On the completeness and Riesz basis property of root subspaces of boundary value problems for first order systems and applications

The paper is concerned with the completeness property of root functions of general boundary value problems for $n \times n$ first order systems of ordinary differential equations on a finite interval. In comparison with the recent paper [45] we substantially relax the assumptions on boundary conditions guarantying the completeness of root vectors, allowing them to be non-weakly regular and even degenerate. Emphasize that in this case the completeness property substantially depends on the values of a potential matrix at the endpoints of the interval. It is also shown that the system of root vectors of the general $n \times n$ Dirac type system subject to certain boundary conditions forms a Riesz basis with parentheses. We also show that arbitrary complete dissipative boundary value problem for Dirac type operator with a summable potential matrix admits the spectral synthesis in $L^2([0,1]; \mathbb{C}^n)$. Finally, we apply our results to investigate completeness and the Riesz basis property of the dynamic generator of spatially non-homogenous damped Timoshenko beam model.