cesàro theorem for the case of positive-order.

In this paper, a general theorem dealing with $\phi -{|C,\alpha ;\delta |}_k$ summability of an infinite series is proved under weaker conditions by using a quasi-$\beta$-power increasing sequence instead of an almost increasing sequence.

Necessary and sufficient conditions for (approximate) null-controllability are obtained for the control system w_tt = w_xx - q²w, w_x(0, t) = u(t), w_x(π, t) = 0, x ∈ (0, π), t ∈ (0, T), where q ≥ 0, T ∈ (0, π], u is a control bounded by a hard constant. These problems are considered in Sobolev spaces. Controls solving them are found explicitly. Continuous controls solving the approximate null-controllability problem are constructed with the aid of the Cesàro means of a Fourier series generated by the data of the control system.