Controllability of [ r]-matrix quasi-differential equations

Abstract

Consider the controllability theory for the linear system x′ = A( t)x + B( t)u, ( l ) where A( t), B( t) are complex-valued matrices in L loc 1 on a real-interval I, and where x( t) is the state and u( t) ϵ L rmloc ∞ is the controller. For an [ r]-matrix control system the components x 1, x 2, …, x n of x, and also u 1, u 2, …, u m of u, as well as the entries of A( t) and B( t), are each r × r matrices, for a fixed integer r ⩾ 1. In the important case of [ r]-matrix quasi-differential control systems (and [ r]-matrix quasi-differential control equations), where there is a prescribed special format for the matrices A( t) and B( t) (as suggested by the theory of scalar control equations), the authors obtain explicit criteria for the full controllability of ( l ) on I. For r = 1 (the classical case) new and explicit controllability criteria are found for linear control systems with continuous (or merely integrable) time-varying coefficients. The methods are also new and rest only on familiar results of real analysis. Analogous results are further obtained for [ r]-matrix linear control systems, for r ⩾ 1. The motivation for considering r > 1 arose from the demand for a level of generality sufficient to deal with the increasingly important matrix linear systems of control theory, such as those associated with matrix Riccati differential equations, and matrix bilinear control systems.