General response formula for continuous \(2-D\) Roesser model and its local controllability (Q1898173)

The 2-D Roesser model can be considered as the straightforward generalization of the classical state space model for linear lumped-parameter systems to system depending on two independent variables where the state vector \(x(t_1, t_2)\) can be divided in two subvectors \(x^1\) and \(x^2\) such that the model consists of a set of linear partial differential equations \(x_{t_k}^k(t_1, t_2) = A^{k_1} x^1 + A^{k_2} x^2 + Bu\) \(k = 1,2\) where subscript \(t_k \) denotes partial differentiation with respect to \(t_k\). Rather simple boundary conditions and an -- equally linear -- equation for the output complete the model. All matrices \(A\) and \(B\) are supposed constant what allows the representation of the transition matrix by convergent series of the type occuring in exponential functions. By this, a general response formula is given and conditions for local controllability are established.