Controllability for distributed bilinear systems

Ball, J. M., J. E. Marsden and M. Slemrod

SIAM J. Cont. and Optim., 20, (1982), 575-597

Abstract:

This paper studies controllability of systems of the form dw/dt = $\mathcal {A}$w + p(t)$\mathcal {B}$w where $\mathcal {A}$ is the infinitesimal generator of a C⁰ semigroup of bounded linear operators e^{$\mathcal {A}$t} on a Banach space X , $\mathcal {B}$ : X$\to$X is a C¹ map, and p $\in$ L¹$\bigl($[0, T] : $\mathbb {R}$$\bigr)$ is a control. The paper (i) gives conditions for elements of X to be accessible from a given initial state w₀ and (ii) shows that controllability to a full neighborhood in X of w₀ is impossible for dimX = $\infty$ . Examples of hyperbolic partial differential equations are provided.