Overdetermined Multidimensional Systems: State Space and Frequency Domain Methods

Abstract

We consider 2D input-state-output linear systems where the evolution of the whole state is specified in two independent directions. The requirement that the value of the state at a given point be independent of the path from the origin chosen to arrive at the given point leads to nontrivial consistency conditions: the transient evolutions (i.e., state evolution with zero inputs) should commute, and the input signal (and then also the output signal) should solve a compatibility partial differential equation. We show that many of the standard structural properties (e.g., controllability, observability, minimality, pairing with adjoint system, feedback coupling, equivalence between conservative systems and Lax-Phillips scattering theory) and standard problems (e.g., pole placement, linear-quadratic-regulator problem, H-infinity-control problems) for 1D linear systems carry over for this setting. There is also a frequency-domain theory for this class of systems: the transfer function is a bundle map between flat vector bundles on a compact Riemann surface, or equivalently, between kernel bundles for determinantal representations of an algebraic curve embedded in C-2 (or rather in the projective plane P-C(2)). The transform from the time domain to the frequency domain C is implemented by a ``Laplace transform along the curve C{''}. Just as optimal control for continuous-time systems leads to control-theoretic interpretations for Hardy-space function theory on the right half plane, control theory for this class of overdetermined systems leads to control-theoretic interpretations for function theory on a finite bordered Riemann surface. We expect such a mathematically rich theory to have use in control applications yet to be discovered as well as applications beyond the scope of traditional system theory; we mention two such possibilities of the latter type: wave-particle duality in quantum mechanics and a mathematical model for DNA chains.