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|Title:||Causal and stable input/output structures on multidimensional behaviors|
|Citation:||SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 43(4), 1493-1520|
|Abstract:||In this work we study multidimensional (nD) linear differential behaviors with a distinguished independent variable, called "time." We define in a natural way causality and stability of input/output structures with respect to this distinguished direction. We make an extension of some results in the theory of partial differential equations, demonstrating that causality is equivalent to a property of the transfer matrix which is essentially hyperbolicity of the P-c operator defining the behavior (B-c)(0,y). We also quote results which in effect characterize time-autonomy for the general systems case. Stability is likewise characterized by a property of the transfer matrix. We prove this result for the 2D case and for the case of a single equation; for the general case it requires solution of an open problem concerning the geometry of a particular set in C-n. In order to characterize input/output stability we also develop new results on inclusions of kernels, freeness of variables, and closure with respect to S, S' and associated spaces, which are of independent interest. We also discuss stability of autonomous behaviors, which we believe to be governed by a corresponding condition.|
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