Mathematics

Partial differential operators in $G \times I$, where $G \subset \R^n$ is a bounded set and $I \subset \R$ an interval of time variable, and their generalizations, pseudo-differential operators, are studied here keeping in mind the goal of modelling physical distributed-parameter phenomena. Control actions of such systems usually take place on the boundary $\partial G$. Symbolic calculus applied gives tools to form e.g. compositions, formal adjoints, generalized right and left inverses and so-called compatibility conditions. The operators form an algebra $\mathcal D$ by using of which typical boundary-value control problems can be formulated. Parametrizability, which is a concept closely related to flatness of ordinary controlled differential systems, means that for a given control system $\L u=0$, where $\L$ is the system operator, and the variable $u$ includes the actual control, state and output variables, one can find an operator $\S$ such that $\L u=0$ if and only if $u = \S f$ for some relevant function $f$. It is required that the components $f_i$ of $f$ are $\mathcal D$-linearly independent. The pseudo-differential operators and boundar-value operators are formed by the matrix-like operators: r^+A+B K \L= T Q where the $KQ$-column operates on the functions defined on $\partial G \times I$, and the first column on functions in $\bar G \times I$. The operator $r^+$ restricts the global operator $A$ to act in $\bar G \times I$. $T$ is the trace operator. In our applications the computation rules in $\mathcal D$ give explicitly the parametrization operator $\S$ subject to certain existence assumptions of related pseudo-differential operators or of one-sided inverse operators. This construction is based on methods of homological algebra. Projective freeness of a certain factor module (defined by the system equations) implies parametrizability. Some examples of partial differential control systems including boundary conditions are presented to illustrate the parametrizability concept and construction of the operator $\S$.