HIGHER-ORDER CONTINUOUS-TIME IMPLICIT SYSTEMS - CONSISTENCY AND WEAK CONSISTENCY, IMPULSE CONTROLLABILITY, GEOMETRIC CONCEPTS, AND INVERTIBILITY PROPERTIES

The paper provides a simple, and fully algebraic, distributional setup for a higher-order linear implicit system, with arbitrary constant co efficients, on the continuous, nonnegative time axis. The distribution al system equation exhibits impulses, depending on arbitrary points in the state space, and for these points several concepts of weak consis tency, of increasing degree, are introduced in such a way that weak co nsistency of degree 0 coincides with the standard concept of consisten cy, whereas weak consistency of the highest degree is related to impul se controllability. All weakly consistent subspaces are expressed in t he consistent subspace, and for the latter space a numerically stable algorithm is presented. Also, we derive for every concept of weak cons istency a condition, in terms of the system coefficients only, for the statement that the associated subspace is as large as possible. In pa rticular, we get a condition for impulse controllability that generali zes the celebrated condition for impulse controllability of a first-or der system. Further, we state two conditions for control solvability, and we specify when distributional state and/or input trajectories are unique. Finally, we define and characterize various subspaces for a h igher-order system, in combination with an output equation, and link t hese spaces to mio of our four invertibility concepts, namely those in the strong sense, and we establish that the above composite system is left- (right-)invertible in the strong sense if and only if the corre sponding system matrix is left- (right-)invertible as a rational matri x, even if the transfer matrix does not exist.