Point to point controllability in linear time invariant systems

We revisit a classical problem of control theory from the pint of view of computability.

Consider a discrete time linear time invariant (LTI) system x(n+1)=Ax(n)+Bu(n) where x(n) is a vector of R^n, A and B are matrices and u(n) is an external input. Given x(0) and y, we want to know if there is a sequence of inputs u(0),..., u(k) such that x(k+1)=y.

This problems models physical systems where x(n) is the state and u(n) is the input controlled the a user or a computer. We are therefore trying to steer the system to achieve some statereach some state y.

In this talk, we will focus on the nontrivial case where the input u(n) is restricted to some convex state. Surprisingly, the problem becomes hard in general but we give positive results in some cases.