Problem 1.12

State and first order representations

Jan C. Willems¹

Department of Electrical Engineering - SCD (SISTA)

University of Leuven

Kasteelpark Arenberg 10

B-3001 Leuven-Heverlee

Belgium

[email protected]

1 DESCRIPTION OF THE PROBLEM

We conjecture that the solution set of a system of linear constant coefficient PDEs is Markovian if and only if it is the solution set of a system of first order PDEs. An analogous conjecture regarding state systems is also made.

Notation

First, we introduce our notation for the solution sets of linear PDEs in the n real independent variables x = (x1, . . . , xn). Let denote, as usual, the set n of real distributions on and wnthe linear subspaces of consisting nof the solutions of a system of linear constant coefficient PDEs in the w real-valued dependent variables w = col(w1, . . . , w_w). More precisely, each element is defined by a polynomial matrix with w columns, but any number of rows, such that

We refer to elements of as linear differential n-D systems. The above PDE is called a kernel representation of . Note that each has many kernel representations. For an in-depth study of , see [1] and [2].

Next, we introduce a class of special three-way partitions of Denote by the following set of partitions of :

Think of S- as the “past”, S₀ as the “present”, and S+ as the “future.”

Markovian means that if two solutions of the PDE agree on the present, then their pasts and futures are compatible, in the sense that the past (and present) of one, concatenated with the (present and) future of the other, is also a solution. In the language of probability: the past and the future are independent given the present.

We come to our first conjecture:

State systems

Think again of S- as the “past”, S0 as the “present”, S-+ as the “future” . State means that if the state components of two solutions agree on the present, then their pasts and futures are compatible, in the sense that the past of one solution (involving both the manifest and the state variables), concatenated with the present and future of the other solution, is also a solution. In the language of probability: the present state “splits” the past and the present plus future of the manifest and the state trajectory combined.

We come to our second conjecture:

2 MOTIVATION AND HISTORY OF THE PROBLEM

These open problems aim at understanding state and state construction for n-D systems. Maxwell’s equations constitute an example of a Markovian system. The diffusion equation and the wave equation are non-examples.

3 AVAILABLE RESULTS

It is straightforward to prove the “if”-part of both conjectures. The conjectures are true for n = 1, i.e., in the ODE case, see [3].

BIBLIOGRAPHY

[1] H. K. Pillai and S. Shankar, “A behavioral approach to control of distributed systems, ” SIAM Journal on Control and Optimization, vol. 37, pp. 388-408, 1999.

[2] U. Oberst, “Multidimensional constant linear systems, ” Acta Applican-dae Mathematicae, vol. 20, pp. 1-175, 1990.

[3] P. Rapisarda and J. C. Willems, “State maps for linear systems, ” SIAM Journal on Control and Optimization, vol. 35, pp. 1053-1091, 1997.

1This research is supported by the Belgian Federal Government under the DWTC program Interuniversity Attraction Poles, Phase V, 2002 - 2006, Dynamical Systems and Control: Computation, Identification and Modelling.