Handbook of Linear Algebra, 2nd Edition

Chapter 80

Control Theory

Peter Benner

Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg

Given a dynamical system described by the ordinary differential equation (ODE)

$\dot{\mathbf{\text{x}}}(t)=\mathbf{\text{f}}(t\text{, }\mathbf{\text{x}}(t)\text{, }\mathbf{\text{u}}(t))\text{, }\mathbf{\text{x}}(t_0)={\mathbf{\text{x}}}^0\text{, }$

where x is the state of the system and u serves as input, the major problem in control theory is to steer the state from x0 to some desired state; i.e., for a given initial value x(t0) = x0 and target x1, can we find a piecewise continuous or L2 (i.e., square-integrable, Lebesgue measurable) control function û such that there exists t1 ≥ t0 with x(t1; û) = x1, where x(t; û) is the solution trajectory of the ODE given above for u ≡ û? Often, the target is x1 = 0, in particular if x describes the deviation from a nominal ...