Control Processes with Distributed Parameters in Unbounded Sets. Approximate Controllability with Variable Initial Locus

Abstract

We consider the following distributed parameter linear control system

$$ z_{xy} + A\left( {x,y} \right)z_x + B\left( {x,y} \right)z_y + C\left( {x,y} \right)z = F\left( {x,y} \right)U\left( {x,y} \right). $$

Here (x, y) ranges over the unbounded set

$$ L_{I,J} = \bigcup\limits_{\left( {u,v} \right) \in I \times J} {l\left( {u,v} \right),} $$

where

$$ l\left( {u,v} \right) = \left( {\left[ {u, + \infty } \right[x\left\{ v \right\}} \right) \cup \left( {\left\{ u \right\} \times \left[ {v, + \infty } \right[} \right), \left( {u,v} \right) \in \mathbb{R}^2 , $$

and I, J are two non-degenerate intervals of ℝ. The state vector function z belongs to the Sobolev type functional space

$$ W_{p,loc}^* \left( {L_{I,J} ,\mathbb{R}^n } \right) = \left\{ {z \in L_{loc}^p \left( {L_{I,J} ,\mathbb{R}^n } \right):z_x ,z_y ,z_{xy} \in L_{loc}^p \left( {L_{I,J} ,\mathbb{R}^n } \right)} \right\} $$

and the control vector function U is in L ^p_loc (L _I,J, ℝ^m). Moreover, for every (u, v) ∈ I × J, the trace of z on l(u, v) is taken as the system state corresponding to the values x = u, y = v of the parameters. All these traces belong to a functional space of Sobolev type, which does not depend on (u,v).

In this setting, given a point (a,b) ∈ I × J, we study the controllability of system (E) from a given initial state, to be taken on the variable initial locus l(a ₀,b ₀), (a ₀,b ₀) ∈ I × J, a ₀ ≤ a,b ₀ ≤ b, to an arbitrary final state, to be taken on the fixed final locus l(a,b). We get a characterization of the approximate controllability when the set of the available controls is the unit ball of L ^∞ (L _I,J, ℝ^m).