Maximum hands-off control without normality assumption

Maximum hands-off control is a control that has the minimum L
⁰
norm among all feasible controls. It is known that the maximum hands-off (or L
⁰
-optimal) control problem is equivalent to the L
¹
-optimal control under the assumption of normality. In this article, we analyze the maximum hands-off control for linear time-invariant systems without the normality assumption. For this purpose, we introduce the Lp-optimal control with 0 <; p <; 1, which is a natural relaxation of the L
⁰
problem. By using this, we investigate the existence and the bang-off-bang property (i.e. the control takes values of ±1 and 0) of the maximum hands-off control. We then describe a general relation between the maximum hands-off control and the L
¹
-optimal control. We also prove the continuity and convexity property of the value function, which plays an important role to prove the stability when the (finite-horizon) control is extended to model predictive control.