Feb 6, 2014

L0/L1 Equivalence in Optimal Control

One of the fundamental results of compressed sensing is that L1 optimization gives the sparsest (or L0-optimal) solution of an inverse problem under some assumptions.

Is it also true with L1 optimal control?
Is L1 optimal control the sparsest one among all admissible controls?

The answer is YES (under some assumptions, of course).
Our recent research
M. Nagahara, D. E. Quevedo, and D. Nesic,
Maximum Hands-Off Control and L1 Optimality,
52nd IEEE Conference on Decision and Control (CDC),
pp. 3825-3830, Dec. 2013.
has revealed that.

It is interesting that the L0 norm in this article is defined for continuous-time signals while most compressed sensing methods uses one for vectors. We call a control with small L0 norm a hands-off control.

The abstract reads:
In this article, we propose a new paradigm of control, called a maximum hands-off control. A hands-off control is defined as a control that has a much shorter support than the horizon length. The maximum hands-off control is the minimum support (or sparsest) control among all admissible controls. We first prove that a solution to an L1-optimal control problem gives a maximum hands-off control, and vice versa. This result rationalizes the use of L1 optimality in computing a maximum hands-off control. The solution has in general the "bang-off-bang" property, and hence the control may be discontinuous. We then propose an L1/L2-optimal control to obtain a continuous hands-off control. Examples are shown to illustrate the effectiveness of the proposed control method.

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